Bridging Asset Pricing and Market Microstructure: Option Valuation in Roll’s Framework

Abstract

We introduce a binary tree for pricing contingent claims when the underlying security prices exhibit history dependence. We apply the model to the specific cases of moving-average and autoregressive behavior that are characteristic of price histories induced by market microstructure behavior. Our model is market-complete and arbitrage-free. When passing to the risk-neutral measure, the model preserves all parameters governing the natural-world price dynamics, including the instantaneous mean of the asset return and the instantaneous probabilities for the direction of asset price movement. This preservation holds for arbitrarily small, but non-zero, time increments characteristic of market microstructure transactions. In the (unrealistic) limit of continuous trading, the model reduces to continuous diffusion price processes, with the concomitant loss of the microstructure information.

1. Introduction

“At the level of transactions prices, …, the random walk conjecture is …a hypothesis that is very easy to reject in most markets even in small data samples. In microstructure, the question is not ‘whether’ transactions prices differ from a random walk, but rather ‘how much’ and ‘why?”’

(Hasbrouck, 1996)

Dynamic asset pricing theory, as introduced1 by Black and Scholes (1973) and Merton (1973) (BSM), is based on the concepts of no-arbitrage opportunity and replicating portfolios, along with a set of assumptions that can be classified into two groups. The first group of assumptions concerns the microstructure of the market: the rules under which trades are performed; the impact of transaction and timing costs; the role of information and its disclosure; discovery and formation of prices; volatility; liquidity; and market-maker and investor behavior. Under the assumptions of the BSM model, any trade is executed without taxes, transaction costs, and amount restriction (the market is frictionless); traders are price takers with symmetric information (a perfectly competitive market) and are able to trade any amount (no liquidity constraints) over any infinitesimally small time interval (continuous trading); and the market is assumed to be efficient (all relevant information is embedded in the market price), liquid (every order is executed instantaneously at the current equilibrium price), and free of arbitrage opportunity. The second group of assumptions is related to the choice of geometric Brownian motion (GBM) as the stochastic process describing the price dynamics of the security underlying the option contract. The assumption of GBM invokes a strong set of restrictions, including constant volatility, the normal distribution of log returns, and absence of long memory.2

The BSM model provides analytic solutions and an elegant machinery for computing the price of a European option; however, many of the hypotheses upon which it is rooted have been shown to be too restrictive. It is well known that many of the empirical properties3 of stock price returns are not consistent with the assumption of GBM. Consequently, a range of alternative models have been proposed to include various stylized facts. Another problem with the original BSM model is that it does not provide solutions for more complex contingent claims, such as those with a path-dependent pay-off (e.g., American options) or those whose underlying risk is not fully priced in financial markets, leading to market incompleteness. The most baffling result of the BSM model is that the option price does not depend on the drift of the underlying security. This puzzle was clarified in the subsequent work of Cox and Ross (1976) and Merton (1976), and then reformulated in terms of the risk-neutral measure by Harrison and Kreps (1979) and Harrison and Pliska (1981). The concept of the risk-neutral price was subsequently accepted and continuous-time models have proliferated.

Subsequent developments to the solution of the continuous-time pricing problem can be interpreted as improvements in one of two directions. The first is directed to the stochastic process driving the price dynamics in order to incorporate more statistical features of real price processes. This direction has produced the following strong result, known as the general version of the fundamental theorem of asset pricing (Delbaen & Schachermayer, 1994): “if a stochastic price process $\mathcal{S}$ is a bounded, real-valued semimartingale, there is an equivalent martingale measure (EMM) for $\mathcal{S}$ if and only if $\mathcal{S}$ satisfies the condition of “no free lunch with vanishing risk” (NFLVR), where NFLVR is a generalization of the no-arbitrage condition.4^,5 One consequence of this continuous-time fundamental theorem is the necessity to work within the confines of stochastic integration theory. A more critical consequence is that the general version of the fundamental theorem of asset pricing does not guarantee a unique EMM. In fact, there can be uncountably many EMMs, allowing for all possible option pricings within natural bounds. In incomplete markets, where perfect replication is not possible, hedgers seek alternative approaches to mitigate risk in a cost-efficient manner. Several alternative strategies exist, each with their own advantages and drawbacks, but they are still not entirely attractive due to their costs and limitations (Dolinsky & Neufeld, 2018; El Karoui & Quenez, 1995; Karatzas, 1997; Löhne & Rudloff, 2014; Rouge & El Karoui, 2000).

The second direction is to develop methods for solving pricing problems having no known analytic solution, due either to the complexity of the stochastic process or the complexity of the pay-off function associated with a contingent claim. The binomial option pricing model proposed by Cox et al. (1979) (CRR) was the first approach to pricing American options without sacrificing the intellectual machinery developed under the BSM model. CRR utilized a discrete-time, binomial lattice graph to describe the evolution of the price process of the underlying security. The discrete process was designed to converge to GBM as the time interval between two successive trades converged to zero. There was no intention in the CRR model to use the discrete setting to incorporate other stylized facts of asset returns. Other discrete models—utilizing binomial or trinomial lattices, or binary trees—have been developed to numerically price contingent claims under more complex assumptions, such as stochastic volatility or jump processes (see, e.g., Boyle, 1986; Derman et al., 1996; Rubinstein, 1994, 1998). Again, these discrete models have been designed to converge to a solution of a continuous-time stochastic process. This is usually ensured by setting moment-matching conditions in order to apply Donsker’s invariance principle (Billingsley, 2013). Using discrete models avoids working explicitly with stochastic integration theory.6

As noted above, the BSM model is very restrictive with regards to its incorporation of the details of market microstructure.7 In seminal work, Roll (1984) showed that, in an efficient market, the effective bid–ask spread can be measured by $\mathrm{spread}=2\sqrt{-\mathrm{cov}}$, where $\mathrm{cov}$ is the first-order serial covariance of price changes. Crucially, Roll’s reasoning was based upon analysis of a discrete-time model. We briefly recapitulate Roll’s model to indicate the connection with our work.8 Roll considered a martingale-efficient price $m_t$ for an asset evolving as $m_t=m_{t-1}+u_t$, where $u_t$ represents independent, identically distributed, mean-zero random variables. A trade (buy or sell) at time t through a dealer results in a transaction price

$$p_1=m_t+q_{t}c,$$

where $q_t=-1$ denotes a sale to a dealer, while $q_t=1$ denotes a purchase from a dealer. The value $2c$ is the bid–ask spread. Under the buy–sell assumptions $\mathbb{E}\left[q_{t_1}{q}_{t_2}\right]=0$ and $\mathbb{E}\left[u_{t_1}{q}_{t_2}\right]=0$ for all $t_1\ne t_2$, the variance and first-order covariance of $\Delta p_t=p_t-p_{t-1}$ are

$$\mathrm{Var}[\Delta p_t]:={\gamma }_0={\sigma }_u^2+2c^2,\mathrm{cov}[\Delta p_t\Delta p_{t-1}]:={\gamma }_1=-c^2.$$

Solving for c produces Roll’s spread formula $c=2\sqrt{-{\gamma }_1}$. Solving for ${\sigma }_u^2$ results in ${\sigma }_u^2={\gamma }_0+2{\gamma }_1$. The values ${\gamma }_0$ and ${\gamma }_1$ can be estimated from historical transaction prices to obtain estimates for the model parameters c and ${\sigma }_u$.

In his treatise on market microstructure, Hasbrouck (2007) describes several discrete-time empirical market microstructure models which build upon Roll’s bid–ask model. The models are designed to capture, in various ways, the price formation process, incorporating the sequence of actions and reactions between market makers and traders. Using the binary-tree model developed in this paper, in Section 7 we develop a base model of binary white noise (BWN). Upon this, we build two binary-tree models which capture, successively, moving-average-of-order-one (MA(1)) and autoregressive-of-order-one (AR(1)) risky-asset price behaviors.

The works of Kim et al. (2016, 2019) and Hu et al. (2020a, 2020b) have shown that binomial pricing trees have sufficient flexibility to capture some of the stylized facts of price dynamics for option pricing in complete discrete-time markets (enabling a unique hedging strategy). These include the preservation, from the natural world to the risk-neutral valuation, of the probabilities of the natural-world stock-price directions; the mean and higher moments of returns; and the effects of noisy, informed, and misinformed traders. However, binomial trees are too simplistic to accommodate either the autoregressive or moving-average behavior of asset prices. Our thesis in this paper is that binary pricing trees are crucial for developing dynamic asset pricing models that incorporate such phenomena.

To further clarify the need for binary pricing trees, recall the fundamental pricing model in continuous time for a market consisting of a single bond and stock. The continuous-time bond price dynamics are given by

$$d{\beta }_t^{\left(\mathrm{cts}\right)}=r_t^{\left(\mathrm{cts}\right)}{\beta }_t^{\left(\mathrm{cts}\right)}dt,t\in [0,T],$$

(1)

where ${\beta }_0^{\left(\mathrm{cts}\right)}={\beta }_0>0$ and $r_t^{\left(\mathrm{cts}\right)}$ is a continuous-time riskless rate (Duffie, 2001, p. 102). The stock’s log-price dynamics $L_t^{\left(\mathrm{cts}\right)}=ln\left(S_t^{\left(\mathrm{cts}\right)}\right)$, with $S_0^{\left(\mathrm{cts}\right)}=S_0>0$, follow a continuous diffusion determined by the Itô process (Aït-Sahalia & Jacod, 2014, Chapter 1):

$$\begin{array}{c}\hfill d{L}_t^{\left(\mathrm{cts}\right)}={\mu }_{t}dt+{\sigma }_{t}d{B}_t,t\in [0,T],\end{array}$$

(2)

where $B_t$, $t\in [0,T]$ is a standard Brownian motion whose trajectories generate a canonical filtered probability space (Duffie, 2001, Chapter 5 and Appendix E):

$$\left(\Omega ,{\mathbb{F}}^{\left(\mathrm{cts}\right)}=\left\{{\mathcal{F}}^{\left(\mathrm{cts}\right)}=\sigma (B_u,0\le u\le t)\right\},\mathbb{P}\right).$$

Inclusion of microstructure features modifies the stock log-price dynamics, which can be written in discrete form as

$$L_{t_n}^{\left(\mathrm{obs}\right)}=L_{t_n}^{\left(\mathrm{cts}\right)}+{ϵ}_{t_n}^{\left(\mathrm{micro}\right)},n=1,\cdots ,m-1,$$

(3)

where $t_n=t_1,\cdots ,t_{m-1}$ indicate the times at which the microstructure features associated with a particular market “actor” (such as a trader) are realized.9 The microstructure dynamics ${ϵ}_{t_n}^{\left(\mathrm{micro}\right)}$, $n=1,\cdots ,m-1$, are determined by (for example) a moving-average process MA(q) of order $q=0,1,\cdots$ (Mills, 2019, Chapter 3):

$${ϵ}_{t_n}^{\left(\mathrm{micro}\right)}=\sum _{k=0}^q{\varphi }_k{\zeta }_{t_{n-k}},{\varphi }_0=1,{\varphi }_k\in \mathbb{R},{\varphi }_k\ne 0,n=1,\cdots ,m-1.$$

(4)

Here, ${\zeta }_{t_{n-k}}=0$ when $k>n$, and ${\zeta }_{t_k}$, $k=0,\cdots ,m-1$ are independent, identically distributed random variables with zero mean and specified variance. As can be seen, for example, as shown by O’Hara (1999, Figure 1), when general microstructure features are included in the observed log-prices, the recombining binomial pricing tree is no longer an appropriate model for the stock-price dynamics and an extension to a binary (i.e., non-recombining) pricing model must be introduced.

The fundamental asset pricing theorem of Delbaen and Schachermayer (1994) requires the ability to trade in continuous time. But market microstructure phenomena occur at discrete times. The resultant observed process (3), being a combination of a semimartingale plus discrete-time microstructure noise, is therefore not a semimartingale, and the fundamental asset pricing theorem cannot be applied. Cheridito (2003) showed that fractional Brownian motion can be used as a price process and still maintain NFLVR by “introducing a minimal amount of time $h>0$ that must lie between two consecutive transactions”. Jarrow et al. (2009) extended this work to show that arbitrage-free price processes can be obtained without reliance on semimartingales provided continuous-time trading is not allowed (although the finite time intervals can be arbitrarily small).

We therefore adopt the discrete-time perspectives of Cheridito and Jarrow10 and present a discrete, binary-tree approach that is general enough to reproduce the statistical properties of real prices and encompass a class of models that are used in microstructure theory (see Easley & O’Hara, 1995, 2003; Hasbrouck, 2007; Fan et al., 2016; Aït-Sahalia & Jacod, 2014, Chapter 2). To this end, we apply the general approach of Hu et al. (2020a, 2020b) to binary-tree option pricing models.11 In Section 2, we develop a discrete binary tree (binary information tree) supporting random walks. In Section 3 and Section 4, we develop discrete-time pricing on this tree for a riskless rate, a riskless bank account, and a risky asset. Using a self-financing portfolio formulation, computation of the risk-neutral measure and pricing of options is discussed in Section 5.

In general, a random walk on a non-recombined binary tree (which is a particular case of an arbitrary branching process) will converge to a measure-valued diffusion (Daley & Vere-Jones, 2003, 2008; Mitov et al., 2009; Skorokhod, 1997). In Section 6, we show that, under the well-known Donsker–Prokhorov invariance principle (for constant instantaneous mean return and variance) or the Davydov–Rotar invariance principle (for time-dependent instantaneous mean return and variance), the restrictions of these invariance principles unfortunately require that a non-recombined random walk approach a classical random walk, which, in the continuum limit, produces price processes such as as GBM (under the Donsker–Prokhorov invariance principle) or continuous diffusion (under the Davydov–Rotar invariance principle), resulting in the concomitant loss of the microstructure information.

The principal contributions of this theoretical paper are the following. We develop a binary-tree framework that supports market-complete, arbitrage-free, option pricing models having path-dependent price processes (Section 2, Section 3, Section 4 and Section 5). We show (Section 6) that the attempt to consider the continuous-time limit of this binary-tree framework is self-defeating, as critical natural world parameters are lost in the continuum limit. We apply (Section 7) this general framework to construct models in which a component driving the random price behavior is described by BWN—mimicking market microstructure. We further incorporate this underlying microstructure into two path models, in which the risky-asset prices display either MA(1) or AR(1) path dependence. In Section 8, we discuss the simplifications inherent in these three models that reduce the complexity of computing option prices. We show that, as a result of partial recombination, the computational complexity of the MA(1) (and consequently of the BWN) model is $O\left(n^2\right)$ rather than the $O\left(2^n\right)$ expected of non-recombining binomial trees.

As the price-direction probabilities are preserved in the risk-neutral dynamics of our framework, we delve (Section 9) into a technical analysis of the probabilities governing sequences of price changes. We compute sequence probabilities empirically, based upon daily closing prices of an ETF and for stocks comprising the Dow Jones Industrial Average. This analysis provides a test of the efficient market hypothesis for daily data. By categorizing price-change sequences, our results indicate that the efficient market hypothesis operates within categories, but not across categories.

A brief consideration of future directions is provided in Section 10.

2. A Binary Information Tree for Pricing

Binomial (recombining binary) trees are described by simple lattice notation. Each node on the binomial lattice is labeled by a time and price-state index. If the lattice has $m+1$ time steps, $n=0,1,\cdots ,m$, then time step n has $n+1$ price states. Each node p, corresponding to a price state on a binomial tree, has the unique label $p_n^k$, $n=0,\cdots ,m$, $k=0,\cdots ,n$. Employing graph theory terminology, there are $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ unique paths from the root to node $p_n^k$, characterized by the number of unique sequences of length n created using k values of “up” and $n-k$ values of “down”. However, the price at node $p_n^k$ depends only on the numbers k and $n-k$ and not on the individual sequences. In describing a binomial tree, it is common practice to provide only the single time interval $n=\{0,1\}$ and refer to the $n=1$ price states as “up” and “down” (along with the formula for computing each).

For binary trees,12 the situation is more complex. Time step n now has $2^n$ price states (nodes), and one would anticipate that the labeling $p_n^k$, $n=0,\cdots ,m$, $k=0,\cdots ,2^n-1$ would be sufficient. However, non-recombining binary trees have the property that there is a unique path between the root ($n=0$) and each node of the tree. Each path is distinguished by a unique sequence of “up” and “down” movements on the tree, and the price at each node generally depends on that unique sequence. We have chosen a notation that makes path labeling possible through a recorded sequence of local up (1) and down (0) movements. This notation also supports the flexibility for parameters, such as the probabilities governing price direction change, to vary along the tree.

Panel (a) in Figure 1 illustrates our labeling for an $m=3$ binary tree. At time step n, each price state (node) on the tree is labeled with a “level” index $l_i$, $i=1,\cdots ,2^{n-1}$, $n=1,\cdots ,m$, $l_0=0$, and a local state index ${ϵ}_n^{\left(l_n\right)}=\{1,0\}$, $n=1,\cdots ,m$, with ${ϵ}_0^{\left(0\right)}=0$. For $n=2$ (Figure 1a) the price states are grouped into two levels, the lower two price states belonging to level $l_2=1$ and the upper two to $l_2=2$. In the $l_2=1$ level, the two price states (nodes) are labeled ${ϵ}_2^{\left(1\right)}=0$ (“down”) and ${ϵ}_2^{\left(1\right)}=1$ (“up”). Analogous labeling occurs for the two $l_2=2$ nodes.

Panel (b) in Figure 1 illustrates path labeling for an $m=3$ binary tree. Each level $l_n$ is reached via a uniquely labeled path through other levels. For example, the $l_3=4$ level (which contains two leaf nodes) is reached via the “level sequence” ${\mathbb{L}}_3=(l_0=0,l_1=1,l_2=2,l_3=4):=(0,1,2,4)$. Each node ${ϵ}_n^{\left(l_n\right)}$ is reached via a uniquely labeled “node sequence” of local up and down price changes. For example, the node ${ϵ}_3^{\left(4\right)}=0$ is reached via the node sequence ${\mathbb{M}}_3=({ϵ}_0^{\left(0\right)}=0,{ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1,{ϵ}_3^{\left(4\right)}=0):=(0,1,1,0)$. We note that, given a specific node sequence ${\mathbb{M}}_n$, it is possible to reconstruct its corresponding level sequence ${\mathbb{L}}_n$. Specifically, for ${\mathbb{M}}_n=(0,m_1,m_2,\cdots ,m_n)$, then ${\mathbb{L}}_n=(l_0,l_1,l_2,\cdots ,l_n)$, with

$$\begin{array}{cc}\hfill l_0=0,l_1& =1,l_2={\left(m_1\right)}_{10}+1,l_3={\left(m_1{m}_2\right)}_{10}+1,\hfill \\ \hfill & \cdots ,l_n={(m_1{m}_2\cdots m_{n-1})}_{10}+1,\hfill \end{array}$$

(5)

where ${(m_1{m}_2\cdots m_k)}_{10}$ denotes the decimal value of the binary integer $m_1{m}_2\cdots m_k$.13 Note, however, that the reverse is not possible. Given the node sequence, ${\mathbb{L}}_n=(l_0,l_1,l_2,\cdots ,l_n)$, it is possible only to compute $m_1$ through $m_{n-1}$, leaving $m_n$ unknown. For simplicity, when employed we will give both the relevant ${\mathbb{M}}_n$ and ${\mathbb{L}}_n$ sequences.

With this discussion of the indexing on tree levels, nodes, and paths, we proceed with the development of the dynamics on the tree. We define a discrete-time-filtered probability space (discrete-time stochastic basis) ${\mathbb{ST}}^{\left(\mathrm{d}\right)}=\left(\Omega ,{\mathbb{F}}^{\left(\mathrm{d}\right)},\mathbb{P}\right)$ with the discrete filtration ${\mathbb{F}}^{\left(\mathrm{d}\right)}=\left\{{\mathcal{F}}^{\left(n\right)},n\in {\mathcal{N}}_0\right\}$, ${\mathcal{N}}_0\stackrel{\mathrm{def}}{=}\left\{0,1,\cdots \right\}$, where ${\mathcal{F}}^{\left(0\right)}=\left\{\varnothing ,\Omega \right\}$. Define $K_n=2^{n-1}$. The sigma fields, ${\mathcal{F}}^{\left(n\right)}=\sigma \left({ϵ}_n^{\left(k\right)},k=1,2,\cdots ,K_n,n\in \mathcal{N}\right)$, are generated by a sequence of dependent binary random variables ${ϵ}_n^{\left(k\right)}$, $k=1,2,\cdots ,K_n$, such that $\mathbb{P}\left({ϵ}_n^{\left(k\right)}=1\right)=1-\mathbb{P}\left({ϵ}_n^{\left(k\right)}=0\right)\in (0,1)$. The triangular array ${\mathcal{E}}_{\mathcal{N}}=\left({ϵ}_n^{\left(k\right)},k=1,2,\cdots ,K_n,n\in \mathcal{N}\right)$ of binary random variables ${ϵ}_n^{\left(k\right)}$ is defined on a probability space $\left(\Omega ,\mathbb{F},\mathbb{P}\right)$, with probability laws

$$\begin{array}{cc}\hfill p_{\left({ϵ}_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}\right)}^{\left(m_n^{\left(1\right)},\cdots ,m_n^{\left(K_n\right)}\right)}& =\mathbb{P}\left({ϵ}_n^{\left(1\right)}=m_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}=m_n^{\left(K_n\right)}\right),\hfill \\ \hfill & m_n^{\left(k\right)}\in \{0,1\},k=1,\cdots ,K_n,n\in \mathcal{N},\hfill \end{array}$$

(6)

satisfying Kolmogorov’s extension theorem (Oksendal, 2013, Theorem 2.1.5, p. 11).

The probability space $(\Omega ,\mathbb{F},\mathbb{P})$ is a standard probability space; without loss of generality, we can assume it is the Lebesgue probability space $\Omega =[0,1]$, $\mathbb{F}=\mathfrak{B}\left(\right[0,1\left]\right)$, $\mathbb{P}=\mathrm{Leb}\left([0,1]\right)$.14 We define the probability law for $\left\{{ϵ}_n^{\left(k\right)},k=1,\cdots ,K_n,n\in \mathcal{N}\right\}$ sequentially in time, defining the dynamics of ${\mathcal{E}}_{\mathcal{N}}$.

For $n=0$, set ${ϵ}_0^{\left(0\right)}=0$, ${\mathcal{E}}_0\stackrel{\mathrm{def}}{=}\left\{{ϵ}_0^{\left(0\right)}\right\}$, and ${\mathcal{F}}^{\left(0\right)}=\sigma \left({\mathcal{E}}_0\right)=\{\varnothing ,\Omega \}$.

For $n=1$, set ${\mathcal{E}}_1\stackrel{\mathrm{def}}{=}\left\{{ϵ}_0^{\left(0\right)},{ϵ}_1^{\left(1\right)}\right\}=\left\{{\mathcal{E}}_0,{ϵ}_1^{\left(1\right)}\right\}$. Then, ${\mathcal{F}}^{\left(1\right)}=\sigma \left({\mathcal{E}}_1\right)$ with

$$\left.\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\right)& =\mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\left|{ϵ}_0^{\left(0\right)}=0\right.\right)\mathbb{P}\left({ϵ}_0^{\left(0\right)}=0\right)\hfill \\ \hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\right)& =\mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\left|{ϵ}_0^{\left(0\right)}=0\right.\right)\mathbb{P}\left({ϵ}_0^{\left(0\right)}=0\right)\hfill \end{array}\right\},\mathrm{where}\mathbb{P}\left({ϵ}_0^{\left(0\right)}=0\right)=1.$$

The conditional probabilities satisfy

$$\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\left|{ϵ}_0^{\left(0\right)}=0\right.\right)& \in (0,1),\hfill \\ \hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\left|{ϵ}_0^{\left(0\right)}=0\right.\right)& =1-\mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\left|{ϵ}_0^{\left(0\right)}=0\right.\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\right):=p_1^{((0,l_1=1),(0,{ϵ}^{\left(1\right)}=1))}:=p_1^{\left(\right(0,1),(0,1\left)\right)}& \in (0,1),\hfill \\ \hfill \mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\right):=p_1^{((0,l_1=1),(0,{ϵ}^{\left(1\right)}=0))}:=p_1^{\left(\right(0,1),(0,0\left)\right)}& =1-p_1^{\left(\right(0,1),(0,1\left)\right)}.\hfill \end{array}$$

For $n>1$, the general case is as follows. (For additional clarity, the sequential definitions for $n=2$ and 3 are provided in Appendix A). Set ${\mathcal{E}}_n\stackrel{\mathrm{def}}{=}\left\{{\mathcal{E}}_{n-1},\left({ϵ}_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}\right)\right\}$. ${\mathcal{E}}_n$ is the triangular array of binary random variables ${ϵ}_l^{\left(k_l\right)}$, $l=1,\cdots ,n$, $k_l=1,\cdots ,K_l$, with ${ϵ}_0^{\left(0\right)}=0$. Then,

$${\mathcal{F}}^{\left(n\right)}=\sigma \left({\mathcal{E}}_n\right),n=0,1,\cdots ,{\mathcal{F}}^{\left(0\right)}=\sigma \left({\mathcal{E}}_0\right)=\{\varnothing ,\Omega \},$$

(7)

and

$$\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},1\right)& =\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)\mathbb{P}\left({\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right)\hfill \\ \hfill & =\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)p_{n-1}^{\left((0,l_1,\cdots ,l_{n-1}),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_2^{\left(l_{n-1}\right)}\right)\right)},\hfill \\ \hfill \mathbb{P}\left({ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},0\right)& =\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=0\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)\mathbb{P}\left({\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right)\hfill \\ \hfill & =\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=0\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)p_{n-1}^{\left((0,l_1,\cdots ,l_{n-1}),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_2^{\left(l_{n-1}\right)}\right)\right)},\hfill \end{array}$$

(8)

where $l_1=1$ and $l_n={\left({ϵ}_1^{\left(l_1\right)}\cdots {ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)}_{10}+1$ for $n>1$. The ${\mathcal{E}}_n$-conditional probabilities satisfy

$$\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)& \in (0,1),\hfill \\ \hfill \mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=0\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right)& =1-\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}=\left(0,{ϵ}_1^{\left(1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}\right)\right.\right).\hfill \end{array}$$

(9)

The unconditional probabilities satisfy

$$\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},1\right)& :=p_n^{\left((0,l_1,\cdots ,l_n),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},1\right)\right)}\in \left(0,p_{n-1}^{\left((0,l_1,\cdots ,l_{n-1}),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_2^{\left(l_{n-1}\right)}\right)\right)}\right),\hfill \\ \hfill \mathbb{P}\left({ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},0\right)& :=p_n^{\left((0,l_1,\cdots ,l_n),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},0\right)\right)}\hfill \\ \hfill & =p_{n-1}^{\left((0,l_1,\cdots ,l_{n-1}),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_2^{\left(l_{n-1}\right)}\right)\right)}-p_n^{\left((0,l_1,\cdots ,l_n),\left(0,{ϵ}_1^{\left(l_1\right)},\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)},1\right)\right)}.\hfill \end{array}$$

(10)

The level sequences

$${\mathbb{L}}_n=(0,l_1,\cdots ,l_j,\cdots ,l_n),l_j=1,\cdots ,K_j,j=1,\cdots ,n,$$

(11)

together with the discrete filtration ${\mathcal{F}}^{\left(n\right)}$ and the conditional probabilities (9) define a particular type of binary random tree which we designate as a binary information tree to level n (${\mathrm{BIT}}_n$).15 Each node sequence

$${\mathbb{M}}_n=(0,m_1,\cdots ,m_j,\cdots ,m_n),m_j=\{0,1\},j=1,\cdots ,n,$$

(12)

defines a unique event

$${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}={\mathcal{E}}_n^{((0,l_1,\cdots ,l_j,\cdots ,l_n),(0,m_1,\cdots ,m_j,\cdots ,m_n))}$$

(13)

on ${\mathrm{BIT}}_n$, where in (13) $l_j={(m_0{m}_1\cdots m_{j-1})}_{10}+1$, $j=1,\cdots ,n$, $m_0=0$. Event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ occurs with probability16

$$\mathbb{P}\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right)=p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}.$$

(14)

Panel (c) in Figure 1 provides an illustration of two specific probabilities of the form $p_3^{({\mathbb{L}}_3,{\mathbb{M}}_3)}$.

To develop a random-price time series simulating microstructure timing, for any given $m\in \mathcal{N}\stackrel{\mathrm{def}}{=}\{1,2,\cdots \}$ we associate the levels of ${\mathrm{BIT}}_m$ with a sequence of time instances $0=t_0<t_1<\cdots <t_{m-1}<t_m=T$ over the finite period $[0,T]$, $T<\infty$. In our application to option pricing, the current time is $t_0=0$ (corresponding to the root event of ${\mathrm{BIT}}_m$), while $t_m=T$ is the terminal time (corresponding to the leaf events of ${\mathrm{BIT}}_m$). Trades of assets occur only at the times $t_1<\cdots <t_{m-1}$. These trading instances are fixed and known at time $t_0$.17 The time intervals $[0,t_1)$, $\left(t_n,t_{n+1}\right)$, $n=1,\cdots ,m-2$, and $(t_{m-1},T]$, over which no trades occur, are denoted inter-trade periods.18 We define $\Delta t_n\stackrel{\mathrm{def}}{=}{t}_n-t_{n-1}$, $n=1,\cdots .,m$.

Associated with ${\mathrm{BIT}}_m$, for all $t\in [0,T]$ we can recursively define the càdlàg node paths ${\mathcal{E}}_{n,t}^{\left({\mathbb{L}}_n\right)}$, $n=0,\cdots ,m-1$, as follows:

$$\begin{array}{cccc}\hfill {\mathcal{E}}_{0,t}^{\left({\mathbb{L}}_0\right)}& ={ϵ}_0^{\left(0\right)}=0\hfill & \hfill & \mathrm{for}t\in [t_0=0,t_1),\hfill \\ \hfill {\mathcal{E}}_{n,t}^{\left({\mathbb{L}}_n\right)}& ={\mathcal{E}}_{n-1,t}^{\left({\mathbb{L}}_{n-1}\right)},{ϵ}_n^{\left(l_n\right)}\hfill & \hfill & \mathrm{for}t\in [t_n,t_{n+1}),1\le n\le m-2,\hfill \\ \hfill {\mathcal{E}}_{m-1,t}^{\left({\mathbb{L}}_m\right)}& ={\mathcal{E}}_{m-2,t}^{\left({\mathbb{L}}_{m-2}\right)},{ϵ}_{m-1}^{\left(l_{m-1}\right)}\hfill & \hfill & \mathrm{for}t\in [t_{m-1},t_m=T].\hfill \end{array}$$

(15)

We define the market information flow ${\mathbb{IF}}_{m;[0,T]}$ as

$${\mathbb{IF}}_{m;[0,T]}=\left\{{\mathcal{E}}_{n,t}^{\left({\mathbb{L}}_n\right)};n=0,\cdots ,m-1,t\in [0,T],{\mathbb{L}}_n\in {\mathcal{L}}_n\right\},$$

(16)

where ${\mathcal{L}}_n$ is the set of all ($n+1$)-tuples, ${\mathbb{L}}_n$. ${\mathbb{IF}}_{m;[0,T]}$ generates the ${\mathrm{BIT}}_m$ stochastic basis ${\mathbb{ST}}_{(m;[0,T\left]\right)}^{\left(\mathrm{d}\right)}=\left(\Omega ,{\mathbb{F}}_{m;[0,T]}^{\left(\mathrm{d}\right)},\mathbb{P}\right)$ on $\left[0,T\right]$, where the filtration is defined by

$${\mathbb{F}}_{m;[0,T]}^{\left(\mathrm{d}\right)}=\left\{{\mathcal{F}}_{0;[0,T]}^{\left(\mathrm{d}\right)}=\left\{\varnothing ,\Omega \right\},{\mathcal{F}}_{n;[0,T]}^{\left(\mathrm{d}\right)}=\sigma \left\{{\mathcal{E}}_{n,t}^{\left({\mathbb{L}}_n\right)},{\mathbb{L}}_n\in {\mathcal{L}}_n\right\},n=0,\cdots ,m-1\right\}.$$

(17)

Given ${\mathrm{BIT}}_m$, specification of ${\mathbb{M}}_m=(0,m_1,\cdots ,m_n,m_{n+1},\cdots ,m_m)=({\mathbb{M}}_n,m_{n+1},\cdots ,m_m)$, $n=0,\cdots ,m-1$, defines a unique, nested set of càdlàg event paths ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, $n=0,\cdots ,m-1$, $t\in [0,T]$, corresponding to the nested node sequences ${\mathbb{L}}_n=(0,l_1,\cdots ,l_n)$, with $l_j={(m_0,m_1\cdots m_{j-1})}_{10}+1$, $j=1,\cdots ,n$, $m_0=0$, and ${ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_n^{\left(l_n\right)}=m_n$. From (15), this unique set of event paths is

$$\begin{array}{cccc}\hfill {\mathcal{E}}_{0,t}^{({\mathbb{L}}_0,{\mathbb{M}}_0)}& =0\hfill & \hfill & \mathrm{for}t\in [t_0=0,t_1),\hfill \\ \hfill {\mathcal{E}}_{1,t}^{({\mathbb{L}}_1,{\mathbb{M}}_1)}& =0,m_1\hfill & \hfill & \mathrm{for}t\in [t_1,t_2),\hfill \\ \hfill {\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}& ={\mathcal{E}}_{n-1,t}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},m_n\hfill & \hfill & \mathrm{for}t\in [t_n,t_{n+1}),2\le n\le m-2,\hfill \\ \hfill {\mathcal{E}}_{m-1,t}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}& ={\mathcal{E}}_{m-2,t}^{({\mathbb{L}}_{m-2},{\mathbb{M}}_{m-2})},m_{m-1}\hfill & \hfill & \mathrm{for}t\in [t_{m-1},t_m=T].\hfill \end{array}$$

(18)

Event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ and event path ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ labels on ${\mathrm{BIT}}_4$ are illustrated in Figure 2.

The ${\mathcal{E}}_{n-1}$-conditional probabilities along path ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ are

$$\mathbb{P}\left({ϵ}_j^{\left(l_j\right)}=m_j\left|{ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{j-1}^{\left(l_{j-1}\right)}=m_{j-1}\right.\right),j=1,\cdots ,n.$$

(19)

The unconditional probability $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ for event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ on path ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ is determined by the sequence of conditional probabilities (19).

As there is no trade at $t_m=T$, there are no new events at time $t_m$ on ${\mathrm{BIT}}_m$; the last events occur at $t_{m-1}$. Similarly, there is no event at $t_0=0$, the first events occur at $t_1$. Thus, on ${\mathrm{BIT}}_m$ the events are labeled from ${\mathcal{E}}_1^{({\mathbb{L}}_1,{\mathbb{M}}_1)}$ to ${\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}$. From (13) and (18), we have the event-path equivalence

$${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\stackrel{\mathrm{EP}}{\sim }{\mathcal{E}}_{n,t_n}^{({\mathbb{L}}_n,{\mathbb{M}}_n)},n=1,\cdots ,m-1,$$

where $\stackrel{\mathrm{EP}}{\sim }$ means that event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ occurs at time point $t_n$ on path ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$. To preserve uniformity of notation on ${\mathrm{BIT}}_m$, we define the pseudo-event ${\mathcal{E}}_0^{({\mathbb{L}}_0,{\mathbb{M}}_0)}={\mathcal{E}}_0^{(0,0)}$ for the root node, leading to the equivalence

$${\mathcal{E}}_0^{(0,0)}\stackrel{\mathrm{EP}}{\sim }{\mathcal{E}}_{0,t_0}^{(0,0)}.$$

As the “leaf” nodes at $t=t_m$ are path termination points, the terminating pseudo-events will be labeled using path notation ${\mathcal{E}}_{m-1,t_m}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}$. We shall refer to all events, pseudo or otherwise, simply as events.

For every fixed $m\in \mathcal{N}$, the set of all ${\sum }_{n=0}^{m-1}{2}^n$ paths ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, $n=0,\cdots ,m-1$, defines an ${\mathbb{F}}^{\left(\mathrm{d}\right)}$-adapted ${\mathrm{BIT}}_m$, which we denote ${\mathbb{BT}}_m$. The probabilities $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ will represent the natural probabilities for the direction of stock movements. ${\mathbb{BT}}_m$ provides the stochastic dynamics of the market information based on the time instances $0=t_0<t_1<\cdots <t_{m-1}<t_m=T$.

Estimation of Probabilities

In our discrete market setting (Section 3, Section 4 and Section 5), $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ will represent the probability of the direction of price changes at $t_n$. For example, given price $S_{t_{n-1}}$ corresponding to event ${\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}$, then $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, with ${\mathbb{L}}_n=({\mathbb{L}}_{n-1},l_n)$, ${\mathbb{M}}_n=({\mathbb{M}}_{n-1},1)$, and $l_n={(m_1\cdots m_{n-1}1)}_{10}+1$ represents the probability of a price increase $S_{t_n}-S_{t_{n-1}}>0$, while $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ with ${\mathbb{M}}_n=({\mathbb{M}}_{n-1},0)$ and $l_n={(m_1\cdots m_{n-1}0)}_{10}+1$ represents the probability of a price decrease $S_{t_n}-S_{t_{n-1}}<0$.

Assuming that a sufficient history ($t<0$) of stock prices is available at $t=0$, one can utilize the historical frequency ${\widehat{p}}_1^{((0,1),(0,1);\Delta t_1)}$ of positive price changes over trading periods of size $\Delta t_1$ as an estimator for $p_1^{\left(\right(0,1),(0,1\left)\right)}$. For example, if $\Delta t_1=5$ min, then ${\widehat{p}}_1^{((0,1),(0,1);\Delta t_1)}$ is the proportion of positive stock returns observed in a historical sample of 5-min returns.19 For $n>1$, one can use the historical frequency ${\widehat{p}}_n^{\left({\mathbb{L}}_n,{\mathbb{M}}_n;\Delta t_{1,n}\right)}$ as an estimator for $p_n^{\left({\mathbb{L}}_n,{\mathbb{M}}_n\right)}$, where $\Delta t_{1,n}\stackrel{\mathrm{def}}{=}\Delta t_1,\cdots ,\Delta t_n$. As an example of the computation of ${\widehat{p}}_n^{\left({\mathbb{L}}_n,{\mathbb{M}}_n;\Delta t_{1,n}\right)}$, assume equally spaced time intervals, $\Delta t_1=\cdots .=\Delta t_n=\Delta t$, where $\Delta t=1$ day. Consider a data set of $\mathcal{T}$ historical daily returns partitioned into V non-overlapping time periods, each period consisting of n days. In each period, the succession of signs of the n daily returns is compared to the succession of signs implied by the ${\mathbb{M}}_n$ indexing of the path ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$. If these two sign successions agree, the trial period is marked as a “success” (otherwise a “failure”). Thus, the partitioning of the data provides a set of V Bernoulli trials from which to compute

$${\widehat{p}}_n^{\left({\mathbb{L}}_n,{\mathbb{M}}_n;\Delta t_{1,n}\right)}={\displaystyle \frac{\mathrm{number}\mathrm{of}\mathrm{successes}\mathrm{in}V\mathrm{trials}}{V}}.$$

(20)

In Section 9, we empirically investigate this procedure for computing the required probabilities. Our results indicate that a prohibitively extensive history $\mathcal{T}=Vn$ would be required to ensure adequate sampling for values of $n\gtrsim 6$. In Section 9, we consider the use of bootstrap resampling to provide adequate samples.

3. Dynamics of the Riskless Rate and Bank Account Value on ${\mathbb{BT}}_{\mathbf{m}}$

We consider first the path-dependent dynamics of the riskless rate $r_t^{\left(\mathrm{d},\mathrm{f}\right)}$ and ${\mathcal{B}}_t$ on ${\mathbb{BT}}_m$. Corresponding to the discrete times $t_n$, $n=1,\cdots ,m$, we consider the ${\mathcal{F}}^{(n-1)}$-measurable riskless rates $r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}$ and, without loss of generality, set $r_{t_0}^{\left(\mathrm{d},\mathrm{f}\right)}=0$.20 For any time $t\in [0,T]$, we define the ${\mathbb{ST}}_{m;[0,T]}^{\left(\mathrm{d}\right)}$-adapted riskless rate $r_t^{\left(\mathrm{d},\mathrm{f}\right)}$ as follows.

For $t\in [0,t_1)$, $r_t^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_0}^{\left(\mathrm{d},\mathrm{f}\right)}=0$.

For $t\in [t_1,t_2)$, $r_t^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_1;{\mathcal{E}}_0^{(0,0)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_1;({ϵ}_0^{\left(0\right)}=0)}^{\left(\mathrm{d},\mathrm{f}\right)}$.

For $t\in [t_2,t_3)$,

$$r_t^{\left(\mathrm{d},\mathrm{f}\right)}=\left\{\begin{array}{c}{r}_{t_2;{\mathcal{E}}_1^{\left(\right(0,1),(0,1\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_2;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,1\left)\right)},\hfill \\ r_{t_2;{\mathcal{E}}_1^{\left(\right(0,1),(0,0\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_2;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,0\left)\right)}.\hfill \end{array}\right.$$

For $t\in [t_3,t_4)$,

$$r_t^{\left(\mathrm{d},\mathrm{f}\right)}=\left\{\begin{array}{c}{r}_{t_3;{\mathcal{E}}_2^{\left(\right(0,1,2),(0,1,1\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,2),(0,1,1\left)\right)},\hfill \\ r_{t_3;{\mathcal{E}}_2^{\left(\right(0,1,2),(0,1,0\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,2),(0,1,0\left)\right)},\hfill \\ r_{t_3;{\mathcal{E}}_2^{\left(\right(0,1,1),(0,0,1\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,1),(0,0,1\left)\right)},\hfill \\ r_{t_3;{\mathcal{E}}_2^{\left(\right(0,1,1),(0,0,0\left)\right)}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,1),(0,0,0\left)\right)}.\hfill \end{array}\right.$$

In general, for path ${\mathcal{E}}_{n-1,t}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}$ and $t\in [t_n,t_{n+1})$, $n=2,\cdots ,m-1$,

$$r_t^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}\equiv r_{t_n;\left({ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}=m_{n-1}\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\mathrm{w}.\mathrm{p}.p_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},$$

(21)

with the understanding that, when $n=m-1$, for any path ${\mathcal{E}}_{m-2,t}^{({\mathbb{L}}_{m-2},{\mathbb{M}}_{m-2})}$, (21) holds for the closed time interval $t\in [t_{m-1},T]$. Figure 3 illustrates the path dependence of the riskless rates $r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}$.

For $t\in [t_n,t_{n+1})$, the riskless rate $r_t^{\left(\mathrm{d},\mathrm{f}\right)}$ is ${\mathcal{F}}^{(n-1)}$-measurable. This definition is consistent with the definition of the riskless rate (short rate) dynamics in continuous time (Duffie, 2001, p. 102). Without loss of generality, we can define the path-dependent, instantaneous riskless rate $r_t^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}>0$, $t\in [t_n,t_{n+1})$ by the relation

$$r_t^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\Delta t_n.$$

(22)

Having determined the dynamics of the riskless rates $r_t^{\left(\mathrm{d},\mathrm{f}\right)}$, $t\in [0,T]$, we now turn our attention to the dynamics of the riskless bank account $\mathcal{B}$. The ${\mathbb{ST}}_{m;[0,T]}^{\left(\mathrm{d}\right)}$ -adapted bank account price ${\beta }_t^{\left(\mathrm{d}\right)}$, $t\in [0,T]$, is defined as follows.

For $t\in [0,t_1)$,

$${\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_0}^{\left(\mathrm{d}\right)}={\beta }_0.$$

For $t\in [t_1,t_2),$

$${\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}={\beta }_0\left(1+r_{t_1}^{\left(\mathrm{d},\mathrm{f}\right)}\right).$$

For $t\in [t_2,t_3),$

$${\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_2}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\left\{\begin{array}{c}{\beta }_{t_2;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,1\left)\right)},\hfill \\ {\beta }_{t_2;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,0\left)\right)}.\hfill \end{array}\right.$$

For $t\in [t_3,t_4)$,

$${\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_3}^{\left(\mathrm{d}\right)}={\beta }_{t_2}^{\left(\mathrm{d}\right)}\left(1+r_{t_3}^{\left(\mathrm{d},\mathrm{f}\right)}\right)=\left\{\begin{array}{c}\begin{array}{cc}\hfill {\beta }_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)& \left(1+r_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\hfill \\ \hfill & \mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,2),(0,1,1\left)\right)},\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill {\beta }_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)& \left(1+r_{t_3;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\hfill \\ \hfill & \mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,2),(0,1,0\left)\right)},\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill {\beta }_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)& \left(1+r_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\hfill \\ \hfill & \mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,1),(0,0,1\left)\right)},\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill {\beta }_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d}\right)}={\beta }_{t_1}^{\left(\mathrm{d}\right)}\left(1+r_{t_2;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)& \left(1+r_{t_3;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right)\hfill \\ \hfill & \mathrm{w}.\mathrm{p}.p_2^{\left(\right(0,1,1),(0,0,0\left)\right)}.\hfill \end{array}\hfill \end{array}\right.$$

For $t\in [t_n,t_{n+1})$, $n=2,\cdots ,m-1$, given path ${\mathcal{E}}_{n-1,t}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}$, the value of the bank account ${\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_n}^{\left(\mathrm{d}\right)}$, $t\in [t_n,t_{n+1})$ is

$$\begin{array}{cc}\hfill {\beta }_t^{\left(\mathrm{d}\right)}={\beta }_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}& ={\beta }_{t_n;\left\{{ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}=m_{n-1}\right\}}^{\left(\mathrm{d}\right)}\hfill \\ \hfill & ={\beta }_{t_{n-1};\left\{{ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{n-2}^{\left(l_{n-2}\right)}=m_{n-2}\right\}}^{\left(\mathrm{d}\right)}\left[1+r_{t_n;\left\{{ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{n-1}^{\left(l_{n-1}\right)}=m_{n-1}\right\}}^{\left(\mathrm{d},\mathrm{f}\right)}\right]\hfill \\ \hfill & ={\beta }_{t_1}^{\left(\mathrm{d}\right)}\prod _{k=2}^n\left[1+r_{t_k;\left({ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{k-1}^{\left(l_{k-1}\right)}=m_{k-1}\right)}^{\left(\mathrm{d},\mathrm{f}\right)}\right]\mathrm{w}.\mathrm{p}.p_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},\hfill \end{array}$$

(23)

with the understanding that, when $n=m-1$, (23) holds for the closed time interval $t\in [t_{m-1},T]$ for any path ${\mathcal{E}}_{m-2,t}^{({\mathbb{L}}_{m-2},{\mathbb{M}}_{m-2})}$.

For $[t_n,t_{n+1})$, the bank account value ${\beta }_t^{\left(\mathrm{d}\right)}$ is $F^{(n-1)}$-measurable. This definition is consistent with the definition of the riskless asset price dynamics ${\beta }_t,t\in [0,T]$ in continuous time. More precisely, as shown in Section 6, in continuous time the riskless asset price ${\beta }_t,t\in [0,T]$ is defined on the filtered probability space $(\Omega ,{\mathbb{F}}_{[0,T]}={\{{\mathcal{F}}_t=\sigma (B_u,0\le u\le t)\}}_{t\in [0,T]},\mathbb{P})$, where $B_t$, $t\in [0,T]$, is a standard Brownian motion on $[0,T]$, and its price dynamics are determined by $d{\beta }_t=r_t{\beta }_{t}dt$, ${\beta }_0>0$, where the short rate $r_t\ge 0$ is ${\mathcal{F}}_t$-measurable with $\mathbb{P}\left({sup}_{t\in [0,T]}\left\{r_t+1/r_t\right\}<\infty \right)=1$. Thus, ${\beta }_t$ is instantaneously riskless; in other words, ${\beta }_u={\beta }_t$ for $u\in \left[t,t+dt\right)$. This definition of the riskless asset pricing in continuous time was the motivation to define the riskless bank account valuation in discrete time by Equation (23).

4. Stock Price Dynamics on ${\mathbb{BT}}_{\mathbf{m}}$

The price dynamics $S_t^{\left(\mathrm{d}\right)},t\in [0,T]$ of $\mathcal{S}$ on the stochastic basis ${\mathbb{ST}}_{m;t\in [0,T]}^{\left(\mathrm{d}\right)}$ is an ${\mathbb{F}}_{m;t\in [0,T]}^{\left(\mathrm{d}\right)}$-adapted process. We define $S_t^{\left(\mathrm{d}\right)}$, $t\in [0,T]$ on ${\mathbb{ST}}_{m;t\in [0,T]}^{\left(\mathrm{d}\right)}$ as follows.

For $t\in [0,t_1)$,

$$S_t^{\left(\mathrm{d}\right)}=S_{t_0}^{\left(\mathrm{d}\right)}=S_0>0.$$

For $t\in [t_1,t_2),$

$$S_t^{\left(\mathrm{d}\right)}=S_{t_1}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}{S}_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}=S_0{s}_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,1\left)\right)},\hfill \\ S_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}^{\left(\mathrm{d}\right)}=S_0{s}_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1),(0,0\left)\right)},\hfill \end{array}\right.$$

(24)

for price ratio values $s_{t_1;\left({ϵ}_1^{\left(1\right)}\right)}>0$. Let

$$r_{t_1}^{\left(\mathrm{d}\right)}={\displaystyle \frac{S_{t_1}^{\left(\mathrm{d}\right)}-S_0}{S_0}}=\left\{\begin{array}{c}{s}_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\right),\hfill \\ s_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\right),\hfill \end{array}\right.$$

(25)

be the discrete (arithmetic) return of the stock at $t_1$. We assume that the mean $\mathbb{E}\left[r_{t_1}^{\left(\mathrm{d}\right)}\right]$ and variance $\mathrm{Var}\left[r_{t_1}^{\left(\mathrm{d}\right)}\right]$ of the return,

$$\begin{array}{cccc}\hfill \mathbb{E}\left[r_{t_1}^{\left(\mathrm{d}\right)}\right]& ={\mu }_{t_1}^{\left(r\right)}\Delta t_1,\hfill & \hfill & \mathrm{for}\mathrm{some}{\mu }_{t_1}^{\left(r\right)}>r_{t_1}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)},\hfill \\ \hfill \mathrm{Var}\left[r_{t_1}^{\left(\mathrm{d}\right)}\right]& ={\left({\sigma }_{t_1}^{\left(r\right)}\right)}^2\Delta t_1,\hfill & \hfill & \mathrm{for}\mathrm{some}{\sigma }_{t_1}^{\left(r\right)}>0,\hfill \end{array}$$

(26)

are known. In (26), we interpret ${\mu }_{t_1}^{\left(r\right)}$ as the instantaneous mean return and ${\left({\sigma }_{t_1}^{\left(r\right)}\right)}^2$ as the instantaneous variance at $t_1$. The moment conditions (26) imply that the price ratios $s_{t_1;\left({ϵ}_1^{\left(1\right)}\right)}$ in (25) are determined by

$$\begin{array}{cc}\hfill s_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}& =1+{\mu }_{t_1}^{\left(r\right)}\Delta t_1+{\sigma }_{t_1}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\right)}{\mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\right)}}\Delta t_1},\hfill \\ \hfill s_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}& =1+{\mu }_{t_1}^{\left(r\right)}\Delta t_1-{\sigma }_{t_1}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_1^{\left(1\right)}=1\right)}{\mathbb{P}\left({ϵ}_1^{\left(1\right)}=0\right)}}\Delta t_1}.\hfill \end{array}$$

For $t\in \left[t_2,t_3\right)$,

$$S_t^{\left(\mathrm{d}\right)}=S_{t_2}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}{S}_{t_2;{\mathcal{E}}_2^{\left(\right(0,1,2),(0,1,1\left)\right)}}^{\left(\mathrm{d}\right)}=S_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1,2),(0,1,1\left)\right)},\hfill \\ S_{t_2;{\mathcal{E}}_2^{\left(\right(0,1,2),(0,1,0\left)\right)}}^{\left(\mathrm{d}\right)}=S_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1,2),(0,1,0\left)\right)},\hfill \\ S_{t_2;{\mathcal{E}}_2^{\left(\right(0,1,1),(0,0,1\left)\right)}}^{\left(\mathrm{d}\right)}=S_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}^{\left(\mathrm{d}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1,1),(0,0,1\left)\right)},\hfill \\ S_{t_2;{\mathcal{E}}_2^{\left(\right(0,1,1),(0,0,0\left)\right)}}^{\left(\mathrm{d}\right)}=S_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}^{\left(\mathrm{d}\right)}\mathrm{w}.\mathrm{p}.p_1^{\left(\right(0,1,1),(0,0,0\left)\right)}.\hfill \end{array}\right.$$

(27)

Note that $S_{t_2;\left({ϵ}_1^{\left(1\right)}=m_1,{ϵ}_2^{\left(2\right)}=m_2\right)}^{\left(\mathrm{d}\right)}\equiv S_{t_2;{\mathcal{E}}_2^{((0,l_1,l_2),(0,m_1,m_2))}}^{\left(\mathrm{d}\right)}$ denotes the stock price along the $[t_2,t_3)$ segment of the path ${\mathcal{E}}_{2,t}^{((0,l_1,l_2),(0,m_1,m_2))}$. Figure 4 illustrates the path dependence of the stock prices $S_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}$ up to $t_3$. In contrast, $S_{t_2}^{\left(\mathrm{d}\right)}$ denotes the state space of prices over the interval $[t_2,t_3)$. We introduce the conditional notation $S_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(\mathrm{d}\right)}$ to designate the two state prices determined by the condition ${ϵ}_1^{\left(1\right)}=1$, while $S_{t_2\left|\left({ϵ}_1^{\left(1\right)}=0\right)\right.}^{\left(\mathrm{d}\right)}$ designates the two state prices determined by the condition ${ϵ}_1^{\left(1\right)}=0$.

From (27), (24), and (A1),

$$S_{t_2}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}{S}_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=1\right)p_1^{\left(\right(0,1),(0,1\left)\right)},\hfill \\ S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=1\right)p_1^{\left(\right(0,1),(0,1\left)\right)},\hfill \\ S_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=0\right)p_1^{\left(\right(0,1),(0,0\left)\right)},\hfill \\ S_{t_1;\left({ϵ}_1^{\left(1\right)}=0\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=0\right)p_1^{\left(\right(0,1),(0,0\left)\right)},\hfill \end{array}\right.$$

(28)

for price ratios $s_{t_2;\left({ϵ}_1^{\left(1\right)},{ϵ}_2^{\left(2\right)}\right)}>0$. Let

$$\begin{array}{cc}\hfill r_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(\mathrm{d}\right)}={\displaystyle \frac{S_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(\mathrm{d}\right)}-S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}{S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}}& =\left\{\begin{array}{c}\hfill {\displaystyle \frac{S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}-S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}{S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}}\\ \hfill {\displaystyle \frac{S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}-S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}{S_{t_1;\left({ϵ}_1^{\left(1\right)}=1\right)}^{\left(\mathrm{d}\right)}}}\end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{c}{s}_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=1\right),\hfill \\ s_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=1\right),\hfill \end{array}\right.\hfill \end{array}$$

be the conditional arithmetic return at $t_2$ given that ${ϵ}_1^{\left(1\right)}=1$. We assume that the conditional mean and conditional variance,

$$\begin{array}{cccc}\hfill \mathbb{E}\left[r_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(\mathrm{d}\right)}\right]& ={\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}\Delta t_2,\hfill & \hfill & {\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}>r_{t_2;\left({ϵ}_1^{\left(1\right)}=k\right)}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)},\hfill \\ \hfill \mathrm{Var}\left[r_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(\mathrm{d}\right)}\right]& ={\left({\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}\right)}^2\Delta t_2,\hfill & \hfill & {\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}>0,\hfill \end{array}$$

(29)

are known. In (29), ${\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}$ is the instantaneous conditional mean return and ${\left({\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=k\right)\right.}^{\left(r\right)}\right)}^2$ is the instantaneous conditional variance of the return at $t_2$ given ${ϵ}_1^{\left(1\right)}=k$. The moment conditions (29) imply that, in (28),

$$\begin{array}{cc}\hfill s_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=1\right)}& =1+{\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(r\right)}\Delta t_2+{\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=1\right)}{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=1\right)}}\Delta t_2},\hfill \\ \hfill s_{t_2;\left({ϵ}_1^{\left(1\right)}=1,{ϵ}_2^{\left(2\right)}=0\right)}& =1+{\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(r\right)}\Delta t_2-{\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=1\right)\right.}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=1\right)}{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=1\right)}}\Delta t_2},\hfill \\ \hfill s_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=1\right)}& =1+{\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=0\right)\right.}^{\left(r\right)}\Delta t_2+{\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=0\right)\right.}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=0\right)}{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=0\right)}}\Delta t_2},\hfill \\ \hfill s_{t_2;\left({ϵ}_1^{\left(1\right)}=0,{ϵ}_2^{\left(2\right)}=0\right)}& =1+{\mu }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=0\right)\right.}^{\left(r\right)}\Delta t_2-{\sigma }_{t_2\left|\left({ϵ}_1^{\left(1\right)}=0\right)\right.}^{\left(r\right)}\sqrt{{\displaystyle \frac{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=1|{ϵ}_1^{\left(1\right)}=0\right)}{\mathbb{P}\left({ϵ}_2^{\left(2\right)}=0|{ϵ}_1^{\left(1\right)}=0\right)}}\Delta t_2}.\hfill \end{array}$$

For $t\in [t_n,t_{n+1})$, $n=2,\cdots ,m-1$, given the path ${\mathcal{E}}_{n-1,t}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}$, the stock price is

$$\begin{array}{c}{S}_t^{\left(\mathrm{d}\right)}=S_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\hfill \\ \hfill =\left\{\begin{array}{c}{S}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)}^{\left(\mathrm{d}\right)}=S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}{s}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)},\hfill \\ S_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)}^{\left(\mathrm{d}\right)}=S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}{s}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)},\hfill \end{array}\right.\end{array}$$

(30)

for price ratios $s_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=k\right)}$, $k=0,1$. Again, it is understood that, when $n=m-1$, for a given path ${\mathcal{E}}_{m-2,t}^{({\mathbb{L}}_{m-2},{\mathbb{M}}_{m-2})}$ (30) holds for the closed time interval $t\in [t_{m-1},T]$. Let

$$\begin{array}{c}{r}_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\hfill \\ \hfill \begin{array}{cc}& ={\displaystyle \frac{S_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}-S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}}\hfill \\ & =\left\{\begin{array}{c}{\displaystyle \frac{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}{s}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)}-S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}}:=r_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n=({\mathbb{M}}_{n-1},1))}}^{\left(\mathrm{d}\right)}\hfill \\ {\displaystyle \frac{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}{s}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)}-S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}}:=r_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n=({\mathbb{M}}_{n-1},0))}}^{\left(\mathrm{d}\right)}\hfill \end{array}\right.\hfill \\ & =\left\{\begin{array}{c}{s}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_n^{\left(l_n\right))}=1\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right),\hfill \\ s_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)}-1\mathrm{w}.\mathrm{p}.\mathbb{P}\left({ϵ}_n^{\left(l_n\right))}=0\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right),\hfill \end{array}\right.\hfill \end{array}\end{array}$$

be the conditional arithmetic return at $t_n$ given the path ${\mathcal{E}}_{n-1,t}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}$.21 We assume the conditional mean and conditional variance of $r_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}$,

$$\begin{array}{cc}\hfill \mathbb{E}\left[r_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\right]& ={\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\Delta t_n,\hfill \\ \hfill \mathrm{Var}\left[r_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\right]& ={\left({\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\right)}^2\Delta t_n,\hfill \end{array}$$

(31)

are known for some

$${\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1}))}\right.}^{\left(r\right)}>r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)},{\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}>0.$$

Conditions (31) imply that, in (30)

$$\begin{array}{cc}\hfill s_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)}=1& +{\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\Delta t_n\hfill \\ \hfill & +{\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\sqrt{\frac{\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=0\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right)}{\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right)}\Delta t_n},\hfill \\ \hfill s_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)}=1& +{\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\Delta t_n\hfill \\ \hfill & -{\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\sqrt{\frac{\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right)}{\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=0\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.\right)}\Delta t_n}.\hfill \end{array}$$

(32)

Note the notational equivalences

$$s_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=m_n\right)}\equiv s_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}},S_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=m_n\right)}\equiv S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}.$$

The first form in each equality is used when we wish to specify the value of $m_n$, the second form refers to the respective price at an arbitratry event in ${\mathrm{BIT}}_m$.

The riskless rate $r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}$, bank account value ${\beta }_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}$, stock price $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}$, price ratio $s_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}$, return $r_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}$, and the conditional moments ${\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}$ and ${\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}$, corresponding to the $[t_n,t_{n+1})$ segments of each of the two event paths ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,({\mathbb{M}}_{n-1},m_n=1))}$ and ${\mathcal{E}}_{n,t}^{({\mathbb{L}}_n,({\mathbb{M}}_{n-1},m_n=0))}$, are illustrated in Figure 5. The bank account value, riskless rate, and the two moments are the same for each of the two path segments.

Stock price dynamics in the natural world are determined by (30) and (31), which contain the following sets of model parameters:

• ${\mathcal{I}}^{\left(\mathbb{P}\right)}$—the probabilities for stock upward direction, $\mathbb{P}\left({ϵ}_n^{\left(l_n\right)}=1\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right)\right.$;
• ${\mathcal{I}}^{\left(\mu \right)}$—the conditional means ${\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}$;
• ${\mathcal{I}}^{\left(\sigma \right)}$—the conditional variances ${\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}$;

on each node of the pricing tree. In Section 5, we show that the risk-neutral tree dynamics of the stock preserves ${\mathcal{I}}^{\left(\mathbb{P}\right)},{\mathcal{I}}^{\left(\mu \right)},{\mathcal{I}}^{\left(\sigma \right)}$. This is important given that the information on ${\mathcal{I}}^{\left(\mathbb{P}\right)}$ and ${\mathcal{I}}^{\left(\mu \right)}$ is lost in passing to the continuous-time limit in the natural world and subsequent use of BSM for risk-neutral valuation. Using a discrete pricing tree rather than a continuous-time pricing model allows us to introduce richer, more flexible models for the price dynamics to accommodate market microstructure features in option pricing.

5. Risk-Neutral Dynamics on ${\mathbb{BT}}_{\mathbf{m}}$: Option Pricing

The option $\mathcal{C}$ has discrete price dynamics $f_{t_n}^{\left(\mathrm{d}\right)}=f\left(S_{t_n}^{\left(\mathrm{d}\right)},t_n\right)$, $n=0,\cdots ,m-1$, on ${\mathbb{BT}}_m$ for some $f(x,t)\in \mathbb{R}$, $x>0$, $t\in [0,T]$, with terminal time $T=t_m$ and terminal value $f_T=g\left(S_T\right)$ for some $g\left(x\right)\in \mathbb{R}$, $x>0$.22 At event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, the option has the price

$$f\left(S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)},t_n\right)\equiv f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}.$$

(33)

Consider the replicating portfolio satisfying

$$f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}=D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}{S}_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}+{\beta }_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}$$

(34)

at event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, where $D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}$ is the delta position. Requiring the usual discrete-time, no-arbitrage conditions, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0& =D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}{S}_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}+{\beta }_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}-f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}\hfill \\ & =D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}{S}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}+{\beta }_{t_{n+1};{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}-f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}\hfill \\ & =D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}{S}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}+{\beta }_{t_{n+1};{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}-f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}.\hfill \end{array}\end{array}$$

(35)

Figure 6 illustrates these path-dependent, no-arbitrage conditions on the binary tree at ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$.

From the second two equations in (35), the delta position is given by

$$\begin{array}{c}\begin{array}{cc}\hfill D_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}& ={\displaystyle \frac{f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}-f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}}{S_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}-S_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}}}\hfill \\ & =\left[{\displaystyle \frac{f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}-f_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}}{S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}{\sigma }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}^{\left(r\right)}\sqrt{\Delta t_{n+1}}}}\right]\hfill \end{array}\hfill \end{array}$$

$$\times \sqrt{\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=1|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right)\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=0|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right)},$$

(36)

where the final equality in (36) is obtained using (30) and (32). From the first equation in (35), and using (36), (23), and (22), we obtain the recurrence relation for the risk-neutral option value at event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$:

$$\begin{array}{c}\begin{array}{cc}\hfill f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}& ={\displaystyle \frac{1}{\left(1+r_{t_{n+1};{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\Delta t_{n+1}\right)}}\hfill \\ & \times \left\{q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}{f}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}\right.\hfill \end{array}\hfill \\ \hfill +\left.q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}{f}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}\right\},\end{array}$$

(37)

where the conditional risk-neutral probabilities are given by

$$\begin{array}{cc}\hfill q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}& =\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\hfill \\ \hfill & -{\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}\sqrt{\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\Delta t_{n+1}},\hfill \\ \hfill q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}& =1-q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{l_{n+1})}=1\right)}^{\left(\mathrm{d}\right)}\hfill \\ \hfill & =\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\hfill \\ \hfill & +{\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}\sqrt{\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\mathbb{P}\left({ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.\right)\Delta t_{n+1}},\hfill \end{array}$$

(38)

with

$${\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}={\displaystyle \frac{{\mu }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}^{\left(r\right)}-r_{t_{n+1};{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}}{{\sigma }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}^{\left(r\right)}}}$$

being the market price of risk. Equations (34) through (38) hold for all events ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ on ${\mathbb{BT}}_m$ when $n=0,\cdots ,m-2$. As $S_{t_{m-1};{\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}}^{\left(\mathrm{d}\right)}$ has a constant value $S_{T;{\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}}$ over the time interval $[t_{m-1},t_m=T]$, the terminal value $g\left(S_{T;{\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}}\right)$ determines each option price $f_{t_{m-1};{\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}}^{\left(\mathrm{d}\right)}$ at the events ${\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}$. These values provide the “initial conditions” for the recurrence relation (37).

Note that (36)–(38) each have an overall form that is familiar from binomial tree models. This should not be too surprising as a binomial tree is a particular form of a binary tree. The path dependence in our model therefore arises not through a change in form of these equations but from the fact that the variables (prices, probabilities, conditional moments) are explicitly path-dependent.

We further note that the natural-world conditional probabilities, and the mean and variance of the return dynamics, of $\mathcal{S}$ are retained in the risk-neutral price dynamics of $\mathcal{C}$. This represents the tremendous advantage of discrete binary option pricing over its limiting continuous-time model; under the latter, information about the probabilities and the mean of the return process is lost (Section 6).

The extension to the pricing of an American option follows from the classical approach for valuation of an American option on a binomial tree (see, e.g., Hull, 2006, Section 12.5). The market value of the American option at event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ is given by

$$f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\mathrm{American}\right)}=max\left(f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)},S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\mathrm{exercise}\right)}\right),$$

(39)

where $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\mathrm{exercise}\right)}$ is the exercise value of the stock, which is known at event ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$. We emphasize that $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\mathrm{exercise}\right)}$ is the market value of the stock and not its fair-holding value $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\text{fair-holding}\right)}$. The risk-neutral tree stock dynamics $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\text{fair-holding}\right)}$, $n=1,\cdots ,m$, are determined via the risk-neutral recursion (37),

$$\begin{array}{c}\begin{array}{cc}\hfill S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\text{fair-holding}\right)}& ={\displaystyle \frac{1}{\left(1+r_{t_{n+1};{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\Delta t_{n+1}\right)}}\hfill \\ & \times \left\{q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}{S}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d};\text{fair-holding}\right)}\right.\hfill \end{array}\hfill \\ \hfill +\left.q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}{S}_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d};\text{fair-holding}\right)}\right\},\end{array}$$

(40)

$n=0,\cdots ,m-2$, based on the terminal value23

$$S_T^{\left(\mathrm{d};\text{fair-holding}\right)}=S_{t_{m-1};{\mathcal{E}}_{m-1}^{({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1})}}^{\left(\mathrm{d};\text{fair-holding}\right)}=S_{T;{\mathcal{E}}_{m-1}^{\left({\mathbb{L}}_{m-1},{\mathbb{M}}_{m-1}\right)}}^{\left(\mathrm{d}\right)}.$$

In contrast to (39), Breen (1991) uses $S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\text{fair-holding}\right)}$ to define the fair value of the American option at the node ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$,

$$f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\mathrm{American};\mathrm{fair}\right)}=max\left(f_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)},S_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d};\text{fair-holding}\right)}\right).$$

(41)

A trader in search of statistical arbitrage opportunities relative to an American option could compare (39) with (41) when seeking potential mispricing in the market value of the option.

6. Limiting Dynamics of Binary Pricing Trees

In this section, we investigate the limiting dynamics of the binary-tree pricing by assuming that the lengths of trading intervals uniformly vanish at a rate $N^{-1}$ as $N\to \infty$. To support this, we generalize our notation for the trading times as follows. For any given $N\in \mathcal{N}=\left\{1,2,\cdots \right\}$, we consider the fixed time instances $0=t_{0,N}<t_{1,N}<\cdots <t_{n_N-1,N}<t_{n_N,N}=T<\infty$. The current time is $t_{0,N}=0$, the terminal time is $t_{n_N,N}=T$ and, as previously, trades of $\mathcal{S}$ and $\mathcal{B}$ occur only at the times $t_{1,N}<\cdots <t_{n_N-1,N}$. The time intervals are denoted $\Delta t_{n,N}=t_{n,N}-t_{n-1,N}$, $n=1,\cdots ,n_N$. It is straightforward to adapt the results of Section 2, Section 3, Section 4 and Section 5 to this notational change for t. The resultant ${\mathbb{F}}^{\left(\mathrm{d}\right)}$-adapted binary information tree is now denoted ${\mathbb{BT}}_{n_N}$. We impose the restriction

$${\Delta }_N=max\{\Delta t_{n,N},n=1,\cdots ,n_N\}=O\left({\displaystyle \frac{1}{N}}\right).$$

(42)

To determine the continuum limit behavior, we apply the Donsker–Prokhorov invariance principle (DPIP) for continuous diffusions.24 To apply the DPIP we assume that, for each $n=1,\cdots ,n_N$, the random variables ${ϵ}_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}$ determining the probabilities (14) and (19) are independent. Therefore, the probabilistic structure of the triangular array ${\mathcal{E}}_{\mathcal{N}}=\left({ϵ}_n^{\left(k\right)},k=1,\cdots ,K_n,n\in \mathcal{N}\right)$ is determined by the probability laws,

$$p_{\left({ϵ}_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}\right)}^{\left(m_n^{\left(1\right)},\cdots ,m_n^{\left(K_n\right)}\right)}=\mathbb{P}\left({ϵ}_n^{\left(1\right)}=m_n^{\left(1\right)},\cdots ,{ϵ}_n^{\left(K_n\right)}=m_n^{\left(K_n\right)}\right)=\prod _{k=1}^{K_n}\mathbb{P}\left({ϵ}_n^{\left(k\right)}=m_n^{\left(k\right)}\right),$$

for $m_n^{\left(k\right)}\in \left\{0,1\right\}$, $k=1,\cdots ,K_n$, $n\in \mathcal{N}$. This assumption of independence results in simplified expressions for (14) and (19):

$$\begin{array}{c}\left.\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_n^{\left(k\right)}=1\right)& \stackrel{\mathrm{def}}{=}{p}_n^{(k;1)},\hfill \\ \hfill \mathbb{P}\left({ϵ}_n^{\left(k\right)}=0\right)& \stackrel{\mathrm{def}}{=}{p}_n^{(k;0)}=1-p_n^{(k;1)},\hfill \end{array}\right\}k=1,\cdots ,K_n,\end{array}$$

(43)

$$\begin{array}{c}\mathbb{P}\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right)=p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}=\prod _{k=1}^n\mathbb{P}\left({ϵ}_k^{\left(l_k\right)}=m_k\right)=\prod _{k=1}^n{p}_k^{(l_k;m_k)},\end{array}$$

(44)

$$\begin{array}{c}\mathbb{P}\left({ϵ}_j^{\left(l_j\right)}=m_j\left|\left({ϵ}_1^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{j-1}^{\left(l_{j-1}\right)}=m_{j-1}\right)\right.\right)=\mathbb{P}\left({ϵ}_j^{\left(l_j\right)}=m_j\right)=p_j^{(l_j;m_j)}.\end{array}$$

(45)

As ${ϵ}_n^{\left(k\right)}\stackrel{d}{=}\mathrm{Bernoulli}\left(p_n^{(k;1)}\right)$, then

$${\delta }_n^{\left(k\right)}\stackrel{\mathrm{def}}{=}{\displaystyle \frac{{ϵ}_n^{\left(k\right)}-p_n^{(k;1)}}{\sqrt{p_n^{(k;0)}}}},k=1,\cdots ,K_n,n=1,\cdots ,n_N,$$

has $\mathbb{E}\left[{\delta }_n^{\left(k\right)}\right]=0$ and $\mathrm{Var}\left[{\delta }_n^{\left(k\right)}\right]=1$. We now view the filtration (7) as generated by the triangular series ${\delta }_n^{\left(1\right)},\cdots ,{\delta }_n^{\left(n\right)},n\in \mathcal{N}$; that is,

$${\mathcal{F}}^{\left(n\right)}=\sigma \left({\delta }_1^{\left(1\right)},\left({\delta }_2^{\left(1\right)},{\delta }_2^{\left(2\right)}\right),\cdots ,\left({\delta }_n^{\left(1\right)},\cdots ,{\delta }_n^{\left(n\right)}\right)\right),n=1,\cdots ,n_N.$$

Next, for a given sequence $\mathbb{L}=\left(l_1,\cdots ,l_n,\cdots \right)$, $l_n=1,\cdots ,n$, $n=1,\cdots ,n_N$, we consider the random walk ${\delta }_n^{\left(\mathbb{L}\right)}={\sum }_{k=1}^n{\delta }_k^{\left(k\right)}$. By the DPIP, the sequence of $D[0,\infty )$ processes

$${\mathbb{B}}_n^{(\mathbb{L};[0,\infty \left)\right)}=\left\{B_{t;n}^{\mathbb{L}}={\displaystyle \frac{{\delta }_{\lfloor nt\rfloor }^{\left(\mathbb{L}\right)}}{\sqrt{n}}},t\ge 0\right\}$$

converges to a standard Brownian motion ${\mathbb{B}}_{[0,\infty )}=\{B_t,t\ge 0\}$ in the Skorokhod J1-topology.25 By denoting the canonical filtration ${\mathfrak{F}}^{{\mathbb{B}}_{[0,\infty )}}=\{\sigma (B_u,0\le u\le t),t\ge 0\}$, we can assume, by the Skorokhod embedding theorem (see, e.g., Kallenberg, 1997, Chapter 14) that ${\mathbb{F}}^{\left(\mathrm{d}\right)}\subset {\mathfrak{F}}^{{\mathbb{B}}_{[0,\infty )}}$ and the triangular series ${ϵ}_1^{\left(1\right)},\left({ϵ}_2^{\left(1\right)},{ϵ}_2^{\left(2\right)}\right),\cdots ,\left({ϵ}_2^{\left(1\right)},\cdots ,{ϵ}_n^{\left(n\right)}\right),\cdots$ are in the same stochastic basis space $\left(\Omega ,{\mathcal{F}}^{{\mathbb{B}}_{[0,\infty )}},\mathbb{P}\right)$.

For the time interval $[0,T]$, we now consider the limiting behavior of the discrete riskless rates $r_{t_{n,N}}^{\left(\mathrm{d},\mathrm{f}\right)}=r_{t_{n,N}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\Delta t_{n,N}$ in (22). We assume that the discrete instantaneous riskless rate process

$$\begin{array}{c}{r}_{[0,T];N}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}=\left\{r_{t,N}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}=r_{t_{n-1,N}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)},t\in [t_{n-1,N},t_{n,N}),n=1,\cdots ,n_N,\right.\hfill \\ \hfill \left.r_{T,N}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}=r_{t_{n_N,N}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\right\}\end{array}$$

converges uniformly to a continuous-time instantaneous riskless rate $r_{[0,T]}^{\left(\mathrm{f}\right)}=\left\{r_t^{\left(\mathrm{f}\right)},t\in [0,T]\right\}$, where26

• $r_{[0,T]}^{\left(\mathrm{f}\right)}$ has strictly positive continuous trajectories on $[0,T]$;
• $sup\left\{|r_t^{\left(\mathrm{f}\right)}-r_{t,N}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}\left|,t\in [0,T]\right.\right\}=O\left({\displaystyle \frac{1}{N}}\right)$.

Then, the discrete bank account value ${\beta }_{t,N}^{\left(\mathrm{d}\right)}$, $t\in [0,T]$ in (23) converges uniformly to the continuous-time riskless asset dynamics

$${\beta }_t={\beta }_0{e}^{{\int }_0^t{r}_u^{\left(\mathrm{f}\right)}du},t\in [0,T],$$

where the deterministic instantaneous riskless-rate (short-rate) process $r_{[0,T]}^{\left(\mathrm{f}\right)}$ is ${\mathfrak{F}}^{{\mathbb{B}}_{[0,T]}}$-adapted.

Consider the discrete mean and volatility processes for $\mathcal{S}$:

$$\begin{array}{cc}\hfill {\mu }_{[0,T];N}& =\left\{\begin{array}{c}{\mu }_{t,N}={\mu }_{t_{n,N}\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.},t\in [t_{n,N},t_{n+1,N}),n=1,\cdots ,n_N-1,\hfill \\ {\mu }_{T,N}={\mu }_{t_{n_N-1,N}\left|{\mathcal{E}}_{n_N-2}^{({\mathbb{L}}_{n_N-2},{\mathbb{M}}_{n_N-2})}\right.},\hfill \end{array}\right.\hfill \\ \hfill {\sigma }_{[0,T];N}& =\left\{\begin{array}{c}{\sigma }_{t,N}={\sigma }_{t_{n,N}\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.},t\in [t_{n,N},t_{n+1,N}),n=1,\cdots ,n_N-1,\hfill \\ {\sigma }_{T;N}={\sigma }_{t_{n_N-1,N}\left|{\mathcal{E}}_{n_N-2}^{({\mathbb{L}}_{n_N-2},{\mathbb{M}}_{n_N-2})}\right.}.\hfill \end{array}\right.\hfill \end{array}$$

Assume that ${\mu }_{[0,T],N}$ and ${\sigma }_{[0,T],N}$ converge uniformly on $[0,T]$ to ${\mu }_t$ and ${\sigma }_t$, respectively, such that

$$sup\left\{\right|{\mu }_{t,N}-{\mu }_t|+|{\sigma }_{t,N}-{\sigma }_t|,t\in [0,T]\}=O\left({\displaystyle \frac{1}{N}}\right),$$

Further, assume that ${\mu }_t$ and ${\sigma }_t$ are ${\mathfrak{F}}^{{\mathbb{B}}_{[0,T]}}$-adapted, with ${\mu }_{[0,T]}=\{{\mu }_t,t\in [0,T]\}$ and ${\sigma }_{[0,T]}=\{{\sigma }_t,t\in [0,T]\}$ having continuous trajectories on $[0,T]$.27 We define the $\mathfrak{D}[0,T]$ price process

$${\mathbb{S}}_{[0,T];N}=\left\{\begin{array}{c}{S}_{t,N}=S_{t_{n,N}}^{\left(\mathrm{d}\right)}=S_{t_{n,N};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)},t\in \left[t_{n,N},t_{n+1,N}\right),n=1,\cdots ,n_N-2,\hfill \\ S_{t,N}=S_{t_{n_N-1,N}}^{\left(\mathrm{d}\right)}=S_{t_{n_N-1,N};{\mathcal{E}}_{n_N-1}^{({\mathbb{L}}_{n_N-1},{\mathbb{M}}_{n_N-1})}}^{\left(\mathrm{d}\right)},t\in \left[t_{n_N-1,N},t_{n_N,N}=T\right].\hfill \end{array}\right.$$

(46)

As in Hu et al. (2020a), a non-standard invariance principle (Davydov & Rotar, 2008) can be used to show that (46) converges weakly in $\mathfrak{D}[0,T]$ topology (Skorokhod, 2005) to a process $S_{[0,T]}=\{S_t,t\in [0,T]\}$ governed by a cumulative return process $R_{[0,T]}=\{R_t,t\in [0,T]\}$ satisfying

$$d{R}_t={\mu }_{t}dt+{\sigma }_{t}d{B}_t,R_0=0,$$

(47)

where $B_t$ is a standard Brownian motion and $d{S}_t=S_{t}d{R}_t$ (see Duffie (2001, Appendix 6D)). By (47), $S_{[0,T]}$ is a continuous diffusion and, if ${\mu }_t$ and ${\sigma }_t$ are constant, $S_{[0,T]}$ is a GBM. In the risk-neutral world, the limiting cumulative return process obeys (47) with ${\mu }_t$ replaced by $r_t^{\left(\mathrm{f}\right)}$. We note that the discrete model is much more informative than the continuous-time model as it preserves the path-dependent probabilities $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$, with no assumption on their (in)dependence.

7. Stock Pricing—Special Cases

The binary-tree pricing model presented above encompasses several time-series processes that have been proposed to model stock prices. We consider three such processes, all of which assume constant time spacing, $\Delta t_n=\Delta t$, $n=1,\cdots ,m$.28 These examples are BWN and the cases in which the first difference of the price process is assumed to display either MA(1) or AR(1) behavior. These latter two time-discrete models are of particular importance in microstructure theory. In the Introduction, we mentioned the analogous models of Hasbrouck (1988), which build upon the seminal Roll model (Roll, 1984). See Hasbrouck (2007, Chapter 8) for an overview and discussion of other closely related models.

7.1. Asymmetric Binary White Noise

Consider the case in which the conditional probabilities at each node of ${\mathrm{BIT}}_m$ are the same:

$$\begin{array}{c}\left.\begin{array}{cc}\hfill \mathbb{P}\left({ϵ}_j^{\left(l_j\right)}=1\left|{\mathcal{E}}_{j-1}=(0,{ϵ}_j^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{j-1}^{\left(l_{j-1}\right)}=m_{j-1})\right.\right)& =p\hfill \\ \hfill \mathbb{P}\left({ϵ}_j^{\left(l_j\right)}=0\left|{\mathcal{E}}_{j-1}=(0,{ϵ}_j^{\left(l_1\right)}=m_1,\cdots ,{ϵ}_{j-1}^{\left(l_{j-1}\right)}=m_{j-1})\right.\right)& =1-p\hfill \end{array}\right\},\hfill \end{array}$$

$$l_j=1,\cdots ,K_j,j=1,\cdots ,m.$$

(48)

We develop a white noise process (Hamilton, 2020, Chapter 3.2) on ${\mathrm{BIT}}_m$ where each node takes on one of only two possible values. This asymmetric binary white noise (ABWN) on ${\mathrm{BIT}}_m$ is the ${\mathbb{F}}^{\left(\mathrm{d}\right)}$-adapted process

$$\begin{array}{cc}\hfill z_0^{\left(\mathrm{d}\right)}& =z_{t_0}^{\left(\mathrm{d}\right)}=0,\hfill \\ \hfill z_t^{\left(\mathrm{d}\right)}& =z_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}{z}_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=1\right)}^{\left(\mathrm{d}\right)}:=z_u\mathrm{w}.\mathrm{p}.p,\hfill \\ z_{t_n;\left({\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},{ϵ}_n^{\left(l_n\right)}=0\right)}^{\left(\mathrm{d}\right)}:=z_d\mathrm{w}.\mathrm{p}.1-p,\hfill \end{array}\right.n=1,\cdots ,m-1.\hfill \end{array}$$

(49)

As the process (49) on ${\mathrm{BIT}}_m$ is path-independent, the ABWN notation can be simplified to $z_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\equiv z_{t_n}^{\left(l_n\right)}$, $l_n=1,\cdots ,K_n$, as indicated in (50).

Using (48) and the white noise moments $\mathbb{E}\left[z_{t_n}^{\left(l_n\right)}\right]=0$ and $\mathrm{Var}\left[z_{t_n}^{\left(l_n\right)}\right]={\sigma }_z^2\Delta t$ with ${\sigma }_z^2$ being finite, (49) becomes

$$\begin{array}{cccc}\hfill z_t^{\left(\mathrm{d}\right)}& =z_0^{\left(0\right)}=0,\hfill & \hfill & t\in [0,t_1=\Delta t),\hfill \\ \hfill z_t^{\left(\mathrm{d}\right)}& =z_{t_n}^{\left(l_n\right)}=\left\{\begin{array}{c}{z}_u={\sigma }_z\sqrt{{\displaystyle \frac{1-p_1}{p_1}}\Delta t}\mathrm{w}.\mathrm{p}.p,\hfill \\ z_d=-{\sigma }_z\sqrt{{\displaystyle \frac{1-p_1}{p_1}}\Delta t}\mathrm{w}.\mathrm{p}.1-p,\hfill \end{array}\right.\hfill & \hfill & t\in [t_n,t_{n+1}=t_n+\Delta t),\hfill \\ \hfill & l_n=1,\cdots ,K_n,n=1,\cdots ,m-1.\hfill & \hfill & \hfill \end{array}$$

(50)

It is straightforward to show that the ABWN process $z_t^{\left(\mathrm{d}\right)}$ has the first-order autocorrelation, $\mathbb{E}\left[z_{t_n}^{\left(l_n\right)}{z}_{t_{n-1}}^{\left(l_{n-1}\right)}\right]={\left(\mathbb{E}\left[z_{t_n}^{\left(l_n\right)}\right]\right)}^2=0$ a property of a white noise process.

7.2. Binary Moving Average of Order 1

We define the random process $\Delta S_t^{\left(\mathrm{d}\right)}=S_t^{\left(\mathrm{d}\right)}-S_{t-\Delta t}^{\left(\mathrm{d}\right)}$ to be a binary moving-average process of order one (BMA(1)) by setting

$$\Delta S_t^{\left(\mathrm{d}\right)}=\Delta S_{t_n;{\mathcal{E}}_n^{\left(({\mathbb{L}}_{n-1},l_n),({\mathbb{M}}_{n-1},m_n)\right)}}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}c+\theta z_{t_{n-1}}^{\left(l_{n-1}\right)}+z_u,\mathrm{if}{m}_n=1,\hfill \\ c+\theta z_{t_{n-1}}^{\left(l_{n-1}\right)}+z_d,\mathrm{if}{m}_n=0,\hfill \end{array}\right.$$

(51)

for $t\in [t_n,t_{n+1})$, $n=1,\cdots ,m-1$, and values $c,\theta \in {\mathbb{R}}^{+}$. In (51), the local index $m_n=1$ occurs with probability p, $m_n=0$ occurs with probability $1-p$, and $z_{t_{n-1}}^{\left(l_{n-1}\right)}$ is the ABWN process (50). The expected value, variance, and first-order auto-covariance of $\Delta S_t^{\left(\mathrm{d}\right)}$ follow the usual MA(1) process:

$$\mathbb{E}\left[\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]=c,\mathrm{Var}\left[\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]=\left(1+{\theta }^2\right){\sigma }_z^2,\mathrm{cov}{\left[\Delta S_{t_k}^{\left(\mathrm{d}\right)}\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]}_{k<n}=\left\{\begin{array}{c}\theta {\sigma }_z^2,\mathrm{if}k=n-1,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

(52)

giving the familiar MA(1) autocorrelation coefficient:

$${\rho }_1={\displaystyle \frac{\mathrm{cov}\left[\Delta S_{t_{n-1}}^{\left(\mathrm{d}\right)}\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]}{\mathrm{Var}\left[\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]}}={\displaystyle \frac{\theta }{1+{\theta }^2}}.$$

In practice, the coefficients $\theta$ and ${\sigma }_z$ of the BMA(1) process can be estimated from historical price differences. Let ${\widehat{\gamma }}_0$ and ${\widehat{\gamma }}_1$ denote, respectively, the empirical variance and first-order auto-covariance of the risky-asset price differences. Setting

$${\widehat{\gamma }}_0=\left(1+{\theta }^2\right){\sigma }_z^2,{\widehat{\gamma }}_1=\theta {\sigma }_z^2,$$

(53)

and solving for $\theta$ and ${\sigma }_z^2$ gives the solution pairs ${\theta }_{+},{\sigma }_{z+}^2$ and ${\theta }_{-},{\sigma }_{z-}^2$,

$$\begin{array}{c}\hfill {\theta }_{+}={\displaystyle \frac{{\widehat{\gamma }}_0+\sqrt{{\widehat{\gamma }}_0^2-4{\widehat{\gamma }}_1^2}}{2{\widehat{\gamma }}_1}},{\sigma }_{z+}^2={\displaystyle \frac{{\widehat{\gamma }}_0-\sqrt{{\widehat{\gamma }}_0^2-4{\widehat{\gamma }}_1^2}}{2}},\\ \hfill {\theta }_{-}={\displaystyle \frac{{\widehat{\gamma }}_0-\sqrt{{\widehat{\gamma }}_0^2-4{\widehat{\gamma }}_1^2}}{2{\widehat{\gamma }}_1}},{\sigma }_{z-}^2={\displaystyle \frac{{\widehat{\gamma }}_0+\sqrt{{\widehat{\gamma }}_0^2-4{\widehat{\gamma }}_1^2}}{2}},\end{array}$$

(54)

which have the following properties: ${\theta }_{+}{\theta }_{-}=1$; ${\sigma }_{z+}^2{\sigma }_{z-}^2={\widehat{\gamma }}_1^2$; ${\theta }_{+}{\widehat{\gamma }}_1={\sigma }_{z-}^2$; and ${\theta }_{-}{\widehat{\gamma }}_1={\sigma }_{z+}^2$. To guarantee ${\sigma }_{z\pm }^2\in \mathbb{R}$ with ${\sigma }_{z\pm }^2>0$, Equation (54) require $|{\widehat{\gamma }}_0/{\widehat{\gamma }}_1|\ge 2$. Addition of the constraint $\left|\theta \right|<1$ to guarantee invertibility of the BMA(1) process restricts the solution of (53) to the pair ${\theta }_{-},{\sigma }_{z-}^2$.

The choice of the BMA(1) tree is motivated by the fact that it represents a generalization (see, e.g., Hasbrouck, 2007, Chapters 4.2 and 8) of the Roll (1984) microstructure model discussed in the Introduction.

7.3. Autoregressive of Order 1

The ABWN process can also be used as a basis to model the first difference of the price process as binary autoregressive of the first-order (BAR(1)). Let

$$\begin{array}{cccc}\hfill \Delta S_t^{\left(\mathrm{d}\right)}& =\Delta S_{t_1;{\mathcal{E}}_1^{(l_1,m_1)}}^{\left(\mathrm{d}\right)}\hfill & \hfill & =\left\{\begin{array}{c}c+z_u,\mathrm{if}{m}_1=1,\hfill \\ c+z_d,\mathrm{if}{m}_1=0,\hfill \end{array}\right.\mathrm{for}t\in [t_1,t_2),\hfill \\ \hfill \Delta S_t^{\left(\mathrm{d}\right)}& =\Delta S_{t_n;{\mathcal{E}}_n^{\left(({\mathbb{L}}_{n-1},l_n),({\mathbb{M}}_{n-1},m_n)\right)}}^{\left(\mathrm{d}\right)}\hfill & \hfill & =\left\{\begin{array}{c}c+\varphi \Delta S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}+z_u,\mathrm{if}{m}_n=1,\hfill \\ c+\varphi \Delta S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}+z_d,\mathrm{if}{m}_n=0,\hfill \end{array}\right.\hfill \\ \hfill & \hfill & \hfill & \mathrm{for}t\in [t_n,t_{n+1}),n=2,\cdots ,m-1,\hfill \end{array}$$

(55)

where $c,\varphi \in \mathbb{R}$. In (55), the local index $m_n=1$ occurs with probability p, while $m_n=0$ occurs with probability $1-p$. Requiring $\varphi \in \left(-1,1\right)$ ensures that the process has MA(∞) representation. The expected value, variance, and auto-covariance of order h for $\Delta S_{t_n}$ are

$$\begin{array}{cc}\hfill \mathbb{E}\left[\Delta S_{t_n}\right]& =c(1+\varphi +{\varphi }^2+\cdots +{\varphi }^{n-1}),\hfill \\ \hfill \mathrm{Var}\left[\Delta S_{t_n}\right]& ={\sigma }_z^2\Delta t\sum _{i=0}^{n-1}{\varphi }^{2i}={\sigma }_z^2\Delta t\left({\displaystyle \frac{1-{\varphi }^{2n}}{1-{\varphi }^2}}\right),\hfill \\ \hfill \mathrm{cov}\left[\Delta S_{t_k}\Delta S_{t_n}\right]& ={\sigma }_z^2\Delta t{\varphi }^{n-k}\sum _{i=0}^{k-1}{\varphi }^{2i}={\sigma }_z^2\Delta t{\varphi }^{n-k}\left({\displaystyle \frac{1-{\varphi }^{2k}}{1-{\varphi }^2}}\right),\mathrm{for}k<n,\hfill \end{array}$$

giving

$${\rho }_h=\underset{n\to \infty ;k=n-h}{lim}\left({\displaystyle \frac{\mathrm{cov}\left[\Delta S_{t_k}\Delta S_{t_n}\right]}{\mathrm{Var}\left[\Delta S_{t_n}\right]}}\right)={\varphi }^h$$

for a finite lag h.

8. Computational Simplifications

Computations on a non-recombining binary tree are well known to be exponentially expensive, both in terms of memory requirements as well as execution time.29 Common methods to reduce the number of time intervals, $0,t_1,\cdots ,t_m=T$, involved in binary-tree option pricing computations include using either a common large time step, $\Delta t_n=\Delta t>1$, $n=1,\cdots ,m$, or a graduated increase in time intervals $\Delta t_n>\Delta t_{n-1}$, $n=1,\cdots ,m$. However, the following features of the three special-case models reduce the computational and storage complexity.

8.1. ABWN Model

As stated by (49) and illustrated in Figure 7, every level $l_n$ on the ABWM binary tree has exactly the same two state values. As all conditional probabilities (48) reduce to either p or $1-p$, the ABWM model reduces to a simple model on a (recombining) binomial tree.

8.2. BMA(1) Model

As a result of (51), the triangular configuration illustrated in Figure 7 represents the fundamental replicating unit for $\Delta S$ in the state space of the BMA(1) tree. By iterating (51) to obtain prices, it can be shown that, at time step n, $n=1,\cdots ,m$, there are $2n$ unique price states, each of the form

$$S_{n,i}=S_0+nc+{\alpha }_{1,i}{z}_u+{\alpha }_{2,i}2z_d+\theta \left({\delta }_{1,i}{z}_u+{\delta }_{2,i}{z}_d\right),i=1,\cdots ,2n.$$

(56)

In (56), $S_0$ is the initial price. The coefficients ${\alpha }_{j,i}$ and ${\delta }_{j,i}$ are listed in

Table 1. Values of the coefficients ${\alpha }_{j,i}$ and ${\delta }_{j,i}$, $j=1,2$, in (56).

	n Even				n Odd
i	${\mathsf{\alpha }}_{\mathbf{1},\mathit{i}}$	${\mathsf{\alpha }}_{\mathbf{2},\mathit{i}}$	${\mathsf{\delta }}_{\mathbf{1},\mathit{i}}$	${\mathsf{\delta }}_{\mathbf{2},\mathit{i}}$	${\mathsf{\alpha }}_{\mathbf{1},\mathit{i}}$	${\mathsf{\alpha }}_{\mathbf{2},\mathit{i}}$	${\mathsf{\delta }}_{\mathbf{1},\mathit{i}}$	${\mathsf{\delta }}_{\mathbf{2},\mathit{i}}$
$2n$	n	0	$n-1$	0	n	0	$n-1$	0
$2n-1$	$n-1$	1	$n-1$	0	$n-1$	1	$n-1$	0
$2n-2$	$n-1$	1	$n-2$	1	$n-1$	1	$n-2$	1
$2n-3$	$n-2$	2	$n-2$	1	$n-2$	2	$n-2$	1
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
$n+1$	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n}{2}}-1$	${\displaystyle \frac{n+1}{2}}$	${\displaystyle \frac{n-1}{2}}$	${\displaystyle \frac{n-1}{2}}$	${\displaystyle \frac{n-1}{2}}$
n	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n}{2}}-1$	${\displaystyle \frac{n}{2}}$	${\displaystyle \frac{n-1}{2}}$	${\displaystyle \frac{n+1}{2}}$	${\displaystyle \frac{n-1}{2}}$	${\displaystyle \frac{n-1}{2}}$
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
4	2	$n-2$	1	$n-2$	2	$n-2$	1	$n-2$
3	1	$n-1$	1	$n-2$	1	$n-1$	1	$n-2$
2	1	$n-1$	0	$n-1$	1	$n-1$	0	$n-1$
1	0	n	0	$n-1$	0	n	0	$n-1$

. The coefficients satisfy ${\alpha }_{1,i}+{\alpha }_{2,i}=n$ and ${\delta }_{1,i}+{\delta }_{2,i}=n-1$, $i=1,\cdots ,2n$, with ${\alpha }_{1,i}={\alpha }_{2,2n-i+1}$ and ${\delta }_{1,i}={\delta }_{2,2n-i+1}$ reflecting the $z_u\leftrightarrow z_d$ symmetry of the binary tree.

The binary-tree description of the BMA(1) model in Section 7.2 uses the path notation $({\mathbb{L}}_n,{\mathbb{M}}_n)$, with ${\mathbb{M}}_n=\{0,m_1,m_2,\cdots ,m_n\}$, to designate a specific node at time n. The partial recombining of the binary tree (from $2^n$ distinct price states to $2n$ price states at time step n) suggests the simpler node (price) labeling of $(n,i)$, $i=1,\cdots ,2n$, of (56). The connection between the two is as follows. Let ${\delta }_1$ denote the number of elements $m_i=1$ in the binary string $m_1{m}_2\cdots m_{n-1}$ and ${\alpha }_1$ denote the number of elements $m_i=1$ in the binary string $m_1{m}_2\cdots m_n$. Then, node $({\mathbb{L}}_n,{\mathbb{M}}_n)$ corresponds to the node $(n,i)$, with $i={\alpha }_1+{\delta }_1+1$ having the price given by the corresponding value of i in Table 1.

As a result of the reduction of the number of nodes, from $2^n$ to $2n$ at time n, the BMA(1) model becomes computationally tractable for large values of n (competitive with trinomial models, which have $2n+1$ nodes at time n). With prices (and therefore nodes) indexed as in (56), each price is uniquely specified by the vector of values $\left(n{\alpha }_1{\alpha }_2{\delta }_1{\delta }_2\right)$ (we ignore the value $S_0$, which is common to all nodes). Using this vector notation, Figure 8 illustrates the prices on a ${\mathrm{BIT}}_5$ BMA(1) tree. The figure also presents the vector denoting the change in the price $\Delta S_{(n-1,i_{n-1}),(n,i_n)}$ along each indicated branch of the tree. It is a feature of the BMA(1) tree that only four price-change vectors occur in the tree: $\left(11010\right)$, $\left(10110\right)$, $\left(11001\right)$, and $\left(10101\right)$. (This is implicitly stated in Figure 7.) Furthermore, these four vectors always occur in this repetitive sequence in the transition from time n to $n+1$.

The BMA(1) model supports the following further simplifications. For simplicity, we express these using the $({\mathbb{L}}_n,{\mathbb{M}}_n)$ path notation of Section 2.

${\mathrm{BIT}}_m$: As the BMA(1) model utilizes ABWN, from (48) we have constant conditional probabilities p and $1-p$.

Riskless rate and bank account dynamics: For all n, if we assume $r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f},\mathrm{inst}\right)}=r$, a constant, then, from (22), $r_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d},\mathrm{f}\right)}=r\Delta t$. Consequently, from (23)

$${\beta }_{t_n;{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}={\beta }_0^{\left(\mathrm{d}\right)}{(1+r\Delta t)}^n\mathrm{w}.\mathrm{p}.p_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})},n=1,\cdots ,m,p_0^{({\mathbb{L}}_0,{\mathbb{M}}_0)}=1.$$

Risky asset: We have the unconditional expectation (52) $\mathbb{E}\left[\Delta S_{t_n}^{\left(\mathrm{d}\right)}\right]=c$, $n=1,\cdots ,m$. From (31),

$$\begin{array}{cc}\hfill \mathbb{E}\left[r_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\right]& ={\mu }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\Delta t=\left\{\begin{array}{c}{\displaystyle \frac{\theta z_u+c}{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}}\mathrm{if}{m}_{n-1}=1,\hfill \\ {\displaystyle \frac{\theta z_d+c}{S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}}}\mathrm{if}{m}_{n-1}=0,\hfill \end{array}\right.\hfill \\ \hfill \mathrm{Var}\left[r_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(\mathrm{d}\right)}\right]& ={\left({\sigma }_{t_n\left|{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}\right.}^{\left(r\right)}\right)}^2\Delta t={\displaystyle \frac{{\sigma }_z^2\Delta t}{{\left(S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}\right)}^2}}.\hfill \end{array}$$

(57)

Risk-neutral dynamics: Equation (38) simplifies to

$$\begin{array}{cc}\hfill q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=1\right)}^{\left(\mathrm{d}\right)}& =p-{\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}\sqrt{p(1-p)\Delta t},\hfill \\ \hfill q_{t_{n+1};\left({\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)},{ϵ}_{n+1}^{\left(l_{n+1}\right)}=0\right)}^{\left(\mathrm{d}\right)}& =(1-p)+{\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}\sqrt{p(1-p)\Delta t}\hfill \end{array}$$

(58)

with

$${\theta }_{t_{n+1}\left|{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}\right.}=\left\{\begin{array}{c}{\displaystyle \frac{\theta z_u+c-r{S}_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}}{{\sigma }_z}}\mathrm{if}{m}_{n+1}=1,\hfill \\ {\displaystyle \frac{\theta z_d+c-r{S}_{t_n;{\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}}^{\left(\mathrm{d}\right)}}{{\sigma }_z}}\mathrm{if}{m}_{n+1}=0.\hfill \end{array}\right.$$

8.3. BAR(1) Model

To understand the simplification of the BAR(1) model, rewrite (55) as

$$\Delta S_t^{\left(\mathrm{d}\right)}=\Delta S_{t_n;{\mathcal{E}}_n^{\left(({\mathbb{L}}_{n-1},l_n),({\mathbb{M}}_{n-1},m_n)\right)}}^{\left(\mathrm{d}\right)}=\left\{\begin{array}{c}\varphi \Delta S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}+U,\mathrm{if}{m}_n=1,\hfill \\ \varphi \Delta S_{t_{n-1};{\mathcal{E}}_{n-1}^{({\mathbb{L}}_{n-1},{\mathbb{M}}_{n-1})}}^{\left(\mathrm{d}\right)}+D,\mathrm{if}{m}_n=0,\hfill \end{array}\right.$$

(59)

for $n=1,\cdots ,m-1$, where $U=c+z_u$, $D=c+z_d$, and $\Delta S_{t_0;{\mathcal{E}}_0^{({\mathbb{L}}_0,{\mathbb{M}}_0)}}^{\left(\mathrm{d}\right)}=0$. By iterating (59), it is straightforward to develop an analytic formula for the $2^n$ possible price states at level n.

Let $U{D}_n$ denote a naturally ordered set of all possible binary sequences of length n, where the binary choice for each element of a sequence is either U or D. Let $U{D}_{n,k}$ denote the k-th sequence in the set $U{D}_n$. The natural ordering in $U{D}_n$ arises from the iteration of (59). Specifically $U{D}_{n+1,2k}=(U{D}_{n,k},U)$ and $U{D}_{n+1,2k-1}=(U{D}_{n,k},D)$. For example,

$$\begin{array}{cc}\hfill U{D}_2& =\left\{(D,D),(D,U),(U,D),(U,U)\right\}\mathrm{where}U{D}_{2,3}=(U,D),\hfill \\ \hfill U{D}_3& =\left\{\right(D,D,D),(D,D,U),(D,U,D),(D,U,U),(U,D,D),(U,D,U),\hfill \\ \hfill & (U,U,D),(U,U,U)\}.\hfill \end{array}$$

Thus, $U{D}_{3,6}=(U,D,U)=(U{D}_{2,3},U)$ and $U{D}_{3,5}=(U,D,D)=(U{D}_{2,3},D)$.

Considering the sequence $U{D}_{n,k}$ to be a vector, at level n each of the $2^n$ possible states can be written as the scalar product:

$${\Phi }_{n-1}·U{D}_{n,k},k=1,\cdots ,2^n,$$

(60)

where ${\Phi }_n$ is the vector $\left({\varphi }^n,{\varphi }^{n-1},\cdots ,\varphi ,1\right)$. For example, the state corresponding to $U{D}_{3,6}$ is

$${\Phi }_2·U{D}_{3,6}={\varphi }^{2}U+\varphi D+U=c({\varphi }^2+\varphi +1)+{\varphi }^2{z}_u+\varphi z_d+z_u.$$

However, under the restriction $\varphi \in (-1,1)$, for a large enough n, ${\varphi }^n$ becomes smaller than the machine precision (machine epsilon) and finite-precision computation of the higher-order terms in sums such as $1+\varphi +{\varphi }^2+\cdots +{\varphi }^n+{\varphi }^{n+1}+\cdots {\varphi }^{n+k}$ becomes meaningless. Thus, beyond a certain value of n, the numerical prices no longer change (the price of each child node remains equal to the price of the parent node). The value of n for which this occurs depends on the magnitudes $\left|\varphi \right|$, $|z_u|$, and $|z_d|$.30

9. Technical Analyses of the Probability Estimates

In Section Estimation of Probabilities, we noted the desirability of estimating the direction-of-price-change probabilities $p_n^{({\mathbb{L}}_n,{\mathbb{M}}_n;\Delta t_{1,n})}$ from historical data using (20) on a set of V non-overlapping binomial sequences, each of length n. However, we also noted that, even for relatively small values of n, a prohibitively extensive history of returns would be required to ensure an adequate sample to determine each of the $2^n$ probabilities for a given value of n.

Table 2. Expected number of occurrences of each sequence in patterns of length n in the DIA return data set covering the period of prices from 20 January 1998 through 5 May 2023.

Pattern	Number of	Number of	Expected Number of
Length	Sequences	Non-Overlapping	Pattern Sequences in
($\mathit{n}$)	in Pattern	Intervals	$\mathit{V}$ Non-Overlapping
	(${\mathbf{2}}^{\mathit{n}}$)	($\mathit{V}$)	Intervals
4	16	1591	99.4
5	32	1273	39.8
6	64	1060	16.6
8	256	795	3.1
10	1024	636	0.6

codifies this problem using daily return data for the SPDR Dow Jones Industrial Average ETF Trust (DIA) covering the period of prices from 20 January 1998 through 5 May 2023. This data set provides 6365 return values; that is, a sequence of 6365 values of 0 (down) or 1 (up) price changes.

From the table it is evident that by $n=6$, V is too small to ensure an adequate sample size for determining each of the $2^6$ possible sequences. It is not clear the sample size is adequate even for $n=5$.

There are models that can be applied to a historical return time series to generate “mimicking” time series. Of course, all such models add additional model error to the process. One approach is to fit the historical return time series to an ARMA($l,m$)-GARCH($p,q$) model combined with a distribution model for the ARMA residuals, and then use the ARMA-GARCH-distribution model with fitted parameters to generate adequate numbers, V, of mimicking return series from which to compute the required probabilities. Such an approach involves fitting a significant number of required parameters. It generates return time series, which is a step removed from the $0,1$ sequences required by (20). We prefer to utilize bootstrap resampling directly on the $0,1$ sequence determined by the historical data set. Bootstrapping has the advantage of constantly resampling the historical sequence.31 For a given value of n, we required $V\ge 10,000\times 2^n$ bootstrapped samples (i.e., each of the $2^n$ probabilities is determined based upon a expectation of 10,000 occurrences for each of the $2^n$ possible sequences).

We compared our bootstrap resampled results against those computed from the historical time series with no resampling (i.e., with expected number of sequence occurrences given in Table 2). For plotting convenience, we have developed the following labeling system for each of the $2^n$ possible $0,1$ sequences of length n. We illustrate the general notation using specific examples. Consider the $n=5$ sequences. They can be mapped to the $2^5=32$ values $x=-15.5,-14.5,\cdots ,-0.5,0.5,\cdots ,14.5,15.5$. For example, 01101 is mapped to $x={\left(01101\right)}_{10}+1-(2^5+1)/2=-2.5$; 00000 is mapped to $x={\left(00000\right)}_{10}+1-(2^5+1)/2=-15.5$; and 11111 to $x={\left(11111\right)}_{10}+1-(2^5+1)/2=15.5$. This labeling has the property that $-x$ and x label binary complement sequences (i.e., 01101 corresponds to $x=-2.5$ and 10010 to $x=2.5$). A positive value of x indicates a binary string beginning with 1, while a negative value of x indicates a binary string beginning with 0.

Using the values x to represent the sequences corresponding to the various events ${\mathcal{E}}_n^{({\mathbb{L}}_n,{\mathbb{M}}_n)}$ with $\Delta t_{1,n}=\Delta t=1$, Figure 9 and Figure 10 compare the results obtained for the probabilities $p_n^{\left(x\right)}$ computed via (20) for the DIA data set without and with bootstrap resampling for $n=4,6,\mathrm{and}8$.

There is reasonable agreement between the results without and with bootstrap resampling for $n=4$; significant differences develop for $n=6$, which are then clearly revealed for $n=8$. In particular, without bootstrap resampling, when $n=8$ the paucity of data results in the probabilities $p_8^{\left(x\right)}$ taking on only nine possible values. The 95% confidence intervals in Figure 9 and Figure 10 are based upon the results in Figure 11 that the probabilities $p_n^{\left(x\right)}$ obtained from bootstrap resampling are well described by a normal distribution.

Analysis of the probabilities of individual sequences falls within the area of technical pattern analysis (Lo et al., 2000). We continue this analysis by examining the highest- and lowest-probability paths. Specifically, for fixed n we consider whether the highest-probability sequences specify paths that are “closely grouped” (with a similar statement for the lowest-probability paths). If the highest-probability paths occur randomly, this would provide further confirmation of the efficient market hypothesis. However, clustered paths suggest the presence of pronounced patterns, as argued by Lo et al. (2000). A related consideration is whether the highest-probability path for $n=n_1$ is a projection of the highest-probability path for $n=n_2>n_1$.

Figure 12 displays the observed results for the grouping of paths. For the $2^n$ sequences of length n, we plot the highest-observed-probability path (colored gold), and the next $n-1$-highest-probability paths (colored blue). Similarly, we plot the lowest-observed-probability path (colored green), and the next $n-1$-lowest-probability paths (colored red). We consider $n=\{4,6,8\}$ and plot results for the data without and with bootstrap resampling. Due to the data limitations with no resampling, there is a more “random” distribution of high- and low-probability paths. (Note in particular the highest- and lowest-probability paths for $n=6$ in the case with no resampling.) With bootstrap resampling improving sample sizes, there is a more distinct grouping of the high- and low-probability paths, with the high-probability paths characterized by more consistent price increases and the low-probability paths characterized by more consistent price decreases.

Figure 13 displays the observed results for the projections of the highest- and lowest-probability paths. The highest-probability path for $n=\{4,5,6,7,8,10\}$ are each plotted on the same graph. Similarly for the respective lowest-probability path for each value of n. It is clear that, over this range of values of n, the lowest-probability path for $n=n_1$ is simply the projection (truncation) of the highest-probability path for $n=n_2>n_1$. In the case of the highest-probability path, there is a “discontinuity” in the projection. For $n<6$, the highest-probability path is a truncation of that for $n=6$. For $n=7,8$, the highest-probability path is almost a truncation of that for $n=10$ (with a slight difference occurring for $n=7$ at $t_2$ and $n=8$ at $t_3$).

To test the dynamic stability of such estimates, we reperformed the probability estimation procedure for the DIA data set using a rolling window of length 15 years (3780 trading days). This generated 2586 windows. For each window, sequence probabilities were computed for $n=\{2,4,6\}$, using bootstrap resampling to ensure adequate sample sizes. (To speed-up the computation, we employed $1000\times 2^n$ bootstrapped samples in each window.) For each choice of n, the rolling windows produced an empirical distribution of probability estimates for each of the $2^n$ sequences. These distributions are summarized as box–whisker plots in Figure 14. Figure 9 and Figure 14 show very similar structures (for $n=4$ and 6), indicating relative stability between the rolling window and global estimates of the sequence probabilities.

We now address the substructure that is apparent in Figure 14 (and in Figure 9 and Figure 10). Figure 15 replots the $n=\{4,6\}$ box–whisker plots with the sequences placed in categories according to the number of zeros (price downturns; equivalently the number of ones (price upturns)) each contains. Within each category, the sequences are still labeled from smallest to largest numerical label, x, as indicated in the top plot of Figure 15. The substructure seen in Figure 14 has largely vanished from Figure 15, indicating that the number of price downturns (equivalently upturns) is the major driver of a sequence’s probabilities. The uniformity of the ranges of the probability distributions within a category is indicative of an efficient market hypothesis operating within each category. It is in the difference in the ranges of the probability distributions between categories that market inefficiencies are seen.32 For $n=4$, there is no overlap between the range of the empirical probability distribution for the sequence 0000 and the range of any distribution for sequences containing two or more upturns. For $n=6$, there is no overlap between the range of the empirical distribution for the sequence 000000 and the range of any other distribution. Furthermore, the range of any distribution for a sequence containing five or six downturns has no overlap with the range of any distribution for a sequence containing four or more upturns.

We compare the sequence probability estimates among different assets using the 30 components comprising (as of 31 August 2020) the Dow Jones Industrial Average (DJIA) index. Price data were used for the period 3 January 2000 through 26 August 2022, with the exceptions of Visa (price data beginning 18 March 2008) and Dow (price data beginning 29 February 2019). This provided 5699 return values for 28 of the assets (3637 returns for Visa and 868 returns for Dow). Sequence probabilities were computed for these assets for $n=\{1,2,3\}$. For these small values of n, the probabilities were computed from the data without bootstrap resampling. The estimated probabilities for sequences of length $n=1,2$ are presented in

Table A1. Path probabilities for sequences of length (columns 2 and 3) $n=1$ and (columns 4 to 7) $n=2$ for assets in the DJIA index and for DIA. Also indicated are the probabilities having significant p-values from the one-sided z-test. *, ** and *** refer to the standard significance levels of 5%, 1% and 0.1%, respectively.

Path Label:	$-0.5$	$0.5$	$-1.5$	$-0.5$	$0.5$	$1.5$
Symbol	Path Probability		Path Probability
AAPL	0.477 ***	0.523 ***	0.230 **	0.248	0.246	0.276 ***
AMGN	0.499	0.501	0.245	0.256	0.253	0.246
AXP	0.490	0.510	0.232 *	0.260	0.257	0.251
BA	0.486 *	0.514 *	0.229 **	0.260	0.254	0.257
CAT	0.486 *	0.514 *	0.239	0.247	0.248	0.266 *
CRM	0.487 *	0.513 *	0.222 ***	0.271 **	0.257	0.249
CSCO	0.484 *	0.516 *	0.229 **	0.262	0.247	0.261
CVX	0.475 ***	0.525 ***	0.221 ***	0.250	0.257	272 **
DIS	0.487 *	0.513 *	0.233 *	0.258	0.251	0.259
DOW	0.486	0.514	0.239	0.268	0.225	0.268
GS	0.488 *	0.512 *	0.229 **	0.256	0.262	0.253
HD	0.480 **	0.520 **	0.229 **	0.248	0.255	0.268 *
HON	0.477 ***	0.523 ***	0.219 ***	0.265 *	0.250	0.266 *
IBM	0.488 *	0.512 *	0.232 *	0.265 *	0.248	0.255
INTC	0.488 *	0.512 *	0.232 *	0.258	0.256	0.255
JNJ	0.486 *	0.514 *	0.225 **	0.265 *	0.257	0.253
JPM	0.493	0.507	0.233 *	0.254	0.267 *	0.246
KO	0.482 **	0.518 **	0.230 **	0.244	0.260	0.266 *
MCD	0.464 ***	0.536 ***	0.207 ***	0.262	0.252	0.279 ***
MMM	0.475 ***	0.525 ***	0.215 ***	0.259	0.261	0.265 *
MRK	0.494	0.506	0.246	0.242	0.253	0.258
MSFT	0.486 *	0.514 *	0.225 **	0.263	0.258	0.253
NKE	0.483 **	0.517 **	0.224 ***	0.253	0.264	0.258
PG	0.481 **	0.519 **	0.225 **	0.262	0.250	0.263
TRV	0.477 ***	0.523 ***	0.216 ***	0.254	0.268 *	0.262
UNH	0.477 ***	0.523 ***	0.221 ***	0.260	0.251	0.267 *
V	0.462 ***	0.538 ***	0.195 ***	0.275 **	0.259	0.270 *
VZ	0.493	0.517	0.234 *	0.264	0.255	0.248
WBA	0.499	0.501	0.244	0.258	0.251	0.246
WMT	0.487 *	0.513 *	0.224 ***	0.263	0.264	0.250
DIA	0.455 ***	0.545 ***	0.201 ***	0.262	0.246	0.291 ***

in Appendix B. The probabilities for sequences of length $n=3$ are presented in

Table A2. Path probabilities for sequences of length $n=3$ for assets in the DJIA index and for DIA. Also indicated are the probabilities having significant p-values from the one-sided z-test. *, ** and *** refer to the standard significance levels of 5%, 1% and 0.1%, respectively.

Path Label:	$-3.5$	$-2.5$	$-1.5$	$-0.5$	$0.5$	$1.5$	$2.5$	$3.5$
Symbol	Path Probability
AAPL	0.102 **	0.125	0.122	0.122	0.128	0.123	0.132	0.146 **
AMGN	0.112	0.119	0.148 *	0.122	0.131	0.121	0.121	0.125
AXP	0.118	0.121	0.132	0.131	0.114	0.132	0.121	0.131
BA	0.111 *	0.134	0.113	0.142 *	0.118	0.130	0.121	0.131
CAT	0.119	0.114	0.124	0.131	0.121	0.126	0.128	0.137
CRM	0.110 *	0.116	0.127	0.127	0.122	0.145 *	0.130	0.124
CSCO	0.095 ***	0.139 *	0.136	0.114	0.128	0.114	0.132	0.142 *
CVX	0.101 ***	0.114	0.128	0.132	0.118	0.144 **	0.128	0.136
DIS	0.112	0.102 **	0.134	0.139 *	0.138	0.115	0.121	0.139 *
DOW	0.115	0.137	0.077 *	0.132	0.150	0.124	0.128	0.137
GS	0.111 *	0.126	0.129	0.132	0.116	0.133	0.125	0.129
HD	0.104 **	0.131	0.120	0.129	0.124	0.129	0.122	0.141 *
HON	0.102 **	0.107 *	0.129	0.140 *	0.122	0.138	0.129	0.132
IBM	0.112 *	0.116	0.123	0.131	0.133	0.125	0.130	0.131
INTC	0.115	0.123	0.126	0.127	0.118	0.126	0.134	0.131
JNJ	0.112	0.135	0.125	0.122	0.117	0.132	0.114	0.144 **
JPM	0.116	0.114	0.131	0.150 ***	0.121	0.127	0.122	0.119
KO	0.116	0.115	0.123	0.130	0.115	0.131	0.132	0.138 *
MCD	0.095 ***	0.108 *	0.107 *	0.132	0.129	0.147 **	0.139 *	0.142 *
MMM	0.102 **	0.112 *	0.118	0.138 *	0.131	0.130	0.129	0.140 *
MRK	0.118	0.132	0.118	0.126	0.130	0.130	0.111 *	0.134
MSFT	0.103 **	0.113	0.136	0.136	0.126	0.134	0.129	0.123
NKE	0.107 *	0.122	0.118	0.132	0.123	0.135	0.136	0.127
PG	0.096 ***	0.122	0.126	0.121	0.135	0.140 *	0.128	0.132
TRV	0.099 ***	0.122	0.124	0.135	0.119	0.148 **	0.120	0.132
UNH	0.101 ***	0.124	0.126	0.121	0.115	0.142 *	0.137	0.135
V	0.079 ***	0.111	0.126	0.144 *	0.119	0.156 ***	0.139	0.125
VZ	0.106 **	0.122	0.136	0.141 *	0.126	0.128	0.122	0.118
WBA	0.111 *	0.133	0.132	0.119	0.132	0.123	0.128	0.122
WMT	0.103 **	0.121	0.127	0.118	0.126	0.146 **	0.139 *	0.119
DIA	0.090 ***	0.109 *	0.118	0.138	0.125	0.129	0.124	0.167 ***

. Significant p-values obtained from the one-sided z-test are also indicated. For comparison, sequence probabilities for the DIA data are also provided in these tables.

For $n=1$, for all 31 assets, the probability of a negative return is smaller than the probability of a positive return. For 24 of these assets, this relationship is significant at a level ≤$5\%$. For assets where the probabilities of the sequences 00 ($n=2$) or 000 ($n=3$) are significant at the level ≤$5\%$, these sequences have the smallest probability. This holds for 26 of 31 assets for $n=2$ and 21 of 31 assets for $n=3$. For the sequences 11 and 111, the results are not as strong. These sequences are significant at the level ≤$5\%$ in 11 of 31 assets for $n=2$ and 9 of 31 assets for $n=3$. However, these sequences represent, respectively, the highest probability for only 10 of the 11, for $n=2$, and 7 of the 9, for $n=3$, of these assets. If, instead, we consider sequences that contain at most a single negative return (a single “0” value), then the highest probability sequence that has significance at the $\le 5\%$ level occurs for 15 of 31, for $n=2$, and 20 of 31, for $n=3$, assets. These observations are consistent with the DIA results in Figure 13 for $n=\{4,5,6,7,8,10\}$.

10. Discussion

This work contains several points that we wish to emphasize.

The most significant is that modeling of market microstructure cannot be achieved using (existing) continuous-time stochastic theory. We provide two critical observations in the Introduction to support this statement. The first is that inclusion of microstructure effects results in price processes that are not semimartingales. The second is to note the work of Jarrow et al. (2009) (and Cheridito (2003)) that shows finite intervals between transaction times are required to maintain arbitrage-free price processes in such cases. (While Roll’s approximation $spread=2\sqrt{-cov}$ resembles a continuous-time statement, it is crucial to recognize that $cov$ is the first-order serial covariance of price changes, which are not instantaneous, but occur over finite market time intervals.) We argue that discrete, binary-tree models are appropriate models to capture microstructure behavior.

Our major theoretical result in this paper is the development a market-complete, arbitrage-free, binary-tree option pricing framework that is capable of capturing price processes exhibiting path-dependent behavior. Consistent with the observations in the previous paragraph, we note that any attempt to consider the continuous-time limit of this binary-tree model is self-defeating, as critical microstructure detail is lost in the continuum limit.

By applying the framework to build specific moving-average MA(1) and autoregressive AR(1) models, we have demonstrated that this binary-tree framework can be used to model market microstructure. In the case of the MA(1) model, we show that the binary price tree is sufficiently recombining that the computational complexity of the model is only $O\left(n^2\right)$. Both the MA(1) and AR(1) models require empirical testing in order to demonstrate their applicability as well as to provide comparison against other models. As this theoretical paper is already lengthy, we made the decision to leave such critical considerations for future investigation.

As it relates directly to the efficient market hypothesis, we have devoted a section of this work to a technical analysis of the direction-of-price-change probabilities that appear as parameters in the binary-tree model. Analysis of price histories for the 30 assets in the Dow Jones Average leads us to conclude that, when fixed-length sequences of daily price changes are categorized according to the number of price downturns in the sequence, it appears that the efficient market hypothesis operates within each category. However, market inefficiencies are evident between categories.