Option Pricing Using a Skew Random Walk Binary Tree

Abstract

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index.

1. Introduction

In the academic literature on dynamic asset pricing theory, discrete-time option pricing models traditionally play a subordinate role to their continuous-time counterparts. This is due to the highly developed theory of semimartingales, Lévy processes, and the fundamental theorem of asset pricing. (

Table 1. Cited literature for selected topics addressed in the text.

Discrete-time option pricing models: Brennan and Schwartz (1978); Rendleman and Bartter (1979); Cox et al. (1979); Jarrow and Rudd (1983, chp. V, VI); Omberg (1987, 1988); Boyle (1988); Boyle et al. (1989); He (1990); Nelson and Ramaswamy (1990); Amin (1991); Pedersen and Shiu (1993); Boyle and Lau (1994); Rogers and Stapleton (1997); Rubinstein (1998); Tian (1999); Korn and Korn (2000, chp. 4); Walsh (2003); Chang and Palmer (2007); Korn and Müller (2010); Simonato (2011); Leccadito et al. (2012); Yuen et al. (2013); Bock and Korn (2016)

Continuous-time option pricing models: Black and Scholes (1973); Merton (1973, 1976); Cox and Ross (1976; Harrison and Kreps (1979); Harrison and Pliska (1981, 1983); Bensoussan (1984); Hull and White (1987); Karatzas (1988); Karatzas and Shreve (1991); Madan and Milne (1991); Heston (1993); Karatzas and Kou (1996); Scott (1997); Campbell et al. (1996, chp. 9); Madan et al. (1998); Chan (1999); Fouque et al. (2000); Duffie (2001); Kou (2002); Markus and Wu (2002); Pan (2002); Schoutens (2003); Björk (2004); Hunt and Kennedy (2004); Bingham and Kiesel (2004); Shreve (2004); Carr and Wu (2004); Applebaum (2004); Huang and Wu (2004); Elliott and Kopp (2005); Eberlein et al. (2008); Barndorff-Nielsen and Stelzer (2011); Rachev et al. (2011); Miyahara (2012); Yeap et al. (2018); Shirvani et al. (2020)

Theory of semimartingales: Emery and Yor (2002); Jacod and Shiryaev (2003); Protter (2004)

Theory of Lévy processes: Bertoin (1996); Sato (1999); Barndorff-Nielsen et al. (2001); Applebaum (2004)

Fundamental theorem of asset pricing: Delbaen and Schachermayer (1994, 1998); Elliott and Kopp (2005, chp. 3); Guasoni et al. (2012); Jarrow (2012)

Donsker Prokhorov invariance principle: Donsker (1951); Prokhorov (1956); Billingsley (1999, sct. 14); Gikhman and Skorokhod (1969, chp. IX); Skorokhod (2005, sct. 5.3.3); Davydov and Rotar (2008)

Trinomial and multinomial pricing trees: Boyle (1986, 1988); Boyle et al. (1989); Milne et al. (1989); Kamrad and Ritchken (1991); Boyle and Lau (1994); Florescu and Viens (2008); Deutsch (2009, chp. 10); Ma and Zhu (2015)

Skew Brownian motion: Itô and McKean (1996, reprint of the 1974 edition, §4.2, Problem 1, p. 115); Harrison and Shepp (1981); Revuz and Yor (1999, chp. VII and X); Lang (1995); Lejay (2006); Corns and Satchell (2007); Cherny et al. (2003); Ramirez (2011); Atar and Budhiraja (2015); Ouknine et al. (2015); Li (2019)

provides a comprehensive list of literature citations for these topics). However, as discussed in Section 2 the standard option pricing approach, based upon passing to the continuous-time limit, results in the loss of the significant information provided by the stylized-fact capturing parameters that can be embedded in discrete-time market models. This loss in the continuum limit produces a dislocation between the information on spot prices that is available to spot traders versus option traders1 (see, e.g., Ross 2015; Audrino et al. 2014). We further note that pure continuum models that attempt to incorporate stylized facts such as the heavy tails and skewness properties of realistic price processes through the use of semimartingale (e.g., Lévy) processes, generally lead to incomplete market models. The choice of the equivalent risk-neutral martingale becomes ad hoc, and any unique relationship between the probability measures driving price processes in the natural and risk-neutral worlds is lost.

This motivates us to propose a primary role for discrete-time option pricing models and proposes that any continuous-time option pricing model should only be considered as a benchmark model for a more precise discrete pricing model that preserves valuable market characteristics2 (see, e.g., Marroni and Perdomo 2014, p. 123; Joshi 2008, p. 197; Rebonato 2004, p. 169). Based on this proposal, we identify the following three fundamental research issues:

• The first is to develop pricing trees that capture valuable market characteristics (stylized facts) in the natural world and retain these in the risk-neutral world.
• The second is to ensure that the model is market-complete.
• In recognition of the role that continuous-time benchmark models play, the third would be to determine whether the pricing tree retains these characteristics in the limit that the transaction time interval $∆t\downarrow 0$.3 (While this step is of interest, it is no longer of fundamental importance if the primacy of discrete-time pricing models is recognized.)

These research issues have been the focus of a series of papers (Hu et al. 2020a, 2020b, 2022). In Hu et al. (2022), the authors developed a market-complete option valuation based on a generalization of the Jarrow–Rudd (JR) pricing model (Jarrow and Rudd 1983). In contrast to other similar works, all of which placed the binomial tree model directly in the risk-neutral world (and were, therefore, unable to state what the determining tree is in the natural world), Hu et al. begin in the natural world and, from that, derived the unique (and hence market-complete) risk-neutral tree. Their generalization involved the use of skew Brownian motion (SBM) (Itô and McKean 1996) with the incorporation of a parameter $\beta$ (in addition to the mean $\mu$ and standard deviation $\sigma$) to capture skewness and kurtosis effects in the real-world pricing tree. These three parameters, $\mu ,\sigma ,$ and $\beta$, carry over into the risk-neutral world. Using the Cherny et al. (2003) invariance principle, Hu et al. (2022) showed that for a fixed, but relatively small, time increment, $∆t$, the pricing tree behavior was geometric SBM (GSBM). However, in the limit $∆t\downarrow 0$, the model converged to geometric BM (GBM), losing the skewness and kurtosis properties of the discrete model.

Corns and Satchell (2007) developed a continuous-time pricing model combining a standard BM and an SBM to capture skewness in option pricing models. Unfortunately, their model, as well as the model by Zhu and He (2017), which provides corrections to the Corns–Satchell (CS) model, is market incomplete since the equivalent martingale obtained in their work only provides hedging against the risks associated with the standard BM. These two papers serve as motivation for the present article on discrete-option pricing.

In this paper, beginning in the natural world with a skewed, heavy-tailed, binomial process, we derive the unique, market-complete, risk-neutral pricing tree that preserves these features. We then show that the continuum-time model preserves these properties. The “cost” to achieve this is the addition of a second Brownian motion to the discrete pricing process. As a result, we show that the risk-free rate in the risk-neutral process cannot be arbitrary but is a rate determined by an appropriate combination of the means and variances governing the returns of the underlying risky assets.

More specifically, we introduce a binary pricing tree, which we refer to as the Itô–McKean skew pricing tree (IMSPT), that can model either standard BM or Itô–McKean SBM. As we ensure hedging against the risks associated with standard BM or SBM in the model, the IMSPT will lead to a complete market model for option pricing. In the IMSPT, the mean return of the underlying stock is preserved in the resulting option pricing model.4 The role of the uptick probability $p_{∆t}$ governing the drift of a standard binomial price process is replaced by a skewness parameter $\alpha \in \left(0,1\right)$ in the IMSPT. The skewness parameter $\alpha$ remains present in both the natural and the risk-neutral world. In contrast to the model of Hu et al. (2022), we show continuous time convergence of the IMSPT model to skew GBM, preserving the skew properties of the pricing tree.

The paper is organized as follows: We begin by reviewing (Section 2) the standard four-step approach used in binomial option pricing, which results in the loss of critical natural world information. In Section 3, we introduce the Itô–McKean SBM, first as represented by Itô and McKean (1996) and then in a representation developed by Corns and Satchell (2007). The conditional probability density governing the Corns–Satchell SBM is a skewed normal distribution. The properties of this distribution are presented in Appendix A.

Section 3.1 and Section 3.2 are devoted to developing preparatory material for option pricing on the IMSPT. Specifically, in Section 3.1, we develop the continuous-time stochastic dynamics of Itô–McKean SBM. In Section 3.2, we develop the risk-free rate for a Black–Scholes–Merton market model where the uncertainty in the price of the risky asset is driven by the Itô–McKean GSBM. The risk-free rate contains a delta-function jump term, which must be eliminated through the addition of a second risky asset in the market model.

In Section 4, we develop the discrete option pricing model on the IMSPT for this market model. In order to remove the aforementioned delta-function term from option (derivative) prices, it is necessary to add a third risky asset to the market model. In Section 4.1, we show that the resultant model is still market-complete. Section 4.2 then develops option pricing for a European call option. Appendix B supplies the formulas for the determinants that appear in the option pricing.

The IMSPT option pricing formulas developed in Section 4.2 require knowledge of the values of the drift, diffusion, and skewness parameters of an asset’s return process. In practice, these parameter values are obtained by fitting a SBM model to a historical return series. However, fitting a return series by either the Corns–Satchell or the original Itô–McKean versions of SBM is impractical. In Section 5, we consider a practical method for generating an Itô–McKean SBM as the limiting process of a skew random walk. Relevant properties of the skew random walk are discussed in Appendix C, and a procedure for fitting this skew random walk to the historical return series is developed in Appendix D. This procedure is demonstrated in Section 5.1 by fitting the parameters of the return process of an exchange-traded fund (ETF) (the SPDR S&P 500 ETF Trust). In Section 5.2, using the IMPST option pricing formulas of Section 4.2, we compute European call and put option prices having, as underlying, three ETFs that track component indices of the S&P Global 1200 Index. Concluding remarks follow in Section 6.

2. The Standard Approach−Loss of Information in Passing to the Continuous Time Limit

Consider the standard four-step approach to binomial option pricing (see, e.g., Cox and Rubinstein 1985, chp. 5; Korn and Korn 2000, sct. 4.3; Bingham and Kiesel 2004, chp. 4; Hull 2012, chps. 12, 20; or Marroni and Perdomo 2014, chp. 4).

Step 1. A binomial pricing tree $\left\{S_{k∆t}, k=0,\dots ,n, n∆t=T\right\}$ is introduced in the natural world $\mathbb{P}$. The price dynamics of the underlying risky asset (stock) follow a binomial tree conditionally on $S_{k∆t}$ as follows:

$$S_{(k+1)∆t}=\left\{\begin{array}{ccc}{S}_{k∆t}{u}_{∆t}& \mathrm{w}.\mathrm{p}.& p_{∆t},\\ S_{k∆t}{d}_{∆t}& \mathrm{w}.\mathrm{p}.& 1-p_{∆t},\end{array}\right\}, p_{∆t}\in \left(0,1\right), k=0, \dots , n-1,$$

(1)

The no-arbitrage condition requires that $u_{∆t}>e^{r∆t}>d_{∆t},$ where $r\ge 0$ is the instantaneous riskless rate. The log-return $R_{(k+1)∆t}=\ln \left(S_{(k+1)∆t}/S_{k∆t}\right)$ is assumed to have a mean $\mathbb{E}\left(R_{(k+1)∆t}\right)=\mu ∆t$ and a variance $\mathrm{V}\mathrm{a}\mathrm{r}\left(R_{(k+1)∆t}\right)={\sigma }^2∆t$ to guarantee that the binomial pricing tree has a weak Gaussian limit (needed in Step 2). With $o\left(∆t\right)=0,$ the pair ($u_{∆t},d_{∆t}$) is then uniquely determined as $u_{∆t}=1+\mu ∆t+\sqrt{\left(1-p_{∆t}\right)/p_{∆t}}\sigma \sqrt{∆t}$ and $d_{∆t}=1+\mu ∆t-\sqrt{p_{∆t}/\left(1-p_{∆t}\right)}\sigma \sqrt{∆t}$ (Rendleman and Bartter 1979).

Step 2. From Step 1 and the Donsker–Prokhorov invariance principle (DPIP), the price process generated by the tree in Step 1 converges weakly in the Skorokhod $\mathfrak{D}\left[0,T\right]$-space to a GBM, $S_t=S_0\mathrm{e}\mathrm{x}\mathrm{p}\left(\left(\mu -{\sigma }^2/2\right)t+\sigma B_t\right), t\in \left[0,T\right],$ where $B_t$ is a BM on $\mathbb{P}$. (For references on DPIP, see Table 1). With this step, the information about the probability $p_{∆t}$ for stock-price uptick that was available in the discrete-time process has been lost.

Step 3. Using the no-arbitrage condition on the continuum model from Step 2, the dynamics of the underlying asset in the risk-neutral world $\mathbb{Q}$ is $S_t=S_0\mathrm{e}\mathrm{x}\mathrm{p}\left(\left(r-{\sigma }^2/2\right)t+\sigma B_t^{\mathbb{Q}}\right), t\in \left[0,T\right],$ where $B_t^{\mathbb{Q}}$ is a BM on $\mathbb{Q}$. Now a hedger, who takes the short position in an option contract and who, in any real trading situation, will trade only in discrete-time instances, has lost information about both the instantaneous mean return $\mu$ and $p_{∆t}$.

In the Black–Scholes–Merton formulation, the loss of $\mu$ in the risk-neutral world means that the instantaneous drift can be arbitrarily large while the option price remains unchanged, a condition known as the option pricing discontinuity puzzle. This puzzle has been addressed by Hu et al. (2020a). The loss of $\mu$ is due to the absurd assumption that the trader can trade continuously in time. We note that including trading costs in continuous-time option pricing cannot retrieve the information lost about $p_{∆t}$ and $\mu$ (Leland 1985; Hodges and Neuberger 1989; Boyle and Vorst 1992; Davis et al. 1993; Edirisinghe et al. 1993; Broadie et al. 1998; Kabanov and Stricker 2001; Zakamouline 2009; Lai and Lim 2009; Guasoni et al. 2012). No real trader can trade in continuous time, with or without transaction costs.

Step 4. Step 3 and the DPIP imply that the risk-neutral, continuous asset price dynamics can be approximated by a risk-neutral binomial tree with price dynamics given by the following:

$$S_{(k+1)∆t}=\left\{\begin{array}{ccc}{S}_{k∆t}{u}_{∆t}^{\left(\mathbb{Q}\right)}& \mathrm{w}.\mathrm{p}.& p_{∆t}^{\left(\mathbb{Q}\right)}\\ S_{k∆t}{d}_{∆t}^{\left(\mathbb{Q}\right)}& \mathrm{w}.\mathrm{p}.& 1-p_{∆t}^{\left(\mathbb{Q}\right)}\end{array}\right\}, k=0, \dots , n-1,$$

(2)

where

$$\begin{gathered} u_{∆t}^{\left(\mathbb{Q}\right)}=1+r∆t+\sqrt{\left(1-p_{∆t}^{\left(\mathbb{Q}\right)}\right)/p_{∆t}^{\left(\mathbb{Q}\right)}}\sigma \sqrt{∆t}, \\ {d}_{∆t}^{\left(\mathbb{Q}\right)}=1+r∆t-\sqrt{p_{∆t}^{\left(\mathbb{Q}\right)}/\left(1-p_{∆t}^{\left(\mathbb{Q}\right)}\right)}\sigma \sqrt{∆t} , \end{gathered}$$

(3)

for any arbitrarily chosen value of $p_{∆t}^{\left(\mathbb{Q}\right)}\in \left(0,1\right).$ Setting

$$p_{∆t}^{\left(\mathbb{Q}\right)}=p_{∆t}^{\left(\mathbb{Q};\mathrm{C}\mathrm{R}\mathrm{R}\right)}\equiv \frac{e^{r∆t}-e^{-\sigma \sqrt{∆t}}}{e^{\sigma \sqrt{∆t}}-e^{-\sigma \sqrt{∆t}}}=\frac{1}{2}+\frac{r-{\sigma }^2/2}{2\sigma }\sqrt{∆t},$$

(4)

results in the CRR (Cox et al. 1979) pricing tree.5 Setting $p_{∆t}^{\left(\mathbb{Q}\right)}=p_{∆t}^{\left(\mathbb{Q};\mathrm{J}\mathrm{R}\right)}=1/2$ results in the JR pricing tree6 (Jarrow and Rudd 1983, chps. V, VI; and Hull 2012, p. 442). By the DPIP, the value of $p_{∆t}^{\left(\mathbb{Q}\right)}\in \left(0,1\right)$ controls the rate of convergence of the binomial option price to its continuous-time limit (Kim et al. 2016)7 and is otherwise irrelevant in that limit. Thus, returning to the realistic case when the hedger trades in discrete-time instances, the pricing tree (which the hedger is supposed to use to hedge his short position in the option contract) is now without the parameters $\mu$ and $p_{∆t}$. This, in turn, leads to a dislocation in the information available to spot traders and to option traders with regard to spot prices.

Remark 1.

There is no need to pass to continuous-time price dynamics when using binomial option pricing.(For various generalizations of the binomial pricing market model in Step 1, see Hu et al. 2020a, 2020b, 2022. See also Jarrow et al. 20098). Proceeding immediately from Step 1 to the discrete risk-neutral pricing tree, the no-arbitrage argument implies that the risk-neutral probability, $q_{∆t}$), over the interval $\left[k∆t,\left(k+1\right)∆t\right], k=0,\dots ,n-1,$ is uniquely determined by $q_{∆t}=p_{∆t}-\theta \sqrt{p_{∆t}\left(1-p_{∆t}\right)}\sqrt{∆t}$, where $\theta =\left(\mu -r\right)/\sigma$ is the market price of risk (Kim et al. 2016, 2019). The risk-neutral asset pricing tree is as follows:

$$S_{(k+1)∆t}=\left\{\begin{array}{ccc}{S}_{k∆t}{u}_{∆t}& \mathrm{w}.\mathrm{p}.& q_{∆t},\\ S_{k∆t}{d}_{∆t}& \mathrm{w}.\mathrm{p}.& 1-q_{∆t},\end{array}\right\}, k=0, \dots , n-1.$$

(5)

The four parameters $\mu , \sigma$, $r,$ and $p_{∆t}$ are preserved in the risk-neutral price process. (The trader’s information about $p_{∆t}$ and $\mu$ could be gathered from stock market data or be based on the trader’s active trading alpha-strategy; see, e.g., Grinold and Kahn (2000, chp. 4)). The resultant binomial market model is an arbitrage-free, complete market model. Therefore, if the option trader had bypassed Steps 2, 3, and 4 and passed directly from Step 1 to the discrete risk-neutral pricing tree, information about $\mu$ and $p_{∆t}$ would be preserved when valuing the option price with the uniquely determined risk-neutral probabilities given by $q_{∆t}=p_{∆t}-\theta \sqrt{p_{∆t}\left(1-p_{∆t}\right)}\sqrt{∆t}$ and $1-q_{∆t}.$

Remark 2.

Trinomial and multinomial pricing trees appearing in the literature are placed directly in the risk-neutral world, bypassing Steps 1, 2, and 3 altogether. (For references on tri- and multinomial trees, see Table 1). These option pricing trees in complete market models do not address the critical question as to which uniquely determined pricing tree in the natural world leads to the selected trinomial or multinomial pricing tree in the risk-neutral world.

3. Itô–McKean Skew Brownian Motion

A SBM ${\mathbb{A}}^{\left(\delta \right)}$ is defined as follows (See Table 1 for references on SBM). Let $\mathbb{B}=\left\{B_t, t\ge 0\right\}$ be a standard BM generating a stochastic basis $\left(\mathsf{\Omega },\mathbb{F}=\left({\mathcal{F}}_t=\sigma \left(B_u,u\le t\right);t\ge 0 \right),\mathbb{P}\right)$. Let $\alpha \in (0,1)$ and set

$$\begin{gathered} {\mathcal{A}}_t^{\left(\alpha \right)}={\int }_0^t({\alpha }^2{I}_{\left\{B_s\ge 0\right\}}+{\left(1-\alpha \right)}^2{I}_{\left\{B_s<0\right\}}) ds, \\ {\tau }_t^{\left(\alpha \right)}=\inf \left\{s\ge 0: {\mathcal{A}}_s^{\left(\alpha \right)}>t\right\}, t\ge 0 , \\ {B}_t^{\left(\alpha \right)}={\phi }_{\alpha }\left(B_{{\tau }_t^{\left(\alpha \right)}}\right),t>0, B_0^{\left(\alpha \right)}=0, \end{gathered}$$

(6)

where ${\phi }_{\alpha }\left(x\right)=\alpha x{I}_{\left\{x\ge 0\right\}}+\left(1-\alpha \right)x{I}_{\left\{x<0\right\}}, x\in R$. The process ${\mathbb{B}}^{\left(\alpha \right)}=\left\{B_t^{\left(\alpha \right)}, t\ge 0\right\}$ is called a SBM with parameter $\alpha$ (Cherny et al. 2003). The SBM ${\mathbb{B}}^{\left(\alpha \right)}$ is a semimartingale satisfying the strong Markov property and ${\mathbb{B}}^{\left(\frac{1}{2}\right)}\stackrel{d}{=}\mathbb{B}$ (Cherny et al. 2003, sct. 4; Corns and Satchell 2007). For every $t\ge 0,$ the density $f_t^{\left(\alpha \right)}\left(x\right), x\in R$ of $B_t^{\left(\alpha \right)}$ is given by:

$$f_t^{\left(\alpha \right)}\left(x\right)=\left\{\begin{array}{cc}\alpha \sqrt{\frac{2}{\pi t}}\exp \left(-\frac{x^2}{2t}\right),& \mathrm{i}\mathrm{f} x\ge 0 ,\hfill \\ \left(1-\alpha \right)\sqrt{\frac{2}{\pi t}}\exp \left(-\frac{x^2}{2t}\right),& \mathrm{i}\mathrm{f} x<0.\hfill \end{array}\right.$$

(7)

As noted by Cherny et al. (2003), there are other ways of defining a SBM, all of which are equal in distribution. Trajectories of $\left\{B_t^{\left(\alpha \right)}, t\ge 0\right\}$ can be simulated using the Itô–McKean representation, ${\mathbb{B}}^{\left(\alpha ,\mathrm{I}\mathrm{M}\right)}=\left\{B_t^{\left(\alpha ,\mathrm{I}\mathrm{M}\right)}, t\ge 0\right\}$, where

$$B_t^{\left(\alpha ,\mathrm{I}\mathrm{M}\right)}=\left\{\begin{array}{c}\left|B_t\right| \mathrm{w}.\mathrm{p}. \alpha ,\\ -\left|B_t\right| \mathrm{w}.\mathrm{p}. 1-\alpha ,\end{array}\right.$$

(8)

with $B_t$ being a standard BM. This representation first appears in Itô and McKean (1996, §4.2, Problem 1, p. 115). Corns and Satchell (2007) (CS) developed a representation which they called Itô–McKean SBM. The Itô–McKean SBM ${\mathbb{A}}^{\left(\delta \right)}=\left\{A_t^{\left(\delta \right)},t\ge 0\right\}$ with parameter $\delta \in (-1,1)$ is defined as follows:

$$A_t^{\left(\delta \right)}=\sqrt{1-{\delta }^2}{B}_{1,t}+\delta \left|B_{2,t}\right|,$$

(9)

where ${\mathbb{B}}_1=\left\{B_{1,t},t\ge 0\right\}$ and ${\mathbb{B}}_2=\{B_{2,t},t\ge 0\}$ are two independent BMs. CS showed that ${\mathbb{A}}^{\left(\delta \right)}$ has the conditional probability density:

$$\mathbb{P}\left(A_t^{\left(\delta \right)}\in dz\right)=\frac{2}{\sqrt{t}}\phi \left(\frac{A_t^{\left(\delta \right)}}{\sqrt{t}}\right)\mathsf{\Phi }\left(\lambda \frac{A_t^{\left(\delta \right)}}{\sqrt{t}}\right)dz,$$

(10)

where $\lambda =\delta /\sqrt{1-{\delta }^2}$, and $\phi \left(·\right)$ and $\mathsf{\Phi }\left(·\right)$ are, respectively, the standard normal density and cumulative distribution functions. CS showed that:

$${\mathbb{A}}^{\left(\delta \right)}\stackrel{d}{=}{\mathbb{B}}^{\left(\alpha ,\mathrm{I}\mathrm{M}\right)} \mathrm{with} \alpha =\left(1+\delta \right)/2.$$

(11)

The density $f_{Z^{\left(\mathit{\lambda }\right)}}\left(z\right)=2\varphi \left(z\right)\mathsf{\Phi }\left(\mathit{\lambda }z\right)$ governing a continuous random variable $Z^{\left(\mathit{\lambda }\right)}$ is referred to as a skewed normal distribution (SND). Relevant properties of an SND are presented in Appendix A. Using the moment properties of an SND, the moments ${\mu }_t^{\left(p,\alpha \right)}=\mathbb{E}\left[{\left(B_t^{\left(\alpha \right)} \right)}^p\right], p>0, t>0$ are given by ${\mu }_t^{\left(p,\alpha \right)}=\sqrt{2^p/\pi } \mathsf{\Gamma }\left((p+1)/2\right)\left[\alpha +{\left(-1\right)}^p\left(1-\alpha \right)\right]t^{p/2}$. In particular, the mean, variance, skewness, and excess kurtosis of $B_t^{\left(\alpha \right)}$ are given by the following:

$$\begin{array}{c}\begin{array}{c}{\mu }_t^{\left(\alpha \right)}=\mathbb{E}\left[B_t^{\left(\alpha \right)}\right]=\left(2\alpha -1\right)\sqrt{2t/\pi } \equiv \sqrt{t}{\mu }_1^{\left(\alpha \right)},\\ V_t^{\left(\alpha \right)}=\mathrm{V}\mathrm{a}\mathrm{r}\left[B_t^{\left(\alpha \right)}\right]=t\left[1-{\left({\mu }_1^{\left(\alpha \right)}\right)}^2\right]\equiv t{V}_1^{\left(\alpha \right)},\end{array}\\ {\gamma }_t^{\left(\alpha \right)}=\frac{\mathbb{E}\left[{\left(B_t^{\left(\alpha \right)}-{\mu }_t^{\left(\alpha \right)}\right)}^3\right]}{{\left(V_t^{\left(\alpha \right)}\right)}^{3/2}}=\frac{\left[2 {\left({\mu }_1^{\left(\alpha \right)}\right)}^2-1\right]{\mu }_1^{\left(\alpha \right)}}{{\left[1-{\left({\mu }_1^{\left(\alpha \right)}\right)}^2\right]}^{3/2}}\equiv {\gamma }_1^{\left(\alpha \right)},\\ {\kappa }_t^{\left(\alpha \right)}=\frac{\mathbb{E}\left[{\left(B_t^{\left(\alpha \right)}-{\mu }_t^{\left(\alpha \right)}\right)}^4\right]}{{\left(V_t^{\left(\alpha \right)}\right)}^2}-3=\frac{2 {\left({\mu }_1^{\left(\alpha \right)}\right)}^2 \left[1-3{\left({\mu }_1^{\left(\alpha \right)}\right)}^2\right]}{{\left[1-{\left({\mu }_1^{\left(\alpha \right)}\right)}^2\right]}^2}\equiv {\kappa }_1^{\left(\alpha \right)}.\end{array}$$

(12)

The terms ${\mu }_1^{\left(\alpha \right)}$, $V_1^{\left(\alpha \right)}, {\gamma }_1^{\left(\alpha \right)},$ and ${\kappa }_1^{\left(\alpha \right)}$ are introduced in (12) to isolate the time dependence of each moment.

We develop asset and option pricing based upon Itô–McKean SBM using the Corn–Satchel version (9) rather than the original version (8). However, neither of these two versions of SBM is useful for estimating the parameter values (which need to be obtained by fitting an SBM model to a real asset return process) required in the option price formulas. Estimation using (8) requires knowledge of trajectories of $B_t$ in the limit $t\uparrow \infty$, while estimation using (9) requires working with two independent Brownian motions as well as a Dirac $\delta$-function (see Section 3.1). In Section 5, we describe a third representation of an Itô–McKean SBM as a limiting process of a single skew random walk. This representation is used in our empirical examples to provide fits to historical return time series to estimate the appropriate parameter values.

3.1. Stochastic Dynamics of Itô–McKean SBM

The goal in this subsection and the next is to motivate the market-complete tree model needed for derivative pricing.9 The stochastic differential equation for ${\mathbb{A}}^{\left(\delta \right)}$ is derived directly from Tanaka’s formula (Chung and Williams 1991, sct. 7.3; Björk 2019, chp. 4). If $\mathbb{B}=\left\{B_t, t\ge 0\right\}$ is a Brownian motion, then

$$\left|B_t\right|={\int }_0^t\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(B_s\right)d{B}_s+L_t^{\left(0,\mathbb{B}\right)}, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(x\right)=\left\{\begin{array}{rrr}1,& \mathrm{i}\mathrm{f}& x>0,\\ 0,& \mathrm{i}\mathrm{f}& x=0,\\ -1,& \mathrm{i}\mathrm{f}& x<0.\end{array}\right.$$

(13)

In (13), ${\mathbb{L}}^{\left(a,\mathbb{B}\right)}=\left\{L_t^{\left(a,\mathbb{B}\right)},t\ge 0\right\}$ is the Brownian motion local time at $a\in R$ defined by

$$L_t^{\left(a,\mathbb{B}\right)}=\underset{\epsilon \downarrow 0}{\lim }\frac{1}{2\epsilon }{\int }_0^t{I}_{\left\{a-\epsilon \le B_s\le a+\epsilon \right\}}ds=\underset{\epsilon \downarrow 0}{\lim }\frac{1}{2\epsilon }\mathrm{L}\mathrm{e}\mathrm{b}\left\{s\in \left[0,t\right]:B_s\in \left(a-\epsilon ,a+\epsilon \right)\right\} ,$$

(14)

where $\mathrm{L}\mathrm{e}\mathrm{b}\left\{·\right\}$ is the Lebesgue measure. Informally (Björk 2019, Proposition 2.4), $L_t^{\left(a;\mathbb{B}\right)}={\int }_0^t{\delta }_a\left(B_s\right)ds,$ where ${\delta }_a\left(x\right), x\in R,$ is the Dirac delta function at $a,$ and

$$d{L}_t^{\left(a,\mathbb{B}\right)}={\delta }_a\left(B_t\right)dt.$$

(15)

For $a\in R,L_{\mathrm{\infty }}^{\left(a;\mathbb{B}\right)}=\mathrm{\infty }$ with probability 1 (Revuz and Yor 1999, Corollary 2.4, chp. VI). If ${\mathbb{M}}^{\left(\mathbb{B}\right)}=\left\{m_t^{\left(\mathbb{B}\right)}=\underset{0\le s\le t}{\max }{B}_s,t\ge 0\right\}$ and ${\mathbb{L}}^{\left(0;\mathbb{B}\right)}=\left\{L_t^{\left(0;\mathbb{B}\right)},t\ge 0\right\}$, then ${\mathbb{M}}^{\left(\mathbb{B}\right)}\stackrel{d}{=}{\mathbb{L}}^{\left(0;\mathbb{B}\right)}$. (See Revuz and Yor 1999, Theorem 2.3, chp. VI for a more general result10). From (13)–(15),

$$d{A}_t^{\left(\delta \right)}=\sqrt{1-{\delta }^2}d{B}_{1,t}+\delta d\left|B_{2,t}\right|=d{B}_{1\leftrightarrow 2,t}+\delta d{L}_t^{\left(0,{\mathbb{B}}_2\right)},$$

(16)

where ${\mathbb{B}}_{1\leftrightarrow 2,t}=\left\{B_{1\leftrightarrow 2,t}=\sqrt{1-{\delta }^2}{B}_{1,t}+\delta \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(B_{2,t}\right)B_{2,t}, t\ge 0\right\}$ is a standard BM. Equation (16) succinctly encapsulates the challenges faced in developing asset and derivative pricing in a model whose risky asset prices are driven by $d{A}_t^{\left(\delta \right)}$. There are two independent Brownian motions $d{B}_{1,t}$ and $d{B}_{2,t}$ as well as the Dirac $\delta$-function $d{L}_t^{\left(0,{\mathbb{B}}_2\right)}$ to be considered in developing an appropriate riskless rate as well as a feasible strategy for a hedging portfolio. We first consider the issue of an appropriate risk-free rate.

3.2. The Risk-Free Rate in a Market with Uncertainties Determined by Itô–McKean SBM

We start from the basic Black–Scholes–Merton market model $\left(\mathcal{S},\mathcal{B},\mathcal{C}\right)$, (Black and Scholes 1973; Merton 1973; Duffie 2001, chps. 5 and 6), with $\mathcal{S},\mathcal{B}$ and $\mathcal{C}$ defined as follows.

(i)

$\mathcal{S}$ is a risky asset (stock) with price dynamics $S_t^{\left(\delta \right)}, t\in \left[0,T\right], 0<T<\infty$ given by Itô–McKean GSBM,

$$S_t^{\left(\delta \right)}=S_0\exp \left(\mu t+\sigma A_t^{\left(\delta \right)}\right), S_0>0, \mu \in R, \sigma >0 .$$

(17)

In (17), ${\mathbb{A}}^{\left(\delta \right)}=\left\{A_t^{\left(\delta \right)},t\ge 0\right\}$ is an Itô–McKean SBM as defined in (9), with dynamics determined by (16), where ${\mathbb{B}}_1=\left\{B_{1,t}, t\ge 0\right\},$ and ${\mathbb{B}}_2=\left\{B_{2,t}, t\ge 0\right\}$ are two independent Brownian motions generating a stochastic basis $\left(\mathsf{\Omega },\mathbb{F}=\left\{{\mathcal{F}}_t=\sigma \left(\left(B_{1,s},B_{2,s}\right), 0\le s\le t\right),t\ge 0\right\},\mathbb{P}\right).$

(ii)

$\mathcal{B}$ is a riskless asset (bond) with price dynamics.

$${\beta }_t={\beta }_0{e}^{rt} , t\in \left[0,T\right], {\beta }_0>0, r\ge 0.$$

(18)

(iii)

$\mathcal{C}$ is a European Contingent Claim (ECC)11 with underlying asset $\mathcal{S}$ and maturity time $T$. The price dynamics of $\mathcal{C}$ is given by the following:

$$f_t=f\left(S_t^{\left(\delta \right)},t\right),t\in \left[0,T\right), f_T=g\left(S_T^{\left(\delta \right)}\right),$$

(19)

where $f\left(x,t\right)\in R, x>0, t\in [0,T)$ and $g\left(x\right), x>0$ satisfy the usual regularity conditions (Duffie 2001, chp. 5).

With the underlying risky asset $\mathcal{S}$ having price dynamics (17), the riskless rate $r\ge 0$ in (18) cannot be chosen arbitrarily. This is due to the delta-function jump in the instantaneous mean return of the stock; that is, from (15)–(17) and the extended Itô formula (Björk 2019, Theorem 4.3),

$$d{S}_t^{\left(\delta \right)}=S_t^{\left(\delta \right)}\left\{\left(\mu +\frac{1}{2}{\sigma }^2+\sigma \delta {\delta }_0\left(B_{2,t}\right)\right)dt+\sigma d{B}_{1\leftrightarrow 2,t}\right\} .$$

(20)

The $\delta$-function ${\delta }_0\left(B_{2,t}\right)$ remains in the model if only a single risky asset is considered. To illustrate how to eliminate the $\delta$-function term in the rate of the riskless asset $\mathcal{B}$, it is necessary to consider a market model with at least two risky assets, ${\mathcal{S}}^{\prime }=\left({\mathcal{S}}^{\left(1\right)},{\mathcal{S}}^{\left(2\right)}\right)$12, each having price dynamics as follows:

$$S_t^{\left(\delta ,i\right)}=S_0^{\left(i\right)}\exp \left({\mu }^{\left(i\right)}t+{\sigma }^{\left(i\right)}{A}_t^{\left({\delta }^{\left(i\right)}\right)}\right), S_0^{\left(i\right)}>0, {\mu }^{\left(i\right)}\in R, {\sigma }^{\left(i\right)}>0, i=\mathrm{1,2} ,$$

(21)

where

$$A_t^{\left({\delta }^{\left(i\right)}\right)}=\sqrt{1-{{\delta }^{\left(i\right)}}^2}{B}_{1,t}+{\delta }^{\left(i\right)}\left|B_{2,t}\right|=B_{1\leftrightarrow 2,t}+{\delta }^{\left(i\right)}{L}_t^{\left(0\right)}\left({\mathbb{B}}_2\right), {\delta }^{\left(i\right)}\in \left(-\mathrm{1,1}\right) .$$

(22)

Then

$$d{S}_t^{\left(\delta ,i\right)}=S_t^{\left(\delta ,i\right)}\left\{\left({\mu }^{\left(i\right)}+\frac{1}{2}{{\sigma }^{\left(i\right)}}^2+{\sigma }^{\left(i\right)}{\delta }^{\left(i\right)}{\delta }_0\left(B_{2,t}\right)\right)dt+{\sigma }^{\left(i\right)}d{B}_{1\leftrightarrow 2,t}\right\} .$$

(23)

To evaluate the rate in the corresponding riskless asset $\mathcal{B}$, we follow the approach in Section 6D of Duffie (2001) (based upon Black 197213). The portfolio $({\mathcal{S}}^{\prime },\mathcal{B},\mathcal{C})$ is free of arbitrage and complete if there exists a unique state price deflator $\mathsf{\Pi }=\left\{{\pi }_t, t\ge 0\right\}$ on $\left(\mathsf{\Omega },\mathbb{F}=\left\{{\mathcal{F}}_t=\sigma \left(\left(B_{1,s},B_{2,s}\right), 0\le s\le t\right),t\ge 0\right\},\mathbb{P}\right)$. We search for $\mathsf{\Pi }$ having the dynamics

$$d{\pi }_t={\pi }_t\left({\mu }_t^{\left(\pi \right)}dt+{\sigma }_t^{\left(\pi \right)}d{B}_{1\leftrightarrow 2,t}\right) ,$$

(24)

such that $S_t^{\left(\delta ,\pi ,i\right)}=S_t^{\left(\delta ,i\right)}{\pi }_t, i=\mathrm{1,2}$ are martingales on $\left(\mathsf{\Omega },\mathbb{F},\mathbb{P}\right)$. The martingale condition implies that

$${\mu }_t^{\left(\pi \right)}=\frac{\left({\mu }^{\left(1\right)}+\frac{{{\sigma }^{\left(1\right)}}^2}{2}+{\sigma }^{\left(1\right)}{\delta }^{\left(1\right)}{\delta }_0\left(B_{2,t}\right)\right){\sigma }^{\left(2\right)}-\left({\mu }^{\left(2\right)}+\frac{{{\sigma }^{\left(2\right)}}^2}{2}+{\sigma }^{\left(2\right)}{\delta }^{\left(2\right)}{\delta }_0\left(B_{2,t}\right)\right){\sigma }^{\left(1\right)}}{{\sigma }^{\left(1\right)}-{\sigma }^{\left(2\right)}}$$

(25a)

and

$${\sigma }_t^{\left(\pi \right)}=\frac{{\mu }^{\left(1\right)}+\frac{{{\sigma }^{\left(1\right)}}^2}{2}+{\sigma }^{\left(1\right)}{\delta }^{\left(1\right)}{\delta }_0\left(B_{2,t}\right)-\left({\mu }^{\left(2\right)}+\frac{{{\sigma }^{\left(2\right)}}^2}{2}+{\sigma }^{\left(2\right)}{\delta }^{\left(2\right)}{\delta }_0\left(B_{2,t}\right)\right)}{{\sigma }^{\left(1\right)}-{\sigma }^{\left(2\right)}} .$$

(25b)

The riskless rate $r_t^{\prime }$ in the market $({\mathcal{S}}^{\prime },\mathcal{B},\mathcal{C})$ is now determined by ${\pi }_t$ via (Duffie 2001, sct. 6D)

$$r_t^{\prime }=-{\mu }_t^{\left(\pi \right)}=\frac{{\mu }^{\left(2\right)}{\sigma }^{\left(1\right)}-{\mu }^{\left(1\right)}{\sigma }^{\left(2\right)}+{\sigma }^{\left(1\right)}{\sigma }^{\left(2\right)}\left[\frac{1}{2}\left({\sigma }^{\left(2\right)}-{\sigma }^{\left(1\right)}\right)+\left({\delta }^{\left(2\right)}-{\delta }^{\left(1\right)}\right){\delta }_0\left(B_{2,t}\right)\right]}{{\sigma }^{\left(1\right)}-{\sigma }^{\left(2\right)}} .$$

(26)

Thus, as long as

$${\delta }^{\left(1\right)}={\delta }^{\left(2\right)}=\delta \in \left(-\mathrm{1,1}\right),$$

(27)

then14

$$r_t^{\prime }=r^{\left(\mathrm{F}\mathrm{B}\right)}=\frac{{\mu }^{\left(2\right)}{\sigma }^{\left(1\right)}-{\mu }^{\left(1\right)}{\sigma }^{\left(2\right)}+\frac{1}{2}{\sigma }^{\left(1\right)}{\sigma }^{\left(2\right)}\left({\sigma }^{\left(2\right)}-{\sigma }^{\left(1\right)}\right)}{{\sigma }^{\left(1\right)}-{\sigma }^{\left(2\right)}} ,$$

(28)

eliminating the $\delta$-function term from the riskless rate. We shall assume that (26) holds in our development of derivative pricing.

4. Derivative Pricing with an Itô–McKean Skew Pricing Tree

While the market model $({\mathcal{S}}^{\prime },\mathcal{B},\mathcal{C})$ is arbitrage-free and complete, and the riskless rate is free of the uncertainties $\left({\mathbb{B}}_1,{\mathbb{B}}_2, {\mathbb{L}}^{\left(0,{\mathbb{B}}_2\right)}\right)$ under (26), as the price uncertainty is determined by ${\mathbb{A}}^{\left(\delta \right)}$, any derivative price $f_t, t\in \left[0,T\right]$ remains affected by the three market uncertainties $\left({\mathbb{B}}_1,{\mathbb{B}}_2, {\mathbb{L}}^{\left(0,{\mathbb{B}}_2\right)}\right).$ To enable a feasible hedging strategy (via a unique replicating portfolio), the $\delta$-function term must be removed from the deterministic drift in the asset prices. To remove this term while retaining market completeness, we add a third risky asset and assume a derivative price dependence:

$$f_t^{\left(\delta \right)}=f\left(S_t^{\left(\delta \right)},t\right), S_t^{\left(\delta \right)}=\left(S_t^{\left(\delta ,1\right)},S_t^{\left(\delta ,2\right)},S_t^{\left(\delta ,3\right)}\right), t\in \left[0,T\right],$$

(29)

where each of the underlying assets has price dynamics determined by:

$$S_t^{\left(\delta ,i\right)}=S_0^{\left(i\right)}\exp \left({\mu }^{\left(i\right)}t+{\sigma }^{\left(i\right)}{A}_t^{\left(\delta \right)}\right), {\mu }^{\left(i\right)}\in R, {\sigma }^{\left(i\right)}\in R, \delta \in \left(-\mathrm{1,1}\right), i=\mathrm{1,2},3.$$

(30)

To guarantee the appropriate riskless rate, consistent with (26) we ensure the same skewness parameter $\delta \in (-1,1)$ for the three price processes.

4.1. Completeness of the IMSPT

We show that the market model $\left(\mathcal{S}=\left({\mathcal{S}}^{\left(1\right)},{\mathcal{S}}^{\left(2\right)},{\mathcal{S}}^{\left(3\right)}\right),\mathcal{B},\mathcal{C}\right)$ determined by (30), (18) and (29) is complete. First, we will derive the price dynamics of $S_t^{\left(\delta \right)}, t\in \left[0,T\right]$, as a limit of two dependent binomial trees. Let ${\xi }_k^{\left(j\right)}, k\in \mathcal{N},j=1, 2$, with ${\xi }_0^{\left(j\right)}=0$, be a sequence of pairs of independent random signs, $\mathbb{P}\left({\xi }_k^{\left(j\right)}=1\right)=1-\mathbb{P}\left({\xi }_k^{\left(j\right)}=-1\right)=1/2$15. Let $∆t=T/n$ and ${\zeta }_k^{\left(j,n\right)}={\sum }_{i=1}^k{\xi }_i^{\left(j\right)}, k=1,\dots ,n, {\zeta }_0^{\left(j,n\right)}=0.$ Consider the $\mathfrak{D}[0,T]$ processes

$${\mathbb{B}}_j^{\left(n\right)}=\left\{\begin{array}{ccc}{B}_{j,t}^{\left(n\right)}=\sqrt{∆t}{\zeta }_k^{\left(j,n\right)}& \mathrm{f}\mathrm{o}\mathrm{r} t\in \left[k∆t,\left(k+1\right)∆t\right),& k=0,\dots ,n-1,\\ B_{j,T}^{\left(n\right)}=\sqrt{∆t}{\zeta }_n^{\left(j,n\right)},& & \end{array}\right\}, j=1,2.$$

(31)

Then, as $n\uparrow \infty$, ${\mathbb{B}}_j^{\left(n\right)}$ converges weakly in $\mathfrak{D}[0,T]$ to ${\mathbb{B}}_i$ by DPIP. Let

$${\mathbb{A}}^{\left(\delta ,n\right)}=\left\{A_t^{\left(\delta ,n\right)}=\sqrt{1-{\delta }^2}{B}_{1,t}^{\left(n\right)}+\delta \left|B_{2,t}^{\left(n\right)}\right| , t\in \left[0,T\right]\right\} .$$

(32)

Then, ${\mathbb{A}}^{\left(\delta ,n\right)}$ converges weakly in $\mathfrak{D}\left[0,T\right]$ to ${\mathbb{A}}^{\left(\delta \right)}.$ Define the filtration ${\mathbb{F}}^{\left(\xi ,n\right)}=\left\{{\mathcal{F}}_k^{\left(\xi ,n\right)}=\sigma \left[\left({\xi }_0^{\left(1\right)},{\xi }_0^{\left(2\right)}\right),\dots ,\left({\xi }_k^{\left(1\right)},{\xi }_k^{\left(2\right)}\right)\right],k=0,\dots ,n\right\}$. For $k=0,\dots ,n-1,$ conditionally on ${\mathcal{F}}_k^{\left(\xi ,n\right)}$,

$$A_{(k+1)∆t}^{\left(\delta ,n\right)}-A_{k∆t}^{\left(\delta ,n\right)}=\left\{\begin{array}{c}\left[\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right]\sqrt{∆t} \mathrm{w}.\mathrm{p}. 1/4,\\ \left[\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right]\sqrt{∆t} \mathrm{w}.\mathrm{p}. 1/4,\\ \begin{array}{c}\left[-\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right]\sqrt{∆t} \mathrm{w}.\mathrm{p}. 1/4,\\ ⌈-\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)⌉\sqrt{∆t} \mathrm{w}.\mathrm{p}. 1/4 .\end{array}\end{array}\right.$$

(33)

Consider the triplet of dependent binomial pricing trees $S_{k∆t}^{\left(\delta ,n\right)}=\left(S_{k∆t}^{\left(\delta ,1,n\right)},S_{k∆t}^{\left(\delta ,2,n\right)},S_{k∆t}^{\left(\delta ,3,n\right)}\right)$, $k=0,\dots ,n,$ where

$$S_{k∆t}^{\left(\delta ,i,n\right)}=S_0^{\left(i\right)}\exp \left({\mu }^{\left(i\right)}k∆t+{\sigma }^{\left(i\right)}{A}_{k∆t}^{\left(\delta ,n\right)}\right), i=1, 2, 3 .$$

(34)

Conditionally on ${\mathcal{F}}_k^{\left(\xi ,n\right)},$

$$\begin{gathered} S_{\left(k+1\right)∆t}^{\left(\delta ,i,n\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left\{{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}\left(A_{\left(k+1\right)∆t}^{\left(\delta ,n\right)}-A_{k∆t}^{\left(\delta ,n\right)}\right)\right\} \\ =\left\{\begin{array}{c}\begin{array}{c}\begin{array}{cc}{S}_{\left(k+1\right)∆t}^{\left(\delta ,i,n;uu\right)}={\psi }_{k+1}^{\left(\delta ,i,uu,n\right)}& \mathrm{w}.\mathrm{p}. 1/4 ,\end{array}\\ \begin{array}{cc}{S}_{\left(k+1\right)∆t}^{\left(\delta ,i,n;ud\right)}={\psi }_{k+1}^{\left(\delta ,i,uu,n\right)}& \mathrm{w}.\mathrm{p}. 1/4 ,\end{array}\end{array}\\ \begin{array}{c}\begin{array}{cc}{S}_{\left(k+1\right)∆t}^{\left(\delta ,i,n;du\right)}={\psi }_{k+1}^{\left(\delta ,i,du,n\right)}& \mathrm{w}.\mathrm{p}. 1/4 ,\end{array}\\ \begin{array}{cc}{S}_{\left(k+1\right)∆t}^{\left(\delta ,i,n;dd\right)}={\psi }_{k+1}^{\left(\delta ,i,dd,n\right)}& \mathrm{w}.\mathrm{p}. 1/4 ,\end{array}\end{array}\end{array}\right. \end{gathered}$$

(35)

where

$$\begin{gathered} {\psi }_{k+1}^{\left(\delta ,i,uu,n\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}\left\{\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right\}\sqrt{∆t}\right] , \\ {\psi }_{k+1}^{\left(\delta ,i,ud,n\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}\left\{\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right\}\sqrt{∆t}\right] , \\ {\psi }_{k+1}^{\left(\delta ,i,du,n\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}\left\{-\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right\}\sqrt{∆t}\right] , \\ {\psi }_{k+1}^{\left(\delta ,i,dd,n\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}\left\{-\sqrt{1-{\delta }^2}+\delta \left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\right\}\sqrt{∆t}\right] , \end{gathered}$$

(36)

for $k=0,\dots ,n,i=1,2,3$.

We call (35) the Itô–McKean skew pricing tree. It generates a $\mathfrak{D}{\left[0,T\right]}^{\times 3}$-valued process ${\mathbb{S}}^{\left(\delta ,n;T\right)}=\left\{\left(S_t^{\left(\delta ,1,n\right)},S_t^{\left(\delta ,2,n\right)},S_t^{\left(\delta ,3,n\right)}\right),t\in \left[0,T\right]\right\}$, where

$$S_t^{\left(\delta ,i,n\right)}=\left\{S_{k∆t}^{\left(\delta ,i,n\right)} \mathrm{f}\mathrm{o}\mathrm{r} t\in \left[k∆t,\left(k+1\right)∆t\right), k=0,\dots ,n-1, S_T^{\left(\delta ,i,n\right)}=S_{n∆t}^{\left(\delta ,i,n\right)}\right\}, i=\mathrm{1,2},3 .$$

(37)

By the DPIP, ${\mathbb{S}}^{\left(\delta ,n:T\right)}$ weakly converges to ${\mathbb{S}}^{\left(\delta ,T\right)}=\left\{S_t^{\left(\delta \right)},t\in \left[0,T\right]\right\}$ as $n\uparrow \mathrm{\infty }.$ As stated in the Introduction, the parameter $\delta$ capturing the skewed behavior of the discrete prices processes in the natural world is therefore retained in the continuum limit.

Remark 3.

In the special case when the second random sign sequence ${\xi }_k^{\left(2\right)}=0, k\in \mathcal{N}$, the SRW tree (35) becomes a single binomial tree with up and down prices given, respectively, by $S_{(k+1)∆t}^{\left(\delta ,i,n,u\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathit{exp}\left\{{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}(\sqrt{1-{\delta }^2}+\delta )\sqrt{∆t} \right\}$ and $S_{(k+1)∆t}^{\left(\delta ,i,n,d\right)}=S_{k∆t}^{\left(\delta ,i,n\right)}\mathit{exp}\left\{{\mu }^{\left(i\right)}∆t+{\sigma }^{\left(i\right)}(-\sqrt{1-{\delta }^2}+\delta )\sqrt{∆t} \right\}$. The upward risk-neutral probability then becomes $q_{\left(k+1\right)∆t}^{\left(n;u\right)}=\left(1-\theta \sqrt{∆t}\right)/2$, where $\theta =\left(\overline{\mu }-{\overline{\sigma }}^2/2-r\right)/\overline{\sigma }$.

4.2. Option Pricing with the IMSPT

Having confirmed in Section 4.1 that the continuum limit of the discrete SBM model retains dependence on the skewness parameter $\delta$, for notational brevity, we drop the superscript $\delta$ on all $\delta$-dependent terms in the remainder of the paper.

Suppose the ECC $\mathcal{C}$, with the triplet of underlying assets $S_t, t\in \left[0,T\right]$ and maturity time $T$, has a price process $f_t=f\left(S_t,t\right), t\in \left[0,T\right)$, where $f\left(x,t\right)$ is a sufficiently smooth, real-valued function on ${\left(0,\infty \right)}^3\times \left[0,T\right).$ The payoff of $\mathcal{C}$ at maturity is $g\left(S_T\right)$ for some real-valued function $g\left(x\right),x\in {\left(0,\infty \right)}^3$. $f(·)$ and $g(·)$ satisfy the usual regularity conditions (Duffie 2001, chps. 5 and 6). Consider the finite price dynamics $f_{k∆t}^{\left(n\right)}=f\left(S_{k∆t}^{\left(n\right)},k∆t\right)$ of $\mathcal{C}$ on the lattice $k=0,\dots ,n.$ Conditionally on ${\mathcal{F}}_k^{\left(\xi ,n\right)},$

$$f_{\left(k+1\right)∆t}^{\left(n\right)}=\left\{\begin{array}{c}{f}_{\left(k+1\right)∆t}^{\left(n;uu\right)}=f\left(S_{\left(k+1\right)∆t}^{\left(n;uu\right)},\left(k+1\right)∆t\right) ,\\ f_{\left(k+1\right)∆t}^{\left(n;ud\right)}=f\left(S_{\left(k+1\right)∆t}^{\left(n;ud\right)},\left(k+1\right)∆t\right) ,\\ \begin{array}{c}{f}_{\left(k+1\right)∆t}^{\left(n;du\right)}=f\left(S_{\left(k+1\right)∆t}^{\left(n;du\right)},\left(k+1\right)∆t\right) ,\\ f_{\left(k+1\right)∆t}^{\left(n;dd\right)}=f\left(S_{\left(k+1\right)∆t}^{\left(n;dd\right)},\left(k+1\right)∆t\right) ,\end{array}\end{array}\right.$$

(38)

where $S_{\left(k+1\right)∆t}^{\left(n;uu\right)}=\left(S_{\left(k+1\right)∆t}^{\left(1,n;uu\right)},S_{\left(k+1\right)∆t}^{\left(2,n;uu\right)},S_{\left(k+1\right)∆t}^{\left(3,n;uu\right)}\right)$, with analogous definitions for $S_{\left(k+1\right)∆t}^{\left(n;ud\right)}$, $S_{\left(k+1\right)∆t}^{\left(n;du\right)}$, and $S_{\left(k+1\right)∆t}^{\left(n;dd\right)}$. For $k=0,1,\dots ,n-1$, consider the replicating risk-neutral portfolio $P_{k∆t}^{\left(n\right)}={\sum }_{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{k∆t}^{\left(i,n\right)}-f_{k∆t}^{\left(n\right)},$ where $D_{k∆t}^{\left(i,n\right)},i=\mathrm{1,2},3$ are the hedging deltas to be determined later on. Then, conditionally on ${\mathcal{F}}_k^{\left(\xi ,n\right)}$, $P_{\left(k+1\right)∆t}^{\left(n\right)}={\sum }_{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n\right)}-f_{\left(k+1\right)∆t}^{\left(n\right)}.$ From the risk-neutrality condition, it follows that

$$P_{\left(k+1\right)∆t}^{\left(n,uu\right)}=P_{\left(k+1\right)∆t}^{\left(n,ud\right)}=P_{\left(k+1\right)∆t}^{\left(n,du\right)}=P_{\left(k+1\right)∆t}^{\left(n,dd\right)} ,$$

(39)

where

$$\begin{array}{c}{P}_{\left(k+1\right)∆t}^{\left(n,uu\right)}={\displaystyle \sum _{i=1}^3}{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;uu\right)}-f_{\left(k+1\right)∆t}^{\left(n;uu\right)}, P_{\left(k+1\right)∆t}^{\left(n,ud\right)}={\displaystyle \sum _{i=1}^3}{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;ud\right)}-f_{\left(k+1\right)∆t}^{\left(n;ud\right)} , \\ P_{\left(k+1\right)∆t}^{\left(n,du\right)}={\displaystyle \sum _{i=1}^3}{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;du\right)}-f_{\left(k+1\right)∆t}^{\left(n;du\right)}, P_{\left(k+1\right)∆t}^{\left(n,dd\right)}={\displaystyle \sum _{i=1}^3}{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;dd\right)}-f_{\left(k+1\right)∆t}^{\left(n;dd\right)}.\end{array}$$

(40)

Then (39) and (40) imply that the hedging deltas $D_{k∆t}^{\left(i,n\right)},i=1, 2, 3 ,$ are given by

$$D_{k∆t}^{\left(i,n\right)}=\frac{{\mathbb{D}}_{k∆t}^{\left(i\right)}}{{\mathbb{D}}_{k∆t}}, i=1, 2, 3,$$

(41)

where the 3 × 3 determinants ${\mathbb{D}}_{k∆t}$, ${\mathbb{D}}_{k∆t}^{\left(i\right)}$, $i=1,2,3$ are defined in Appendix B. Given ${\mathcal{F}}_k^{\left(\xi ,n\right)}$, portfolio $P_{\left(k+1\right)∆t}^{\left(n\right)}={\sum }_{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{(k+1)∆t}^{\left(i,n\right)}-f_{\left(k+1\right)∆t}^{\left(n\right)}$ is riskless. This leads to the following:

$$P_{k∆t}^{\left(n\right)}=\sum _{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{k∆t}^{\left(i,n\right)}-f_{k∆t}^{\left(n\right)}=e^{-r∆t}{P}_{\left(k+1\right)∆t}^{\left(n,uu\right)}=e^{-r∆t}\sum _{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;uu\right)}-f_{\left(k+1\right)∆t}^{\left(n;uu\right)} .$$

(42)

Thus, given ${\mathcal{F}}_k^{\left(\xi ,n\right)},$ the derivative price $f_{k∆t}^{\left(n\right)}$ at $k∆t$ is given by

$$f_{k∆t}^{\left(n\right)}=\sum _{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{k∆t}^{\left(i,n\right)}-e^{-r∆t}\sum _{i=1}^3{D}_{k∆t}^{\left(i,n\right)}{S}_{\left(k+1\right)∆t}^{\left(i,n;uu\right)}+f_{\left(k+1\right)∆t}^{\left(n;uu\right)} .$$

(43)

Now, (41) and (43) imply that

$$f_{k∆t}^{\left(n\right)}=e^{-r∆t} \left(q_{\left(k+1\right)∆t}^{\left(n;uu\right)}{f}_{\left(k+1\right)∆t}^{\left(n;uu\right)}+q_{\left(k+1\right)∆t}^{\left(n;ud\right)}{f}_{\left(k+1\right)∆t}^{\left(n;ud\right)}+q_{\left(k+1\right)∆t}^{\left(n;du\right)}{f}_{\left(k+1\right)∆t}^{\left(n;du\right)}+q_{\left(k+1\right)∆t}^{\left(n;du\right)}{f}_{\left(k+1\right)∆t}^{\left(n;du\right)}\right).$$

(44)

In (44) the conditional-on-${\mathcal{F}}_k^{\left(\xi ,n\right)}$ risk-neutral probabilities $q_{\left(k+1\right)∆t}^{\left(n;uu\right)}, q_{\left(k+1\right)∆t}^{\left(n;ud\right)}, q_{\left(k+1\right)∆t}^{\left(n;du\right)}, \mathrm{a}\mathrm{n}\mathrm{d} q_{\left(k+1\right)∆t}^{\left(n;du\right)}$, have the form,

$$q_{\left(k+1\right)∆t}^{\left(n;uu\right)}=\frac{{\mathbb{q}}_{(k+1)∆t}^{\left(n;uu\right)}}{{\mathbb{q}}_{(k+1)∆t}}, q_{\left(k+1\right)∆t}^{\left(n;ud\right)}=\frac{{\mathbb{q}}_{(k+1)∆t}^{\left(n;ud\right)}}{{\mathbb{q}}_{(k+1)∆t}}, q_{\left(k+1\right)∆t}^{\left(n;du\right)}=\frac{{\mathbb{q}}_{(k+1)∆t}^{\left(n;du\right)}}{{\mathbb{q}}_{(k+1)∆t}}, q_{\left(k+1\right)∆t}^{\left(n;dd\right)}=\frac{{\mathbb{q}}_{(k+1)∆t}^{\left(n;dd\right)}}{{\mathbb{q}}_{(k+1)∆t}} .$$

(45)

where the terms in (45) are defined in Appendix B.

Remark 4.

Consider the special case $\delta =0.$ Then the underlying price process follows the Jarrow-Rudd binomial tree (Jarrow and Rudd 1983, chps. V, VI; Hull 2012, p. 442; Kim et al. 2016, 2019),

$$S_{(k+1)∆t}^{\left(1,n\right)}=S_{k∆t}^{\left(1,n\right)}\left\{\begin{array}{ccc}\exp \left({\mu }^{\left(1\right)}∆t+{\sigma }^{\left(1\right)}\sqrt{∆t}\right)& \mathrm{i}\mathrm{f}& {\xi }_k^{\left(1\right)}=1,\\ \exp \left({\mu }^{\left(1\right)}∆t-{\sigma }^{\left(1\right)}\sqrt{∆t}\right)& \mathrm{i}\mathrm{f}& {\xi }_k^{\left(1\right)}=-1,\end{array}\right.$$

(46)

having risk-neutral probabilities

$$q_{\left(k+1\right)∆t}^{\left(n;u\right)}=\frac{e^{{(r-\mu }^{\left(1\right)}) ∆t}-e^{{-\sigma }^{\left(1\right)}\sqrt{∆t}}}{e^{{\sigma }^{\left(1\right)}\sqrt{∆t}}-e^{-{\sigma }^{\left(1\right)}\sqrt{∆t}}}, q_{\left(k+1\right)∆t}^{\left(n;d\right)}=1-q_{\left(k+1\right)∆t}^{\left(n;u\right)}.$$

(47)

With $o\left(∆t\right)=0,q_{\left(k+1\right)∆t}^{\left(n;u\right)}=\left(1-{\theta }^{\left(1\right)}\sqrt{∆t}\right)/2$, where ${\theta }^{\left(1\right)}=\left({\mu }^{\left(1\right)}-{{\sigma }^{\left(1\right)}}^2/2-r\right)/{\sigma }^{\left(1\right)}$ is the market price of risk (Kim et al. 2016, 2019).

Remark 5.

Consider the special case $\delta =1.$ Then the underlying price process follows the path-dependent binomial tree; conditionally on${\mathcal{F}}_k^{\left({\xi }^{\left(2\right)},n\right)}=\sigma \left({\xi }_0^{\left(2\right)},\dots ,{\xi }_k^{\left(2\right)}\right),$

$$S_{(k+1)∆t}^{\left(2,n\right)}=S_{k∆t}^{(2,n)}\left\{\begin{array}{c}\exp \left({\mu }^{\left(2\right)}∆t+{\sigma }^{\left(2\right)}\left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\sqrt{∆t}\right) \mathrm{w}.\mathrm{p}. 1/2 ,\\ \exp \left({\mu }^{\left(2\right)}∆t+{\sigma }^{\left(2\right)}\left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\sqrt{∆t}\right) \mathrm{w}.\mathrm{p}. 1/2 ,\end{array}\right.$$

(48)

for $k=0, 1,\dots ,n-1.$ In this case, the risk-neutral probabilities are

$$\begin{gathered} q_{\left(k+1\right)∆t}^{\left(n;u\right)}=\frac{e^{{(r-\mu }^{\left(2\right)}) ∆t}-e^{{\sigma }^{\left(2\right)}\left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\sqrt{∆t}}}{e^{{\sigma }^{\left(2\right)}\left(\left|{\zeta }_k^{\left(2,n\right)}+1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\sqrt{∆t}}-e^{{\sigma }^{\left(2\right)}\left(\left|{\zeta }_k^{\left(2,n\right)}-1\right|-\left|{\zeta }_k^{\left(2,n\right)}\right|\right)\sqrt{∆t}}} , \\ {q}_{\left(k+1\right)∆t}^{\left(n;d\right)}=1-q_{\left(k+1\right)∆t}^{\left(n;u\right)} , \end{gathered}$$

(49)

given ${\mathcal{F}}_k^{\left({\xi }^{\left(2\right)},n\right)}=\sigma \left({\xi }_0^{\left(2\right)},\dots ,{\xi }_k^{\left(2\right)}\right), k=0,\dots ,n-1.$

5. Empirical Examples

In Section 5.1, we show how to generate an SBM based upon a skew random walk, which is equal in distribution to Itô–McKean SBM. In Section 5.2, we describe how to fit an empirical return series to this skew random walk model to obtain the required parameter values for the discrete option pricing model of Section 4.2. An empirical example of the generation of option prices is given in Section 5.3.

5.1. Generating an SBM via a Skew Random Walk

Using the Harrison–Shepp functional limit theorem (HSFLT: Harrison and Shepp 1981; see also Cherny et al. 2003 for a more general principle), we elucidate a discrete-time process defined on a finite time interval $\left[0,T\right]$ that converges weakly in $\mathcal{D}[0,T]$ (Skorokhod 1956; Billingsley 1999, chp. 3) to Itô–McKean SBM16. Let ${\mathbb{M}}^{\left(\alpha \right)}=\left\{M_k^{\left(\alpha \right)}\in \mathbb{Z},k\in {\mathcal{N}}_0\right\}, \alpha \in \left(0,1\right), {\mathcal{N}}_0=\left\{0,1,\dots \right\},\mathbb{Z}=\left\{0,\pm 1,\pm 2,\right\}$ be a Markov chain, with $M_0^{\left(\alpha \right)}=0$, having the transition probabilities:

$$\begin{gathered} \mathbb{P}\left(M_{k+1 }^{\left(\alpha \right)}=i+1 | M_k^{\left(\alpha \right)}=i\right)=\left\{\begin{array}{ccc}1/2& \mathrm{i}\mathrm{f}& i\ne 0 ,\\ \alpha & \mathrm{i}\mathrm{f}& i=0 ,\end{array}\right. \\ \mathbb{P}\left(M_{k+1 }^{\left(\alpha \right)}=i-1 | M_k^{\left(\alpha \right)}=i\right)=\left\{\begin{array}{ccc}1/2& \mathrm{i}\mathrm{f}& i\ne 0 ,\\ 1-\alpha & \mathrm{i}\mathrm{f}& i=0 ,\end{array}\right. \end{gathered}$$

(50)

for $k\in {\mathcal{N}}_0$. ${\mathbb{M}}^{\left(\alpha \right)}$ is called a skew random walk (SRW) with parameter $\alpha \in \left(0, 1\right)$ (Harrison and Shepp 1981; Cherny et al. 2003). If $\alpha =1/2$, ${\mathbb{M}}^{\left(\alpha \right)}$ is a random walk. Relevant properties of the SRW ${\mathbb{M}}^{\left(\alpha \right)}$ are discussed in Appendix C. With $n∆t=T,$ let $X_{k∆t}^{\left(\alpha \right)}=\sqrt{∆t}{M}_k^{\left(\alpha \right)}, k=1,\dots ,n, \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} X_0^{\left(\alpha \right)}=0$. Consider the following processes in $\mathcal{D}[0,T]$,

$${\mathbb{B}}_{[0,T]}^{\left(\alpha , n\right)}=\left\{\begin{array}{lll}{\mathbb{B}}_{\left[0,T\right]}^{\left(\alpha ,n\right)}\left(t\right)=X_{k∆t}^{\left(\alpha \right)} ,& \mathrm{f}\mathrm{o}\mathrm{r} t\in \left[k∆t,\left(k+1\right)∆t\right),& k=0,\dots ,n-1,\\ {\mathbb{B}}_{\left[0,T\right]}^{\left(\alpha ,n\right)}\left(T\right)=X_{n∆t}^{\left(\alpha \right)} . & & \end{array}\right\}$$

(51)

Then HSFLT states that, as $n\uparrow \mathrm{\infty }$, ${\mathbb{B}}_{\left[0,T\right]}^{\left(\alpha ,n\right)}$ converges weakly in $\mathcal{D}\left[0,T\right]$ to the SBM ${\mathbb{B}}_{[0,T]}^{\left(\alpha \right)}=\left\{B_t^{\left(\alpha \right)},t\in \left[0,T\right]\right\}$.

Consider the pricing tree $S_{k∆t}^{\left(\alpha ,n\right)},k=0,1,\dots ,n$ given by (Hu et al. 2022):

$$\begin{gathered} S_{k∆t}^{\left(\alpha ,n\right)}=S_0\exp \left\{\left({\mu }_{\alpha }-\frac{{{\sigma }_{\alpha }}^2}{2}\right)k∆t+{\sigma }_{\alpha }{M}_k^{\left(\alpha \right)}\sqrt{∆t}\right\} , \\ {\mu }_{\alpha }\in R, {\sigma }_{\alpha }\in R, k=0,\dots ,n, n∆t=T , \end{gathered}$$

(52)

adapted to the filtration ${\mathbb{F}}^{\left(\alpha \right)}=\left\{{\mathcal{F}}_k^{\left(\alpha \right)}=\sigma \left(M_j^{\left(\alpha \right)};j=1,\dots ,k\right),k\in \mathcal{N},{\mathcal{F}}_0^{\left(\alpha \right)}=\left\{\mathrm{\varnothing },\mathsf{\Omega }\right\}\right\}$. Define the $\mathcal{D}\left[0,T\right]$-process:

$${\mathbb{S}}_{\left[0,T\right]}^{\left(\alpha , n\right)}=\left\{\begin{array}{lll}{\mathbb{S}}_{\left[0,T\right]}^{\left(\alpha , n\right)}\left(t\right)=S_{k∆t}^{\left(\alpha ,n\right)} ,& \mathrm{f}\mathrm{o}\mathrm{r} t\in \left[k∆t,\left(k+1\right)∆t\right),& k=0,\dots ,n-1 ,\\ {\mathbb{S}}_{\left[0,T\right]}^{\left(\alpha , n\right)}\left(T\right)=S_{n∆t}^{\left(\alpha ,n\right)} . & & \end{array}\right\}$$

(53)

Then, by HSFLT, ${\mathbb{S}}_{\left[0,T\right]}^{\left(\alpha , n\right)}$ converges weakly in $\mathcal{D}\left[0,T\right]$ to a GSBM.

$${\mathbb{S}}_{\left[0,T\right]}^{\left(\alpha \right)}=\left\{S_t^{\left(\alpha \right)}=S_0\exp \left\{\left({\mu }_{\alpha }-\frac{{{\sigma }_{\alpha }}^2}{2}\right)t+{\sigma }_{\alpha }{B}_t^{\left(\alpha \right)}\right\}, t\in [0,T]\right\}.$$

(54)

With ${\mu }^{\left(i\right)}\equiv {\mu }_{\alpha }^{\left(i\right)}-{\left({\mu }_{\alpha }^{\left(i\right)}\right)}^2/2$ and $\alpha =\left(1+\delta \right)/2$, then by (11) the price process (54) is equal in distribution to the price process (30) driven by Itô–McKean SBM.

Note that the coefficient ${\sigma }_{\alpha }$ appearing in (54) and in its discrete version (52) (and hence in (30) and its discrete version (35)) is not the usual volatility, which appears when the driving random motion is Brownian. When the driving motion is Brownian ($\alpha =1/2$), ${\sigma }_{1/2}{B}_t^{\left(1/2\right)}$ and $-{\sigma }_{1/2}{B}_t^{\left(1/2\right)}$ have the same distribution. However, when the driving motion is skew Brownian ($\alpha \ne 1/2$), then ${\sigma }_{\alpha }{B}_t^{\left(\alpha \right)}$ and $-{\sigma }_{\alpha }{B}_t^{\left(\alpha \right)}$ will not have the same distribution. Thus, when the driving motion is Brownian, the restriction ${\sigma }_{1/2}>0$ can be imposed, since any negative sign can be inverted by the change $B_t^{\left(1/2\right)}\to -B_t^{\left(1/2\right)}$, with no change in the resulting random distribution. For SBM, this is not possible and the sign of ${\sigma }_{\alpha }$ is important. Therefore, ${\sigma }_{\alpha }$ is not volatility in the usual sense but rather a “diffusion” parameter, ${\sigma }_{\alpha }\in R$.

5.2. Fitting Itô–McKean SBM to ETF Return Values

Let $S_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$ denote an empirical price series and $R_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$ and $r_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$ denote, respectively, its cumulative and daily log-returns. Fitting the price series to (52) involves solving both of the following:

$$\begin{gathered} R_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}=\left(\mu -\frac{{\sigma }^2}{2}\right)k∆t+\sigma M_k^{\left(\alpha \right)}\sqrt{∆t} \equiv R_{k∆t}^{\left(\mathrm{t}\mathrm{h}\right)}(\mu ,\sigma ,\alpha ), \\ {r}_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}=\left(\mu -\frac{{\sigma }^2}{2}\right)∆t+\sigma ∆M_k^{\left(\alpha \right)}\sqrt{∆t} \equiv r_{k∆t}^{\left(\mathrm{t}\mathrm{h}\right)}(\mu ,\sigma ,\alpha ), \end{gathered}$$

(55)

for the parameters $\mu$, $\sigma$ and $\alpha$. Fitting (55) to data requires identifying best fit values for the parameters $\alpha$, ${\mu }_{\alpha }$, ${\sigma }_{\alpha }$ over an ensemble of BM trajectories, $M_k^{\left(\alpha \right)}$. Appendix D presents a procedure for fitting daily closing prices. That appendix also presents the results of testing the procedure on synthetic data sets produced from (50) and (52) with known values of $\alpha$, ${\mu }_{\alpha }$ and ${\sigma }_{\alpha }$. The test results show that ${\sigma }_{\alpha }$ can be computed very accurately for any single price trajectory. Since the standard deviation of daily returns is larger than the mean value $\mu$, estimation of $\mu$ from a single price trajectory is much less accurate. Estimation of $\alpha$ from a single price trace is slightly better than for estimation of $\mu$; critically, it is possible to infer whether $\alpha$ differs from 0.5, and in what direction.

To apply the fitting procedure to an empirical data set, we consider data (source: Bloomberg Professional Services, accessed 24 August 2021, 10:00 PM EDT) for the SPDR S&P 500 ETF Trust (SPY) covering the period 3 January 2000 to 17 February 2022. The results of the fits, using a 252-day moving window, to the parameters ${\widehat{\sigma }}_t$, ${\widehat{\alpha }}_t$ and ${\widehat{\mu }}_t$ for SPY are given in Figure 1. The parameters ${\widehat{\alpha }}_t$ and ${\widehat{\mu }}_t$ were computed using the minimization (A4) of Appendix D, with ${\sigma }^{*}={\sigma }_t^{*}$ set to the realized average value of ${\widehat{\sigma }}_t$ over each respective window.17 The time series of mean square errors (MSE) for this minimization are also plotted. To smooth the results for ${\widehat{\alpha }}_t$ and ${\widehat{\mu }}_t$, we computed the median value time series, ${\widehat{\alpha }}_t^{\left(\mathrm{M}\mathrm{e}\mathrm{d}\right)}$ and ${\widehat{\mu }}_t^{\left(\mathrm{M}\mathrm{e}\mathrm{d}\right)}$ using a 1-month moving window. The smoothed series are also shown in Figure 1. The effects of the financial crises of 2008 and the COVID-19 pandemic are discernible in each time series, although they are almost “washed out” of the smoothed time series ${\widehat{\alpha }}_t^{\left(\mathrm{M}\mathrm{e}\mathrm{d}\right)}$ and ${\widehat{\mu }}_t^{\left(\mathrm{M}\mathrm{e}\mathrm{d}\right)}$. Mean and standard deviation values for each of these parameter time series are given in

Table 2. Mean values and standard deviations for the fitted parameters.

	${\widehat{\mathit{\sigma }}}_{\mathit{t}}$	${\widehat{\mathit{\mu }}}_{\mathit{t}}$	${\widehat{\mathit{\alpha }}}_{\mathit{t}}$	${\widehat{\mathit{\mu }}}_{\mathit{t}}^{\left(\mathbf{M}\mathbf{e}\mathbf{d}\right)}$	${\widehat{\mathit{\alpha }}}_{\mathit{t}}^{\left(\mathbf{M}\mathbf{e}\mathbf{d}\right)}$	MSE
Mean	0.177	0.001	0.549	0.001	0.549	0.0009
St. Dev.	0.083	0.021	0.004	0.010	0.002	0.0010

. To reasonably high precision, ${\widehat{\alpha }}_t$ is consistent with a value of 0.549, indicating that SPY acted as an SBM with positive skewness over this long time period.

We desire to construct the pricing tree

$$S_{k∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}=S_{(k-1)∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}\exp \left\{\left(\widehat{\mu }-\frac{{\widehat{\sigma }}^2}{2}\right)∆t+\widehat{\sigma }\sqrt{∆t}\left(M_k^{\left(\widehat{\alpha }\right)}-M_{k-1}^{\left(\widehat{\alpha }\right)}\right)\right\} ,$$

(56)

for $k=1,\dots ,n, n∆t=T$, with $S_0^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}=S_0^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$ that approximates an empirical price series $S_{k∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$. To do so, we need the “best fit” Markov trajectory $\left\{M_0^{\left(\widehat{\alpha }\right)}=0, M_1^{\left(\widehat{\alpha }\right)},\dots , M_{k-1}^{\left(\widehat{\alpha }\right)}\right\}$. Consider the daily log-returns defined by (56):

$$\begin{gathered} r_{j∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}=\ln \left(S_{j∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}/S_{\left(j-1\right)∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}\right)=\left(\widehat{\mu }-\frac{{\widehat{\sigma }}^2}{2}\right)∆t+\widehat{\sigma }\sqrt{∆t}\left(M_j^{\left(\widehat{\alpha }\right)}-M_{j-1}^{\left(\widehat{\alpha }\right)}\right) , \\ j=1,\dots ,k, r_0^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}=0. \end{gathered}$$

(57)

This implies that

$$M_j^{\left(\widehat{\alpha }\right)}=M_{j-1}^{\left(\widehat{\alpha }\right)}+\frac{r_{j∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}-\left(\widehat{\mu }-{\widehat{\sigma }}^2/2\right)∆t}{\widehat{\sigma }\sqrt{∆t}}, j=0,\dots ,k, M_0^{\left(\widehat{\alpha }\right)}=0.$$

(58)

Equating $r_{j∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}$ with the observed daily returns, $r_{j∆t}^{\left(\widehat{\alpha },\widehat{\mu },\widehat{\sigma },n\right)}=r_{j∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}=R_{j∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}-R_{j-1∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)},j=2,3,\dots ,k,$ $r_{1∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}=R_{1∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}$, enables computation of a sequence, $M_j^{\left(\widehat{\alpha }\right)},j=1,\dots ,k$ from (58). However, the values $M_j^{\left(\widehat{\alpha }\right)}$ will (undoubtedly) not be in $\mathbb{Z}=\left\{0,\pm 1,\pm 2,\dots \right\}$. Consequently, we estimate the chain $M_j^{\left(\widehat{\alpha }\right)}$ using the revised formula, as follows:

$${\widehat{M}}_j^{\left(\widehat{\alpha }\right)}={\widehat{M}}_{j-1}^{\left(\widehat{\alpha }\right)}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(\frac{r_{j∆t}^{\left(\mathrm{e}\mathrm{m}\mathrm{p}\right)}-\left(\widehat{\mu }-{\widehat{\sigma }}^2/2\right)∆t}{\widehat{\sigma }\sqrt{∆t}}\right), j=0,\dots ,k, {\widehat{M}}_0^{\left(\widehat{\alpha }\right)}=0 ,$$

(59)

where $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(·\right)$ is defined in (13). Use of (59) enables computation of a data-fitting Markov chain for $j=1,\dots ,n,$ where $n∆t=T$ is the size of the window.

To illustrate, we used the parameter estimates18 $\widehat{\mu }=0.0256$ and $\widehat{\sigma }=0.1486$ obtained for 24 August 2021 to generate the pricing tree via (56) and (59) over the period 25 August 2020 to 24 August 202119. The resulting price series $S^{\left(\widehat{\mu },\widehat{\sigma },n\right)}$ is shown in Figure 2, where it is compared to that of the empirical SPY price, $S^{\left(\mathrm{S}\mathrm{P}\mathrm{Y}\right)}$. The R-squared value of the fit is 0.94, the RMSE is 10.2.

5.3. Option Pricing

We consider an empirical example of option pricing under the IMSPT model. As noted, we assume the same skewness parameter $\delta$ for all three risky assets. To ensure the accuracy of this assumption, we choose three ETFs that track sub-indices of the same composite index. Specifically, we consider the S&P Global 1200 index (SPG1200) and the ETFs: SPY, iShares Europe ETF (IEV), and iShares JPX-NIKKEI 400 ETF (JPXN), which track the three major components20 of this index.

There are strong numerical considerations behind our choice of three ETFs tracking sub-indices of a composite index. Consider an alternate choice consisting of the S&P 500 index and the three ETFs: SPY, IVV (iShares Core S&P 500 ETF), and VOO (Vanguard S&P 500 ETF) each of which tracks the full index. The similarity of the price processes governing these three ETFs leads to numerical instability (division by very small denominators) in the computations of the risk neutral probabilities (45). This can be seen more immediately in the case of two-risky assets. Computation of the riskless rate (45) diverges whenever ${\sigma }^{\left(1\right)}-{\sigma }^{\left(2\right)}\downarrow 0$, which happens frequently when the ETFs closely track the same index. By choosing ETFs that track separate components of a composite index, we ensure that the volatilities of each ETF have both a systemic and an idiosyncratic component. In practice, the (differing) systemic components will ensure that the values of the required denominators remain sufficiently large compared to the values of their numerator terms.

We compute $\delta$ assuming that the dynamics of the SPG1200 index follows the GSBM

$$S_t^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}=S_0^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}\exp \left({\mu }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}t+{\sigma }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}{A}_t^{\left({\delta }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}\right)}\right), t\in \left[0,T\right], {\mu }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}\in R, {\sigma }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}>0 ,$$

(60)

where $A_t^{\left({\delta }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}\right)}$ is given by (9). As noted in Section 3, ${\mathbb{A}}^{\left(\delta \right)}\stackrel{d}{=}{\mathbb{B}}^{\left(\alpha =\left(1+\delta \right)/2\right)}$ for $\delta \in (-1, 1)$. As the processes on a time interval $[0,T]$ defined by either (60) or (54) generate the same probability law (Corns and Satchell 2007), we use the procedure outlined in the previous empirical example on the SPG1200 index values over the period 13 September 2016 through 14 September 2021 using 1-year moving windows to generate a time series ${\widehat{\delta }}_t^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}=2{\widehat{\alpha }}_t^{(\mathrm{S}\mathrm{P}\mathrm{G},\mathrm{M}\mathrm{e}\mathrm{d})}-1$, where, as above, “Med” refers to a moving 1-month median value. As in the previous empirical example, we restrict values of $\widehat{\alpha }$ to the range [0.45, 0.55]. The resultant time series, ${\widehat{\delta }}_t^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}$, is shown in Figure 3.

We model the price dynamics of each ETF over the one-year (252 trading day) period 15 September 2020–14 September 2021 by the following:

$$S_t^{\left(i\right)}=S_0^{\left(i\right)}\exp \left({\mu }^{\left(i\right)}t+{\sigma }^{\left(i\right)}{A}_t^{\left({\delta }^{\left(\mathrm{S}\mathrm{P}\mathrm{G}\right)}\right)}\right), i=\mathrm{S}\mathrm{P}\mathrm{Y},\mathrm{I}\mathrm{E}\mathrm{V},\mathrm{J}\mathrm{P}\mathrm{X}\mathrm{N},$$

(61)

where ${\delta }^{\left(SPG\right)}=0.102$ is the value obtained for 14 September 2021. To compute the values ${\mu }^{\left(i\right)}$ and ${\sigma }^{\left(i\right)}$, we perform least-squares minimization on the relation:

$$R_{k∆t}^{\left(i;\mathrm{e}\mathrm{m}\mathrm{p}\right)}-{\mu }^{\left(i\right)}k∆t-{\sigma }^{\left(i\right)}{\delta }^{\left(SPG\right)}\sqrt{\frac{2k∆t}{\pi }}={\epsilon }_{k∆t}^{\left(i\right)} .$$

(62)

Table 3. The constrained minimization results from (62).

	SPY	IEV	JPXN
$\widehat{\mu }$	$0.32$	$0.31$	$-0.069$
$\widehat{\sigma }$	$-0.090$	$-0.23$	$2.8$
$R^2$	$0.96$	$0.89$	$0.64$
RMSE	$0.018$	$0.032$	$0.033$

presents the fitted estimates ${\widehat{\mu }}^{\left(i\right)}$ and ${\widehat{\sigma }}^{\left(i\right)}, i=\mathrm{S}\mathrm{P}\mathrm{Y},\mathrm{I}\mathrm{E}\mathrm{V},\mathrm{J}\mathrm{P}\mathrm{X}\mathrm{N},$ and the $R^2$ and root mean square error (RMSE) values for each minimization. Note that, for SPY and IEV, the fitted value $\widehat{\sigma }$ is negative; their cumulative returns have a negative dependence on ${\delta }^{\left(SPG\right)}$. Plots of these fits are shown (and briefly discussed) in Appendix E.

Consider a European put option with payoff $f_T^{\left(\mathrm{p}\mathrm{u}\mathrm{t}\right)}=\max \left(0,K-\min \left(S_T^{\left(1\right)},S_T^{\left(2\right)},S_T^{\left(3\right)}\right)\right)$. Using (dividend adjusted) starting prices for SPY ($432.51), IEV ($52.25) and JPXN ($76.09) for 14 September 2021, we compute the risk neutral probabilities (45) and form the risk-neutral tree (44) over a time period $\left[0,T\right]$, where $T$ is the time to maturity in days. In Figure 4, we plot the put option price $f_0^{\left(\mathrm{p}\mathrm{u}\mathrm{t}\right)}$ as a function of $T$ and moneyness $M^{\left(\mathrm{p}\right)}=K/\min \left(S_0^{\left(1\right)},S_0^{\left(2\right)},S_0^{\left(3\right)}\right)$ over the range $T\in \left(\mathrm{0,100}\right)$, $M^{\left(\mathrm{p}\right)}\in \left(\mathrm{0.5,1.5}\right)$. Repeating this for a European call with payoff $f_T^{\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\right)}=\max \left(0,\max \left(S_T^{\left(1\right)},S_T^{\left(2\right)},S_T^{\left(3\right)}\right)-K\right)$, Figure 4 also plots $f_0^{\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\right)}$ as a function of $T$ and moneyness $M^{\left(\mathrm{c}\right)}=K/\max \left(S_0^{\left(1\right)},S_0^{\left(2\right)},S_0^{\left(3\right)}\right)$, over the same range of maturity and moneyness values. The difference in price range for the put and call options represents the disparity in the computed prices of the ETFs at maturity, which in turn is based upon differences in their starting prices, drift, $\widehat{\mu }$, and diffusion parameter, $\widehat{\sigma }$, values. For the specific values in Table 3, the ETFs IEV and JPXN dominate the computation of $f_0^{\left(\mathrm{p}\mathrm{u}\mathrm{t}\right)}$, while SPY dominates the computation of $f_0^{\left(\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\right)}$.

6. Conclusions

Continuous-time formulations of derivative pricing, while attractive in terms of their theoretical bases (semimartingales, Lévy processes, and the fundamental theorem of asset pricing) and the convenience of smoothly varying, closed-form solutions, are generally limited in their ability to capture many of the stylized facts of real asset dynamics and/or their ability to achieve arbitrage-free, market-complete models. It is our view that any continuous time model should only be considered as a benchmark model for a more precise discrete pricing model that is: defined in the natural world; capable of capturing valuable market characteristics; extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities with no loss of natural world parameters; and market-complete. Previous work (Hu et al. 2020a, 2020b, 2022; Kim et al. 2016, 2019) has variously demonstrated the capabilities of discrete models to achieve these goals.

We have presented here a binomial tree model based upon a single Itô–McKean skew random walk (IMSRW) driving process. Although an arbitrage-free, market-complete model is achieved by requiring a Black–Scholes–Merton market containing two risky assets, to achieve a practical hedging strategy, it is necessary to include three risky assets in the replicating portfolio. The requirement of a single IMSRW driving process ensures that the riskless asset return rate is independent of the IMSRW. The model contains seven parameters: the drift ${\mu }^{\left(i\right)}$ and diffusion values ${\sigma }^{\left(i\right)}$, $i=1, 2, 3$, for the risky asset price processes (30) and the parameter $\delta \in \left(-1, 1\right)$ (equivalently $\alpha =(1+\delta )/2$) controlling the deviation of the IMSRW from a standard random walk. The parameter $\delta$ therefore controls the central moments of the IMSRW driving process, providing the ability to match central moments (mean through kurtosis) of realistic price processes. As a result, in our arbitrage-free, market-complete model, the underlying price processes exhibit the stylized facts of skewness and distributional heavy tails of asset returns, without passing to an incomplete market model.

Using an empirical example, we demonstrate the computation of option prices from the theoretical model. It is natural to wonder how such prices compare to existing approaches. Such a comparison is not straightforward, as most existing models (e.g., the standard Black–Scholes–Merton model, or binomial models of the Cox–Ross–Rubenstein or Jarrow–Rudd style) are designed for a market containing only a single risky asset. Most existing approaches, such as the three mentioned above, are based on the assumption that log returns are normally distributed. It is well known that this is not the case and that the return distributions are generally skewed and have heavier tails than the normal distribution. As our model is developed specifically to capture skewness and kurtosis behavior, it offers greater flexibility and, consequently, accuracy. Exact details depend on the asset considered. It would require a carefully designed study with appropriately considered assets to quantify the improvement that our proposed method provides against others, which we leave for a future investigation.

Our development was initially motivated by the continuous-time pricing model of Corns and Satchell (2007), which combined standard Brownian motion with skew Brownian motion. As the equivalent martingale obtained in their work only provides hedging against the risks associated with the standard Brownian motion, their model is market-incomplete. The approach to achieving market completeness in our discrete formulation indicates how one might achieve the same in the Corns–Satchell formulation, should one wish to do so.