The Pricing Formula for Exotic Options Based on a Discrete Investment Strategy

Abstract

This paper proposes an option pricing model under a discrete investment strategy. From the perspective of investors’ initiative, we assume that the investor can take early actions before the underlying asset price reaches the strike level and trade a certain proportion of the underlying asset within the validity period of the option. Building on the Black–Scholes framework, we derive a closed-form pricing formula for options under the discrete investment strategy and compare the resulting prices with those obtained from the classical Black–Scholes model. Furthermore, after parameter calibration, our model exhibits smaller fitting errors when applied to market data. By incorporating investor agency into the pricing framework, this study further extends option pricing theory and offers theoretical insights into the prevention of financial risks.

1. Introduction

People are transcending the confines of classic economic paradigms, such as the rational person and efficient market hypothesis. Classic economic models prove inadequate in navigating the dynamic landscape of the global economy and its associated risks, and people are also changing their behavior to cope with risks. Options, as a hedging tool, are commonly used. The research on option pricing has undergone a long period of development. The earliest option pricing theory was proposed by French economist Bachelier in 1900 [1]. Bachelier first introduced the random walk concept to model the stochastic nature of stock price movements. He assumed that the stock price changes followed a standard Brownian motion, characterized by no drift and a variance of ${\mathsf{\sigma }}^2$ per time unit. Based on the assumption, he derived the call option pricing formula as follows

$$\mathrm{P}[\mathrm{S},\mathrm{t}]=\mathrm{S}\times \mathsf{\theta }\left[{\displaystyle \frac{\mathrm{S}-\mathrm{X}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right]-\mathrm{X}\times \mathsf{\theta }\left[{\displaystyle \frac{\mathrm{S}-\mathrm{X}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right]+\mathsf{\sigma }\sqrt{\mathrm{t}}\times \mathsf{\phi }\left[{\displaystyle \frac{\mathrm{S}-\mathrm{X}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right],$$

where $\mathrm{P}[\mathrm{S},\mathrm{t}]$ denotes the option price, where $\mathrm{t}$ is the time, $\mathrm{S}$ denotes the stock price, $\mathrm{X}$ denotes the strike price, $\mathsf{\theta }[·]$ and $\mathsf{\phi }[·]$ denote the standard normal distribution and its probability density function, respectively. However, his option pricing model had some limitations. Especially when the stock price changes conform to absolute Brownian motion, this model permits the stock price to take negative values and assumes that the expected price change on average is zero, which does not align with reality. Therefore, further refinement and development of option pricing models are necessary.

In 1961, Sprenkle assumed that stock prices followed a lognormal distribution with a constant mean and variance, allowing for positive drift in stock prices [2]. Consequently, in this scenario, positive interest rates and risk aversion exist. Based on this, the call option pricing formula takes the form

$$\mathrm{P}[\mathrm{S},\mathrm{t}]=\mathrm{S}\times {\mathrm{e}}^{\mathsf{\alpha }\mathrm{t}}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }+\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right]-(1-\mathsf{\pi })\times \mathrm{X}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }-\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right],$$

where $\mathsf{\pi }$ is the adjustment amount for market “price leverage”, and $\mathsf{\alpha }$ is the expected stock return rate. Like Bachelier’s option pricing model, Sprenkle’s model also overlooks the impact of the time value of money.

In 1964, Boness incorporated the time value of money into option pricing, resulting in the call option pricing formula taking the following form [3], i.e.,

$$\mathrm{P}[\mathrm{S},\mathrm{t}]=\mathrm{S}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }+\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right]-{\mathrm{e}}^{-\mathsf{\alpha }\mathrm{r}}\times \mathrm{X}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }-\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right].$$

Although Boness’ option pricing model made significant strides compared to previous research, it overlooked the need for different levels of risk measurement between stocks and options. Therefore, it still has certain limitations.

In 1965, Samuelson assumed that stock price movements adhere to a Brownian motion with a growth rate $\mathrm{p}$ ($\mathrm{p}>0$), implying positive interest rates and a risk premium. He also considered different risk levels for stocks and options [4]. Based on this, the call option pricing formula takes the following form

$$\mathrm{P}[\mathrm{S},\mathrm{t}]={\mathrm{e}}^{\mathsf{\alpha }-\mathsf{\beta }}\times \mathrm{S}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }+\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right]-{\mathrm{e}}^{-\mathsf{\beta }\mathrm{r}}\times \mathrm{X}\times \mathsf{\theta }\left[{\displaystyle \frac{ln\left[\frac{\mathrm{S}}{\mathrm{X}}\right]+(\mathsf{\alpha }-\frac{1}{2}{\mathsf{\sigma }}^2)\mathrm{t}}{\mathsf{\sigma }\sqrt{\mathrm{t}}}}\right],$$

where $\mathsf{\alpha }$ represents the expected return rate of the stock, and $\mathsf{\beta }$ denotes the expected return rate of the option, with $0<\mathsf{\alpha }<\mathsf{\beta }$, $\mathsf{\alpha }$ and $\mathsf{\beta }$ depend on the investor’s personal preferences. When $\mathsf{\alpha }=\mathsf{\beta }$, Samuelson’s option pricing model becomes identical to Boness’ option pricing model.

These pricing models laid the foundation for subsequent research. The Black–Scholes model proposed by Fischer Black and Myron Scholes in 1973 is widely regarded as the most widely used option pricing model [5]. By constructing a hedging portfolio that combines positions in the underlying asset and the derivative at a given point in time so that the portfolio’s return matches the risk-free rate, they derived the Black–Scholes differential equation describing the dynamic behavior of option prices. Its solution is given by

$$\mathrm{V}[\mathrm{S},\mathrm{t}]=\mathrm{SN}[-{\mathrm{d}}_1]-{\mathrm{X}}_{\mathrm{e}}{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\mathrm{N}[-{\mathrm{d}}_2],$$

where

$${\mathrm{d}}_2={\displaystyle \frac{ln\left[\frac{{\mathrm{X}}_{\mathrm{e}}}{\mathrm{S}}\right]-(\mathrm{T}-\mathrm{t})\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}},{\mathrm{d}}_1={\mathrm{d}}_2-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}},$$

and $\mathrm{S}$ denotes the underlying asset price, ${\mathrm{X}}_{\mathrm{e}}$ is the strike price, $\mathrm{T}$ represents the option maturity, $\mathrm{r}$ stands for the risk-free interest rate, and $\mathsf{\sigma }$ is the volatility of the underlying asset price. Here $\mathrm{N}[·]$ denotes the cumulative distribution function of a standard normal random variable, and $\mathrm{N}[-{\mathrm{d}}_{\mathrm{i}}]=1-\mathrm{N}\left[{\mathrm{d}}_{\mathrm{i}}\right])$ for $\mathrm{i}=1,2$. The Black–Scholes model is based on the fundamental principle of no-arbitrage pricing, which assumes the absence of arbitrage opportunities in an efficient market. It relies on several stringent assumptions, including zero transaction costs, no tax, and a constant interest rate, among others. However, numerous scholars argue that these assumptions may not accurately reflect real-world capital markets. Consequently, much research begins by exploring hypothetical conditions.

To better align with real-world conditions, firstly, many scholars have proposed improved option pricing models that account for transaction costs [6,7,8]. Secondly, some scholars have conducted in-depth research on option pricing formulas under non-constant volatility [9,10,11,12,13]. Moreover, several models have already incorporated fluctuating interest rates into option pricing [14,15]. Additionally, considering that the underlying asset often distributes dividends throughout the validity period, many scholars have discussed the pricing of options with dividend payments [16,17,18,19].

However, much of the research on option pricing starts with relaxing assumptions and establishing new models, with few in-depth analyses from the perspective of investment strategies. Qiao et al. proposed an option pricing model under a continuous investment strategy from the perspective of investment strategies in 2022 [20]. In their proposed investment strategy, the threshold values are set higher than the option’s exercise price, meaning that investors will only take action when the stock price surpasses the strike price. However, the lower the stock’s acquisition cost, the more profit investors can gain from the stock’s appreciation. Furthermore, investor behavior does not necessarily occur continuously. Building on the preceding discussions, this paper attempts to explore the following aspects from the perspective of discrete investment strategies:

(1) Is it effective for investors to take action before the stock price reaches the strike price?

(2) Can we assume the investors must act during the validity period?

This paper is structured as follows: Section 2 introduces the modelling assumptions and specifies a discrete investment strategy under which investors can take actions before the underlying asset price reaches the strike price and during the validity period of the option. Building upon this framework, we derive the loss function. Section 3 uses the Black–Scholes model to derive an analytical solution for the discrete investment strategy. Section 4 presents a three-dimensional comparison between the option pricing model with the discrete strategy and the classical model, and it examines how the option prices under the discrete strategy vary with the relevant factors. In Section 5, we calibrate the parameters of the discrete-strategy option pricing model using market data. Section 5 contains a summary.

2. Exotic Option with a Discrete Investment Strategy

2.1. Fundamental Assumptions

This paper takes European call options as the object of analysis. In classic option pricing models, investors do not trade the underlying asset over the life of the option, which is inconsistent with real-world investor behavior. In real option markets, investors face transaction costs, risk limits, margin requirements and liquidity constraints. As a consequence, they typically rebalance their positions only at discrete times and tend to start adjusting their hedging portfolios once the underlying price comes sufficiently close to the strike, rather than waiting until the strike is exactly reached. As the underlying price gradually approaches the strike, risk measures such as the option’s $\mathsf{\delta }$ become significantly amplified. If investors wait until the price actually hits the strike to rebalance, they are often exposed to larger jump risks and insufficient counterparty liquidity. Therefore, in practice they tend to start rebalancing their positions somewhat earlier, once the underlying is already close to the strike. The assumption in the paper is a simplified representation of this kind of forward-looking risk management behavior. Building on classical option pricing models while incorporating these features, this paper proposes the following assumptions for the option pricing model under discrete investment strategies:

(1)

The investors predict a positive market and take early action before reaching the strike price.

(2)

The option holders choose to purchase or sell the underlying asset through a discrete investment strategy during the validity period.

(3)

The evolution of primary asset prices follows the geometric Brown motion.

$${\displaystyle \frac{{\mathrm{dS}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}=\mathsf{\mu }\mathrm{dt}+\mathsf{\sigma }{\mathrm{dW}}_{\mathrm{t}},}$$

(1)

where ${\mathrm{S}}_{\mathrm{t}}$ represents the underlying asset price, ${\mathrm{dW}}_{\mathrm{t}}$ denotes the standard Brown motion, $\mathsf{\mu }$ signifies the expected rate of return, while $\mathsf{\sigma }$ represents the volatility of stock price. One has

$$\mathrm{E}\left({\mathrm{dW}}_{\mathrm{t}}\right)=0, \mathrm{Var}\left({\mathrm{dW}}_{\mathrm{t}}\right)=\mathrm{dt}.$$

(4)

The underlying asset does not distribute any dividends.

(5)

There are no transactions and taxes.

(6)

The risk-free interest rate $\mathrm{r}$ remains fixed.

2.2. Constructing a Discrete Investment Strategy

Building on the assumptions outlined above, we first introduce the following notation. Let ${\mathrm{X}}_{\mathrm{e}}$ denote the strike price, $\mathrm{S}$ the underlying asset price, $\mathrm{A}$ the initial capital, $\mathsf{\beta }$ the maximum utilization rate of the initial capital $\mathrm{A}$, $\mathsf{\delta }$ the investment interval, $\mathrm{N}$ the number of purchases of the underlying asset, and $\mathsf{ϵ}$ the investment sensitivity (investors take action before ${\mathrm{X}}_{\mathrm{e}}$; the larger $\mathsf{ϵ}$ is, the earlier they act). The definitions of these parameters are summarized in

Table 1. Description of the parameters.

Parameters	Description
${\mathrm{X}}_{\mathrm{e}}$	the strike price
S	the underlying asset price
A	the initial capital
$\mathsf{\beta }$	the highest rate of capital utilization for the initial capital A
$\mathsf{\delta }$	the investment interval
N	the number of times to purchase the underlying asset
$\mathsf{ϵ}$	the investment sensitivity

. We then propose a discrete investment strategy, illustrated in Figure 1. In contrast to the classical option setting, in which no trading in the underlying asset takes place over the life of the option, we assume that investors possess acute market insight and act preemptively. Specifically, they start purchasing the underlying asset before the price reaches ${\mathrm{X}}_{\mathrm{e}}$. Subsequently, as the underlying asset price reaches ${\mathrm{S}}_1,{\mathrm{S}}_2,{\mathrm{S}}_3,\dots ,{\mathrm{S}}_{\mathrm{m}},\dots ,{\mathrm{S}}_{\mathrm{N}}({\mathrm{S}}_1<{\mathrm{S}}_2<{\mathrm{S}}_3,<\dots ,<{\mathrm{S}}_{\mathrm{m}},<\dots ,<{\mathrm{S}}_{\mathrm{N}})$, investors successively increase their holdings of the underlying asset, each time purchasing a fraction $\mathsf{\beta }/\mathrm{N}$.

2.3. Explanation of the Loss Function

In the context of European call options, in classic option theory(where the underlying asset trading does not occur during the option’s validity period), the investor incurs a loss of ${\displaystyle \frac{\mathrm{A}}{{\mathrm{X}}_{\mathrm{e}}}}(\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})$ when the underlying asset price rises to $\mathrm{S}$ ($\mathrm{S}>{\mathrm{X}}_{\mathrm{e}}$). Under the discrete investment strategy, we assume that the investor does not open any position when the underlying price is below the first trigger level ${\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}$. Once the price enters the interval $[{\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ},{\mathrm{X}}_{\mathrm{e}})$, the investor allocates $\mathsf{\beta }\mathrm{A}/\mathrm{N}$ units of capital to purchase $\mathsf{\beta }/\mathrm{N}$ units of the underlying asset. When the price moves into $[{\mathrm{X}}_{\mathrm{e}},{\mathrm{S}}_1)$, the investor adds another position of the same size, and so on. The sequence of trigger levels ${\left\{{\mathrm{S}}_{\mathrm{n}}\right\}}_{\mathrm{n}=1}^{\mathrm{N}}$ is a monotonically increasing sequence that specifies the preset hedging rule describing the investor’s gradual, stepwise build-up of the position. Given this discrete entry rule, we explicitly derive the loss function in each price interval.

When the investor adopts the discrete investment strategy illustrated in Figure 1, the loss is zero whenever the underlying asset price $\mathrm{S}$ is below ${\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}$. Once the price enters the interval ${\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}\le \mathrm{S}<{\mathrm{X}}_{\mathrm{e}}$, the investor purchases a fraction $\mathsf{\beta }/\mathrm{N}$ of the underlying asset. In this case, the investor’s loss is given by

$$\begin{array}{c}\hfill {\displaystyle \mathrm{L}\left(\mathrm{S}\right)=\frac{\mathrm{A}}{{\mathrm{X}}_{\mathrm{e}}}(\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times \mathrm{A}(\mathrm{S}-({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}))}{\mathrm{N}\times {\mathrm{X}}_{\mathrm{e}}},{\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}\le \mathrm{S}<{\mathrm{X}}_{\mathrm{e}}.}\end{array}$$

For ${\mathrm{X}}_{\mathrm{e}}\le \mathrm{S}<{\mathrm{S}}_1$, the investor’s loss can be written as

$$\begin{array}{c}\hfill {\displaystyle \mathrm{L}\left(\mathrm{S}\right)=\frac{\mathrm{A}}{{\mathrm{X}}_{\mathrm{e}}}(\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times \mathrm{A}(\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{\mathrm{N}\times {\mathrm{X}}_{\mathrm{e}}},{\mathrm{X}}_{\mathrm{e}}\le \mathrm{S}<{\mathrm{S}}_1.}\end{array}$$

Assuming ${\mathrm{S}}_{\mathrm{m}}\le \mathrm{S}<{\mathrm{S}}_{\mathrm{m}+1}$ ($1\le \mathrm{m}\le \mathrm{N}$), under the discrete investment strategy the investor purchases the underlying asset at the trigger levels ${\mathrm{S}}_{\mathrm{n}}$, which are specified as

$${\displaystyle {\mathrm{S}}_{\mathrm{n}}={\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta },\mathrm{n}=1,2,\dots ,\mathrm{N},}$$

with an amount of capital $\left(\mathsf{\beta }\mathrm{A}\right)/\mathrm{N}$ allocated to each purchase. If the underlying asset price subsequently rises to $\mathrm{S}$($\mathrm{S}>{\mathrm{S}}_{\mathrm{n}}$), the return generated by this tranche is ${\displaystyle \frac{\mathsf{\beta }\mathrm{A}}{\mathrm{N}{\mathrm{S}}_{\mathrm{n}}}}(\mathrm{S}-{\mathrm{S}}_{\mathrm{n}})$.

As the underlying asset price increases from ${\mathrm{S}}_1$ to ${\mathrm{S}}_{\mathrm{m}}$, the cumulative return is denoted by

$$\mathrm{R}=\sum _{\mathrm{n}=1}^{\mathrm{m}}{\displaystyle \frac{\mathsf{\beta }\times \mathrm{A}}{\mathrm{N}\times {\mathrm{S}}_{\mathrm{n}}}}(\mathrm{S}-{\mathrm{S}}_{\mathrm{n}}).$$

The resulting loss function is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{A}\times \mathrm{S}}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+\left(\frac{\mathsf{\beta }\times \mathrm{m}}{\mathrm{N}}-1\right)\times \mathrm{A},{\mathrm{S}}_{\mathrm{m}}\le \mathrm{S}<{\mathrm{S}}_{\mathrm{m}+1}.}\end{array}$$

For $\mathrm{m}>\mathrm{N}$, the investor’s loss is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{A}\times \mathrm{S}}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+(\mathsf{\beta }-1)\times \mathrm{A},\mathrm{S}\ge {\mathrm{S}}_{\mathrm{N}}.}\end{array}$$

Therefore, the loss function can be expressed as

$${\displaystyle \mathrm{L}\left(\mathrm{S}\right)=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle \mathrm{S}<{\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ},}\hfill \\ {\displaystyle \frac{\mathrm{A}\times (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times (\mathrm{S}-({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}))}{\mathrm{N}\times {\mathrm{X}}_{\mathrm{e}}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}\le \mathrm{S}<{\mathrm{X}}_{\mathrm{e}},}\hfill \\ {\displaystyle \frac{\mathrm{A}\times (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{\mathrm{N}\times {\mathrm{X}}_{\mathrm{e}}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}\le \mathrm{S}<{\mathrm{S}}_1,}\hfill \\ {\displaystyle \frac{\mathrm{A}\times \mathrm{S}}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+\left(\frac{\mathsf{\beta }\times \mathrm{m}}{\mathrm{N}}-1\right)\times \mathrm{A},}\hfill & {\displaystyle {\mathrm{S}}_{\mathrm{m}}\le \mathrm{S}<{\mathrm{S}}_{\mathrm{m}+1},}\hfill \\ {\displaystyle \frac{\mathrm{A}\times \mathrm{S}}{{\mathrm{X}}_{\mathrm{e}}}-\frac{\mathsf{\beta }\times \mathrm{A}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+(\mathsf{\beta }-1)\times \mathrm{A},}\hfill & {\displaystyle \mathrm{S}\ge {\mathrm{S}}_{\mathrm{N}}.}\hfill \end{array}\right.}$$

(2)

Using Equation (2), the loss per share can be expressed as

$${\displaystyle {\mathrm{L}}^{\prime }=\frac{\mathrm{L}\times {\mathrm{X}}_{\mathrm{e}}}{\mathrm{A}}}$$

(3)

$$=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle \mathrm{S}<{\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ},}\hfill \\ {\displaystyle (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times (\mathrm{S}-({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}))}{\mathrm{N}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}\le \mathrm{S}<{\mathrm{X}}_{\mathrm{e}},}\hfill \\ {\displaystyle (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{\mathrm{N}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}\le \mathrm{S}<{\mathrm{S}}_1,}\hfill \\ {\displaystyle \mathrm{S}-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+\left(\frac{\mathsf{\beta }\times \mathrm{m}}{\mathrm{N}}-1\right)\times {\mathrm{X}}_{\mathrm{e}},}\hfill & {\displaystyle {\mathrm{S}}_{\mathrm{m}}\le \mathrm{S}<{\mathrm{S}}_{\mathrm{m}+1},}\hfill \\ {\displaystyle \mathrm{S}-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+(\mathsf{\beta }-1)\times {\mathrm{X}}_{\mathrm{e}},}\hfill & {\displaystyle \mathrm{S}\ge {\mathrm{S}}_{\mathrm{N}}.}\hfill \end{array}\right.$$

For a call option, the intrinsic value function $\mathrm{f}\left(\mathrm{S}\right)$ is exactly the per-share loss function in (3), namely,

$$\mathrm{f}\left(\mathrm{S}\right)=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle \mathrm{S}<{\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ},}\hfill \\ {\displaystyle (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times (\mathrm{S}-({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}))}{\mathrm{N}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}\le \mathrm{S}<{\mathrm{X}}_{\mathrm{e}},}\hfill \\ {\displaystyle (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})-\frac{\mathsf{\beta }\times (\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}{\mathrm{N}},}\hfill & {\displaystyle {\mathrm{X}}_{\mathrm{e}}\le \mathrm{S}<{\mathrm{S}}_1,}\hfill \\ {\displaystyle \mathrm{S}-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+\left(\frac{\mathsf{\beta }\times \mathrm{m}}{\mathrm{N}}-1\right)\times {\mathrm{X}}_{\mathrm{e}},}\hfill & {\displaystyle {\mathrm{S}}_{\mathrm{m}}\le \mathrm{S}<{\mathrm{S}}_{\mathrm{m}+1},}\hfill \\ {\displaystyle \mathrm{S}-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}\times \mathrm{S}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}+(\mathsf{\beta }-1)\times {\mathrm{X}}_{\mathrm{e}},}\hfill & {\displaystyle \mathrm{S}\ge {\mathrm{S}}_{\mathrm{N}},}\hfill \end{array}\right.$$

(4)

which coincides with the option value at $\mathrm{t}=\mathrm{T}$ (i.e., at maturity).

3. Exotic Option Pricing Formula

3.1. The Price Dynamics of Assets Following Brownian Motion

Considering the fractional Brownian motion, the random process $\{{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right),\mathrm{t}\ge 0\}$ is a Gaussian process with Hurst parameter $\mathrm{H}$, $0<\mathrm{H}<1$. For $\mathrm{t}\in \mathbb{R}$, $\mathbb{E}\left[{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)\right]=0$ and its covariance is given in [21] by

$$\mathbb{E}\left[{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right){\mathrm{W}}_{\mathrm{H}}\left(\mathrm{s}\right)\right]={\displaystyle \frac{{\mathsf{\sigma }}^2}{2}}\left({\left|\mathrm{t}\right|}^{2\mathrm{H}}+{\left|\mathrm{s}\right|}^{2\mathrm{H}}-{|\mathrm{t}-\mathrm{s}|}^{2\mathrm{H}}\right),$$

(5)

where $\mathsf{\sigma }$ is the variance parameter, and $\mathbb{E}\left[{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right){\mathrm{W}}_{\mathrm{H}}\left(\mathrm{s}\right)\right]$ represents the covariance of the fractional Brownian motion ${\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)$ at two different time points $\mathrm{t}$ and $\mathrm{s}$. This formula can be used to model the covariance of the underlying asset price processes influenced by fractional Brownian motion. When $\mathrm{H}=1/2$, ${\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)$ reduces to the standard Brownian motion $\mathrm{W}\left(\mathrm{t}\right)$. Then the fractional stochastic differential equation for the asset price is

$$\mathrm{dS}\left[\mathrm{t}\right]=\mathsf{\mu }\mathrm{S}\left[\mathrm{t}\right]\mathrm{dt}+\mathsf{\sigma }\mathrm{S}\left[\mathrm{t}\right]{\mathrm{dW}}_{\mathrm{H}}\left[\mathrm{t}\right],\mathrm{S}\left[0\right]=\mathrm{b}>0,$$

(6)

where $\mathrm{S}\left[\mathrm{t}\right]$ represents the asset price at time $\mathrm{t}$, $\mathsf{\mu }$ represents the expected return rate, and $\mathsf{\sigma }$ is the volatility. Its solution can be written as

$$\mathrm{S}\left[\mathrm{t}\right]=\mathrm{b}exp\left(\mathsf{\sigma }{\mathrm{W}}_{\mathrm{H}}\left[\mathrm{t}\right]+\mathsf{\mu }\mathrm{t}-{\displaystyle \frac{1}{2}}{\mathsf{\sigma }}^2{\mathrm{t}}^{2\mathrm{H}}\right).$$

(7)

For ${\mathrm{t}}_1,{\mathrm{t}}_2$ with ${\mathrm{t}}_1\ne {\mathrm{t}}_2$, under the risk-free condition $\mathsf{\mu }=\mathrm{r}$, the expression takes the following form:

$${\displaystyle \frac{\mathrm{S}\left[{\mathrm{t}}_2\right]}{\mathrm{S}\left[{\mathrm{t}}_1\right]}}=exp\left(\mathrm{r}({\mathrm{t}}_2-{\mathrm{t}}_1)-{\displaystyle \frac{1}{2}}{\mathsf{\sigma }}^2({\mathrm{t}}_2-{\mathrm{t}}_1)+\mathsf{\sigma }\left(\mathrm{W}\left[{\mathrm{t}}_2\right]-\mathrm{W}\left[{\mathrm{t}}_1\right]\right)\right),\mathrm{H}={\displaystyle \frac{1}{2}}.$$

(8)

This is the explicit solution to the geometric Brownian motion given in (1) on the interval $[{\mathrm{t}}_1,{\mathrm{t}}_2]$ under the risk-neutral measure.

3.2. Defining the Intrinsic Value Function

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \mathrm{V}}{\partial \mathrm{t}}+\mathrm{r}\mathrm{S}\frac{\partial \mathrm{V}}{\partial \mathrm{S}}+\frac{1}{2}{\mathsf{\sigma }}^2{\mathrm{S}}^2\frac{{\partial }^2\mathrm{V}}{\partial {\mathrm{S}}^2}=\mathrm{r}\mathrm{V},}\hfill \\ {\displaystyle \mathrm{V}\left(\mathrm{S}\right)=\mathrm{f}\left[\mathrm{S}\left(\mathrm{T}\right)\right]\left\{\begin{array}{c}{\displaystyle {(\mathrm{S}-{\mathrm{X}}_{\mathrm{e}})}^{+},\left(\mathrm{Call}\mathrm{option}\right)}\hfill \\ {\displaystyle {({\mathrm{X}}_{\mathrm{e}}-\mathrm{S})}^{-},\left(\mathrm{Put}\mathrm{option}\right)}\hfill \end{array}\right.}\hfill \end{array}\right.$$

(9)

where $\mathrm{V}$ represents the intrinsic value of the option, $\mathrm{r}$ denotes the risk-free interest rate, $\mathsf{\sigma }$ stands for the stock price volatility, $\mathrm{t}$ denotes time, and $\mathrm{S}$ represents the underlying asset price.

Lemma 1.

Assume that $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ is such that $\tilde{\mathbb{E}}\left[\left|\mathrm{f}\right({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)\left)\right|\right]<\infty$. Then, for each $\mathrm{t}<\mathrm{T}$,

$$\begin{array}{c}\hfill {\tilde{\mathbb{E}}}_{\mathrm{t}}\left[\mathrm{f}\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)\right)\right]={\int }_{\mathbb{R}}{\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\left(\mathrm{s}-{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)\right)\mathrm{f}\left(\mathrm{s}\right)\mathrm{ds},\end{array}$$

where

$$\begin{array}{c}\hfill {\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\left(\mathrm{x}\right)={\displaystyle \frac{1}{\sqrt{2\mathsf{\pi }\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)}}}exp\left(-{\displaystyle \frac{{\mathrm{x}}^2}{2\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)}}\right),\end{array}$$

and ${\tilde{\mathbb{E}}}_{\mathrm{t}}[·]=\tilde{\mathbb{E}}[·\mid {\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)]$ denotes the conditional expectation given ${\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)$ at time $\mathrm{t}$.

Proof. 

Let $\mathcal{F}\left[\mathrm{f}\right](ø)$ denote the Fourier transform of $\mathrm{f}$, i.e.,

$$\begin{array}{c}\hfill \mathcal{F}\left[\mathrm{f}\right]\left(\mathsf{\tau }\right)={\int }_{\mathbb{R}}{\mathrm{e}}^{-\mathrm{is}\mathsf{\tau }}\mathrm{f}\left(\mathrm{s}\right)\mathrm{ds},\mathsf{\tau }\in \mathbb{R}.\end{array}$$

By the inversion formula, we have

$$\begin{array}{c}\hfill \mathrm{f}\left(\mathrm{x}\right)={\displaystyle \frac{1}{2\mathsf{\pi }}}{\int }_{\mathbb{R}}{\mathrm{e}}^{\mathrm{ix}\mathsf{\tau }}\mathcal{F}\left[\mathrm{f}\right]\left(\mathsf{\tau }\right)\mathrm{d}\mathsf{\tau },\mathrm{x}\in \mathbb{R}.\end{array}$$

By inserting $\mathrm{x}={\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)$ into the above equation, we obtain

$$\begin{array}{c}\hfill \mathrm{f}\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)\right)={\displaystyle \frac{1}{2\mathsf{\pi }}}{\int }_{\mathbb{R}}{\mathrm{e}}^{\mathrm{i}ø{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)}\mathcal{F}\left[\mathrm{f}\right]\left(\mathsf{\tau }\right)\mathrm{d}\mathsf{\tau }.\end{array}$$

Hence

$$\begin{array}{cc}\hfill {\tilde{\mathbb{E}}}_{\mathrm{t}}\left[\mathrm{f}\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)\right)\right]& ={\displaystyle \frac{1}{2\mathsf{\pi }}}{\int }_{\mathbb{R}}{\tilde{\mathbb{E}}}_{\mathrm{t}}\left[{\mathrm{e}}^{\mathrm{i}ø{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)}\right]\mathcal{F}\left[\mathrm{f}\right](ø)\mathrm{d}ø\hfill \\ & ={\displaystyle \frac{1}{2\mathsf{\pi }}}{\int }_{\mathbb{R}}exp\left(\mathrm{i}ø{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)-{\displaystyle \frac{{ø}^2}{2}}\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)\right)\mathcal{F}\left[\mathrm{f}\right](ø)\mathrm{d}ø\hfill \\ & ={\displaystyle \frac{1}{2\mathsf{\pi }}}{\int }_{\mathbb{R}}{\mathrm{e}}^{\mathrm{i}ø{\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)}[{\mathrm{e}}^{-{\displaystyle \frac{{ø}^2}{2}}\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)}\mathcal{F}\left[\mathrm{f}\right](ø)]\mathrm{d}ø.\hfill \end{array}$$

Define

$$\begin{array}{c}\hfill {\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\left(\mathrm{x}\right)={\displaystyle \frac{1}{\sqrt{2\mathsf{\pi }\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)}}}exp\left(-{\displaystyle \frac{{\mathrm{x}}^2}{2\left({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}}\right)}}\right).\end{array}$$

A straightforward calculation shows that its Fourier transform is

$$\begin{array}{c}\hfill \mathcal{F}\left[{\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\right]\left(\mathsf{\tau }\right)={\mathrm{e}}^{-{\displaystyle \frac{{\mathsf{\tau }}^2}{2}}({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}})},\mathsf{\tau }\in \mathbb{R}.\end{array}$$

Therefore, it has

$$\begin{array}{c}\hfill {\mathrm{e}}^{-{\displaystyle \frac{{\mathsf{\tau }}^2}{2}}({\mathrm{T}}^{2\mathrm{H}}-{\mathrm{t}}^{2\mathrm{H}})}\mathcal{F}\left[\mathrm{f}\right]\left(\mathsf{\tau }\right)=\mathcal{F}\left[\mathrm{f}\right]\left(\mathsf{\tau }\right)\mathcal{F}\left[{\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\right]\left(\mathsf{\tau }\right).\end{array}$$

By the convolution theorem and Fourier inversion, we obtain

$$\begin{array}{cc}\hfill {\tilde{\mathbb{E}}}_{\mathrm{t}}\left[\mathrm{f}\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{T}\right)\right)\right]& ={\mathcal{F}}^{-1}\left(\mathcal{F}\left[\mathrm{f}\right]\mathcal{F}\left[{\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\right]\right)\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)\right)\hfill \\ \hfill & =(\mathrm{f}\ast {\mathsf{\eta }}_{\mathrm{t},\mathrm{T}})\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)\right)\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\mathsf{\eta }}_{\mathrm{t},\mathrm{T}}\left({\mathrm{W}}_{\mathrm{H}}\left(\mathrm{t}\right)-\mathrm{y}\right)\mathrm{f}\left(\mathrm{y}\right)\mathrm{dy},\hfill \end{array}$$

When $\mathrm{H}=\frac{1}{2}$, we recover the classical conditional expectation formula for standard Brownian motion. □

Theorem 1.

The analytic solution to the Black–Scholes Equation (9) is expressed as

$$\mathrm{V}[\mathrm{S},\mathrm{t}]={\displaystyle \frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}}{\int }_{-\infty }^{\infty }\mathrm{f}[{\mathrm{Se}}^{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}\mathrm{Z}+(\mathrm{T}-\mathrm{t})(\mathrm{r}-{\displaystyle \frac{{\mathsf{\sigma }}^2}{2}})}]{\mathrm{e}}^{-{\displaystyle \frac{{\mathrm{Z}}^2}{2}}}\mathrm{dZ},$$

(10)

where $\mathrm{T}$ represents the mature date and $\mathrm{f}[·]$ denotes the option’s intrinsic value function given in (4).

Proof. 

According to the Feynman-Kac theorem, the solution to (9) can be represented in terms of a conditional expectation and has the following form:

$$\mathrm{V}(\mathrm{S},\mathrm{t})={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_{\mathrm{t}}^{\mathrm{T}}\mathrm{r} \mathrm{ds}}\mathrm{f}\left[\mathrm{S}\left(\mathrm{T}\right)\right]\right].$$

(11)

By substituting (8) into (11), applying Lemma 1, and setting $\mathrm{Z}=(\mathrm{S}-{\mathrm{W}}_{\mathrm{t}})/\sqrt{\mathrm{T}-\mathrm{t}}$, we obtain (10). □

3.3. The Pricing Formula for the Exotic Option

For a European call option, we partition the integration domain into the intervals $[0,{\mathrm{S}}_0-\mathsf{ϵ}]$, $[{\mathrm{S}}_0-\mathsf{ϵ},{\mathrm{S}}_0]$, $[{\mathrm{S}}_0,{\mathrm{S}}_1]$, $[{\mathrm{S}}_{\mathrm{m}},{\mathrm{S}}_{\mathrm{m}+1}]$ and $[{\mathrm{S}}_{\mathrm{N}},+\infty )$, where ${\mathrm{S}}_0={\mathrm{X}}_{\mathrm{e}}$ and ${\mathrm{S}}_{\mathrm{m}}={\mathrm{X}}_{\mathrm{e}}+\mathrm{m}\mathsf{\delta }$$(\mathrm{m}=1,\dots ,\mathrm{N}-1)$. By combining the intrinsic value function (4) with the analytic solution (10) and then summing the resulting contributions piecewise, we obtain the closed-form pricing formula. The analytic solution can be expressed as

$${\displaystyle \mathrm{V}={\mathrm{V}}_{\mathrm{t}1}+{\mathrm{V}}_{\mathrm{t}2}+{\mathrm{V}}_{\mathrm{t}3}+{\mathrm{V}}_{\mathrm{t}4}+{\mathrm{V}}_{\mathrm{t}5}.}$$

(12)

Let $\mathrm{U}={\mathrm{Se}}^{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}\mathrm{Z}+\left(\mathrm{T}-\mathrm{t}\right)(\mathrm{r}-{\displaystyle \frac{{\mathsf{\sigma }}^2}{2}})}$. If ${\mathrm{S}}_{\mathrm{m}}\le \mathrm{U}<{\mathrm{S}}_{\mathrm{m}+1}$, then we have ${\mathrm{S}}_{\mathrm{m}}\le {\mathrm{Se}}^{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}\mathrm{Z}+\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-{\displaystyle \frac{{\mathsf{\sigma }}^2}{2}}\right)}<{\mathrm{S}}_{\mathrm{m}+1}$, where

$$\begin{array}{cc}\hfill & {\displaystyle \frac{ln\left[\frac{{\mathrm{S}}_{\mathrm{m}}}{\mathrm{S}}\right]-\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}\le \mathrm{Z}<\frac{ln\left[\frac{{\mathrm{S}}_{\mathrm{m}+1}}{\mathrm{S}}\right]-\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}},}\hfill \end{array}$$

marking ${\mathrm{Z}}_{\mathrm{m}}={\displaystyle \frac{ln\left[\frac{{\mathrm{S}}_{\mathrm{m}}}{\mathrm{S}}\right]-\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}},{\mathrm{Z}}_{\mathrm{m}+1}={\displaystyle \frac{ln\left[\frac{{\mathrm{S}}_{\mathrm{m}+1}}{\mathrm{S}}\right]-\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}}$, the above equation is equivalent to

$${\mathrm{Z}}_{\mathrm{m}}\le \mathrm{Z}<{\mathrm{Z}}_{\mathrm{m}+1}.$$

Similarly, when ${\mathrm{S}}_{\mathrm{N}}\le \mathrm{U}$, we have ${\mathrm{S}}_{\mathrm{N}}\le {\mathrm{Se}}^{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}\mathrm{Z}+\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-{\displaystyle \frac{{\mathsf{\sigma }}^2}{2}}\right)}$. It can be expressed as

$$\mathrm{Z}\ge {\displaystyle \frac{ln\left[\frac{{\mathrm{S}}_{\mathrm{N}}}{\mathrm{S}}\right]-\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}},Z_N={\displaystyle \frac{ln\left[\frac{S_N}{S}\right]-\left(T-t\right)\left(r-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{T-t}}}.$$

When $\mathrm{Z}\ge {\mathrm{Z}}_{\mathrm{N}}$, to solve Equation (12), we substitute the intrinsic value function of the option (4) into Equation (10). Let $\sum =\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}\mathrm{Z}+\left(\mathrm{T}-\mathrm{t}\right)\left(\mathrm{r}-{\displaystyle \frac{{\mathsf{\sigma }}^2}{2}}\right)$. Since ${\mathrm{V}}_{\mathrm{t}1}=0$, the option value function (12) is

$$\begin{array}{cc}{\displaystyle \mathrm{V}(\mathrm{t},\mathrm{s})}\hfill & ={\mathrm{V}}_{\mathrm{t}2}+{\mathrm{V}}_{\mathrm{t}3}+{\mathrm{V}}_{\mathrm{t}4}+{\mathrm{V}}_{\mathrm{t}5},\hfill \\ \hfill & {\displaystyle =\frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}\left[{\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}}^{{\mathrm{Z}}_0}\mathrm{f}\left(\mathrm{S}\times {\mathrm{e}}^{\sum }\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}+{\int }_{{\mathrm{Z}}_0}^{{\mathrm{Z}}_1}\mathrm{f}\left(\mathrm{S}\times {\mathrm{e}}^{\sum }\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}\right.}\hfill \\ \hfill & {\displaystyle +\sum _{\mathrm{m}=1}^{\mathrm{N}-1}{\int }_{{\mathrm{Z}}_{\mathrm{m}}}^{{\mathrm{Z}}_{\mathrm{m}+1}}\mathrm{f}\left(\mathrm{S}\times {\mathrm{e}}^{\sum }\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}\left.+{\int }_{{\mathrm{Z}}_{\mathrm{N}}}^{+\infty }\mathrm{f}\left(\mathrm{S}\times {\mathrm{e}}^{\sum }\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}\right].}\hfill \end{array}$$

(13)

To solve the Equation (13), we simplify it further by dividing it into four distinct parts: ${\mathrm{V}}_2,{\mathrm{V}}_3$, ${\mathrm{V}}_4$, and ${\mathrm{V}}_5$, i.e.,

$$\begin{array}{cc}{\displaystyle {\mathrm{V}}_2}\hfill & ={\displaystyle \frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}}{\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}}^{{\mathrm{Z}}_0}\left(\mathrm{S}{\mathrm{e}}^{\sum }-{\mathrm{X}}_{\mathrm{e}}-{\displaystyle \frac{\mathsf{\beta }\left({\mathrm{Se}}^{\sum }-({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ})\right)}{\mathrm{N}}}\right){\mathrm{e}}^{-{\displaystyle \frac{{\mathrm{Z}}^2}{2}}}\mathrm{dZ}\hfill \\ \hfill & {\displaystyle =\frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}{\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}}^{{\mathrm{Z}}_0}\left(\mathrm{S}\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right){\mathrm{e}}^{\sum }+\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}-\frac{\mathsf{\beta }\mathsf{ϵ}}{\mathrm{N}}\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}}\hfill \\ \hfill & {\displaystyle =\frac{1}{\sqrt{2\mathsf{\pi }}}\mathrm{S}\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right){\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}^{{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}{\mathrm{e}}^{-\frac{{(\mathrm{z}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})}^2}{2}}\mathrm{d}(\mathrm{z}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}{\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}}^{{\mathrm{Z}}_0}{\mathrm{e}}^{-\frac{{\mathrm{z}}^2}{2}}\mathrm{dZ}-\frac{\mathsf{\beta }\mathsf{ϵ}}{\mathrm{N}}{\int }_{{\mathrm{Z}}_0-\mathsf{ϵ}}^{{\mathrm{Z}}_0}{\mathrm{e}}^{-\frac{{\mathrm{z}}^2}{2}}\mathrm{dZ}}\hfill \\ \hfill & {\displaystyle =\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right)\mathrm{S}[\mathcal{N}[{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{ϵ}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}-\frac{\mathsf{\beta }\mathsf{ϵ}}{\mathrm{N}}\right)[\mathcal{N}\left[{\mathrm{Z}}_0\right]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{ϵ}]].}\hfill \end{array}$$

$$\begin{array}{cc}{\displaystyle {\mathrm{V}}_3}\hfill & ={\displaystyle \frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}}{\int }_{{\mathrm{Z}}_0}^{{\mathrm{Z}}_1}\left(\mathrm{S}{\mathrm{e}}^{\sum }-{\mathrm{X}}_{\mathrm{e}}-{\displaystyle \frac{\mathsf{\beta }({\mathrm{Se}}^{\sum }-{\mathrm{X}}_{\mathrm{e}})}{\mathrm{N}}}\right){\mathrm{e}}^{-{\displaystyle \frac{{\mathrm{Z}}^2}{2}}}\mathrm{dZ}\hfill \\ \hfill & {\displaystyle =\frac{{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}}{\sqrt{2\mathsf{\pi }}}{\int }_{{\mathrm{Z}}_0}^{{\mathrm{Z}}_1}\left(\mathrm{S}\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right){\mathrm{e}}^{\sum }+\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}\right){\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}}\hfill \\ \hfill & {\displaystyle =\frac{1}{\sqrt{2\mathsf{\pi }}}\mathrm{S}\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right){\int }_{{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}^{{\mathrm{Z}}_1-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}{\mathrm{e}}^{-\frac{{(\mathrm{Z}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})}^2}{2}}\mathrm{d}(\mathrm{Z}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}{\int }_{{\mathrm{Z}}_0}^{{\mathrm{Z}}_1}{\mathrm{e}}^{-\frac{{\mathrm{z}}^2}{2}}\mathrm{dz}}\hfill \\ \hfill & {\displaystyle =\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right)\mathrm{S}[\mathcal{N}[{\mathrm{Z}}_1-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}[\mathcal{N}\left[{\mathrm{Z}}_1\right]-\mathcal{N}\left[{\mathrm{Z}}_0\right]].}\hfill \end{array}$$

$$\begin{array}{cc}{\displaystyle {\mathrm{V}}_4}\hfill & =\sum _{\mathrm{m}=1}^{\mathrm{N}-1}{\int }_{{\mathrm{Z}}_{\mathrm{m}}}^{{\mathrm{Z}}_{\mathrm{m}+1}}\left({\mathrm{Se}}^{\sum }-{\displaystyle \frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}\mathrm{S}{\mathrm{e}}^{\sum }}{\mathrm{N}}}\sum _{\mathrm{n}=1}^{\mathrm{N}}{\displaystyle \frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta }}}+({\displaystyle \frac{\mathsf{\beta }\mathrm{m}}{\mathrm{N}}}-1){\mathrm{X}}_{\mathrm{e}}\right){\mathrm{e}}^{-{\displaystyle \frac{{\mathrm{Z}}^2}{2}}}\mathrm{dZ}\hfill \\ \hfill & {\displaystyle =\sum _{\mathrm{m}=1}^{\mathrm{N}-1}\left\{{\int }_{{\mathrm{Z}}_{\mathrm{m}}}^{{\mathrm{Z}}_{\mathrm{m}+1}}\mathrm{S}\left(1-\frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta }}\right){\mathrm{e}}^{\sum }{\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}+{\int }_{{\mathrm{Z}}_{\mathrm{m}}}^{{\mathrm{Z}}_{\mathrm{m}+1}}\left(\frac{\mathsf{\beta }\mathrm{m}}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}{\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}\right\}}\hfill \\ \hfill & {\displaystyle =\sum _{\mathrm{m}=1}^{\mathrm{N}-1}\left\{\mathrm{S}\left(1-\frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta }}\right)[\mathcal{N}[{\mathrm{Z}}_{\mathrm{m}+1}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_{\mathrm{m}}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]\right.}\hfill \\ & {\displaystyle \left.+e^{-r(T-t)}\left(\frac{\mathsf{\beta }m}{N}-1\right)X_e[\mathcal{N}\left[Z_{m+1}\right]-\mathcal{N}\left[Z_m\right]]\right\}}\hfill \end{array}$$

$$\begin{array}{cc}{\displaystyle {\mathrm{V}}_5}\hfill & ={\int }_{{\mathrm{Z}}_{\mathrm{N}}}^{+\infty }[{\mathrm{Se}}^{\sum }-{\displaystyle \frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}{\mathrm{Se}}^{\sum }}{\mathrm{N}}}\sum _{\mathrm{n}=1}^{\mathrm{N}}{\displaystyle \frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta }}}+(\mathsf{\beta }-1){\mathrm{X}}_{\mathrm{e}}]{\mathrm{e}}^{-{\displaystyle \frac{{\mathrm{Z}}^2}{2}}}\mathrm{dZ}\hfill \\ \hfill & {\displaystyle ={\int }_{{\mathrm{Z}}_{\mathrm{N}}}^{+\infty }\left\{\mathrm{S}\left(1-\frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\mathsf{\delta }}\right){\mathrm{e}}^{\sum }{\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}+(\mathsf{\beta }-1){\mathrm{X}}_{\mathrm{e}}{\mathrm{e}}^{-\frac{{\mathrm{Z}}^2}{2}}\mathrm{dZ}\right\}}\hfill \\ \hfill & {\displaystyle =\mathrm{S}\left(1-\frac{\mathsf{\beta }{\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }}\right)\mathcal{N}[-({\mathrm{Z}}_{\mathrm{N}}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})]+{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}(\mathsf{\beta }-1){\mathrm{X}}_{\mathrm{e}}\mathcal{N}[-{\mathrm{Z}}_{\mathrm{N}}].}\hfill \end{array}$$

Based on the calculation results of the above four parts, the option value is

$$\begin{array}{cc}\hfill {\displaystyle \mathrm{V}(\mathrm{t},\mathrm{s})}& =(V_2+V_3+V_4+V_5)\hfill \\ \hfill & {\displaystyle =\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right)\mathrm{S}[\mathcal{N}[{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{ϵ}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}-\frac{\mathsf{\beta }\mathsf{ϵ}}{\mathrm{N}}\right)[\mathcal{N}\left[{\mathrm{Z}}_0\right]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{ϵ}]]}\hfill \\ \hfill & {\displaystyle +\left(1-\frac{\mathsf{\beta }}{\mathrm{N}}\right)\mathrm{S}[\mathcal{N}[{\mathrm{Z}}_1-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]}\hfill \\ \hfill & {\displaystyle +{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}\left(\frac{\mathsf{\beta }}{\mathrm{N}}-1\right){\mathrm{X}}_{\mathrm{e}}[\mathcal{N}\left[{\mathrm{Z}}_1\right]-\mathcal{N}\left[{\mathrm{Z}}_0\right]]}\hfill \\ \hfill & {\displaystyle +\sum _{\mathrm{m}=1}^{\mathrm{N}-1}\left\{\mathrm{S}\mathrm{Q}[\mathcal{N}[{\mathrm{Z}}_{\mathrm{m}+1}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]-\mathcal{N}[{\mathrm{Z}}_{\mathrm{m}}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}]]\right.}\hfill \\ \hfill & \left.+e^{-r(T-t)}\left({\displaystyle \frac{\mathsf{\beta }m}{N}}-1\right)X_e[\mathcal{N}\left[Z_{m+1}\right]-\mathcal{N}\left[Z_m\right]]\right\}\hfill \\ \hfill & {\displaystyle +\mathrm{S}\mathrm{R}\mathcal{N}[-({\mathrm{Z}}_{\mathrm{N}}-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}})]+{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}(\mathsf{\beta }-1){\mathrm{X}}_{\mathrm{e}}\mathcal{N}[-{\mathrm{Z}}_{\mathrm{N}}],}\hfill \end{array}$$

(14)

where

$$\begin{array}{c}{\displaystyle \mathrm{Q}=1-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{m}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }},}\hfill \\ {\displaystyle \mathrm{R}=1-\frac{\mathsf{\beta }\times {\mathrm{X}}_{\mathrm{e}}}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\mathrm{X}}_{\mathrm{e}}+\mathrm{n}\times \mathsf{\delta }},}\hfill \end{array}$$

and $\mathcal{N}[·]$ represents the normal distribution.

When the parameters are set to $\mathsf{\beta }=0$, $\mathsf{ϵ}=0$, and $\mathrm{N}=2$, meaning that the investor neither takes any early action nor implements discrete investment but only trades at the initial time t and the maturity date T, the strategy reduces to a one-time buy-and-hold position as in the standard Black–Scholes model. In this case, the Equation (14) collapses to the classical Black–Scholes model. The solution can be written as

$$\mathrm{V}=\mathrm{S}\mathcal{N}[-{\mathrm{W}}_0]-{\mathrm{e}}^{-\mathrm{r}(\mathrm{T}-\mathrm{t})}{\mathrm{X}}_{\mathrm{e}}\mathcal{N}[-{\mathrm{Z}}_0],$$

(15)

where ${\mathrm{W}}_0={\mathrm{Z}}_0-\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}},{\mathrm{Z}}_0={\displaystyle \frac{ln\left[\frac{{\mathrm{X}}_{\mathrm{e}}}{\mathrm{S}}\right]-(\mathrm{T}-\mathrm{t})\left(\mathrm{r}-\frac{{\mathsf{\sigma }}^2}{2}\right)}{\mathsf{\sigma }\sqrt{\mathrm{T}-\mathrm{t}}}}$.

The option pricing model under the discrete strategy offers a lower price compared to the classic option pricing model, enabling investors to mitigate a portion of their losses. The classic option theory assumes that there is no trading of stocks during the option’s validity period, while the discrete investment strategy allows investors to take early action before the stock price reaches the strike price and trade a certain proportion of stocks within the validity period to reduce losses caused by market changes.

4. Sensitivity Analysis of the Exotic Option and the Classic Option

4.1. Comparison Between the Exotic Option and the Classic Option

From Equation (4), it can be seen that a higher maximum capital allocation ratio $\mathsf{\beta }$ reduces the intrinsic value function of the option, thereby leading to a lower resulting option price. We define a price ratio as ${\mathrm{P}}_{\mathrm{R}}={\displaystyle \frac{{\mathrm{P}}_{\mathrm{N}}}{{\mathrm{P}}_{\mathrm{C}}}}$, where ${\mathrm{P}}_{\mathrm{N}}$ is the option value under the discrete Strategy and ${\mathrm{P}}_{\mathrm{C}}$ is the value of classic options. It is found that, within the changing parameter range [21], the ratio of option price between the discrete strategy and classic option is always less than 1. This indicates that the discrete investment strategy effectively reduces investor losses by lowering the option prices.

Figure 2 illustrates the price ratio between the exotic option (14) and the classic option (15) over the $(\mathsf{\delta },\mathrm{T})$-plane for ${\mathrm{X}}_{\mathrm{e}}=40$, $\mathsf{\beta }=1$, $\mathrm{t}=0$, $\mathrm{r}=0.05$, $\mathsf{\sigma }=0.3$, $\mathsf{\alpha }=0.2$, and $\mathsf{ϵ}=0.2$. The maturity $\mathrm{T}$ ranges from 0.5 to 1, while the investment interval $\mathsf{\delta }$ ranges from 0 to 8. The figure shows that the price ratio remains consistently below 1 over the whole domain. Furthermore, for a fixed $\mathrm{T}$, the price ratio increases as the investment interval $\mathsf{\delta }$ increases. Thus, when the investment interval is small, the difference between the price of the classic option and that under the discrete strategy becomes more pronounced, implying a greater reduction in losses for investors. For example, when $\mathrm{T}=0.5$ is fixed and $\mathsf{\delta }=2$, the classic option price is 60.9876, whereas the option price under the discrete strategy is 12.7122, giving a price ratio of 0.2084. When $\mathsf{\delta }=5$, the classic option price remains 60.9876, because the pricing formula does not depend on $\mathsf{\delta }$, while the option price under the discrete strategy is 26.0187, resulting in a price ratio of 0.4266. When $\mathsf{\delta }$ reaches 8, the option price under the discrete strategy increases to 35.2747, and the corresponding price ratio is 0.5784. Hence, although the price advantage of the discrete strategy gradually decreases as the investment interval widens, it is still more favorable than the classic option, allowing investors to obtain risk protection with relatively smaller losses.

Figure 3 plots the price ratio between the exotic option (14) and the classic option (15) for parameters $\mathrm{T}=0.5$, $\mathsf{\beta }=1$, $\mathrm{t}=0$, $\mathrm{r}=0.05$, $\mathsf{\sigma }=0.3$, $\mathsf{\alpha }=0.2$, and $\mathsf{ϵ}=0.2$. In this figure, the exercise price ${\mathrm{X}}_{\mathrm{e}}$ varies from 15 to 50, while the investment interval $\mathsf{\delta }$ ranges from 0 to 8. Notably, the price ratio depicted in Figure 3 indicates that option prices under the discrete investment strategy are consistently lower than those of the classic option. This suggests that investors can obtain equivalent risk protection at a lower cost by adopting the discrete investment strategy, making it more attractive. For instance, with $\mathsf{\delta }$ fixed at 3 and ${\mathrm{X}}_{\mathrm{e}}$ taking values 10, 30 and 40, the price ratio decreases from 0.4931, 0.3131 and then to 0.2910. Under the discrete strategy, the option prices decline from 44.4982 to 22.1478 and then to 17.7457, while the classic option prices decrease from 90.2469 to 70.7407 and 60.9876, respectively. Thus, the option prices decrease as ${\mathrm{X}}_{\mathrm{e}}$ increases. Similarly, when ${\mathrm{X}}_{\mathrm{e}}$ is set at 30 and $\mathsf{\delta }$ increases from 3 to 5 and 8, the price ratio rises from 0.3131 to 0.4469 and 0.5906, respectively, indicating that the price ratio increases with $\mathsf{\delta }$.

Figure 4 illustrates the correlation between the price ratio and the variations in the investment interval $\mathsf{\delta }$ and the maximum capital utilization rate $\mathsf{\beta }$ of the initial capital. We present both the exotic option (14) and the classic option (15) in the ($\mathsf{\delta }$, $\mathsf{\beta }$)-plane where $\mathrm{T}=0.5$, ${\mathrm{X}}_{\mathrm{e}}=40$, $\mathrm{t}=0$, $\mathrm{r}=0.05$, $\mathsf{\sigma }=0.3$, $\mathsf{\alpha }=0.2$, and $\mathsf{ϵ}=0.2$. The price ratio fluctuates within the defined range of $\mathsf{\delta }$ and $\mathsf{\beta }$, where $\mathsf{\delta }$ ranges from 0 to 8 and $\mathsf{\beta }$ ranges from 0 to 1. The figure illustrates that fixing $\mathsf{\beta }$ and increasing $\mathsf{\delta }$ results in a higher price ratio. Conversely, when $\mathsf{\delta }$ remains constant, the price ratio decreases with increasing $\mathsf{\beta }$. However, in both cases, the price ratio consistently remains below 1. This demonstrates to investors that employing a discrete strategy is effective, as it offers comparable risk protection at lower option prices. To illustrate, with $\mathsf{\beta }$ fixed at 0.8 and $\mathsf{\delta }$ varying from 3 to 5 to 8, the classic option price remains constant at 60.9876. However, under the discrete strategy, the option price increases from 26.3941 to 33.0125 and then to 40.4173, resulting in corresponding price ratios of 0.4328, 0.5413, and 0.6627, respectively. In another scenario, when $\mathsf{\delta }$ is set at 4 and $\mathsf{\beta }$ at 0.6, 0.8, and 1, the classic option price remains unchanged at 60.9876. Conversely, the option price under the discrete strategy decreases from 37.6802 to 29.9110 and then to 22.1419, resulting in price ratios of 0.6178, 0.4904, and 0.3631, respectively.

Figure 5 illustrates how the price ratio varies with the investment interval $\mathsf{\delta }$ and the volatility $\mathsf{\sigma }$. The price ratios of the exotic option (14) relative to the classic option (15) are plotted in the ($\mathsf{\delta }$,$\mathsf{\sigma }$)-plane with parameters $\mathrm{T}=0.5$, $\mathsf{\beta }=1$, $\mathrm{t}=0$, $\mathrm{r}=0.05$, ${\mathrm{X}}_{\mathrm{e}}=40$, $\mathsf{\alpha }=0.2$, and $\mathsf{ϵ}=0.2$. The ranges of $\mathsf{\sigma }$ and $\mathsf{\delta }$ are $[0,0.6]$ and $[0,8]$, respectively. For a given $\mathsf{\delta }$, the price ratio changes only slightly as $\mathsf{\sigma }$ increases, whereas for fixed $\mathsf{\sigma }$ it rises with $\mathsf{\delta }$. For example, when $\mathsf{\sigma }=0.3$ and $\mathsf{\delta }$ increases from 3 to 5 and then to 8, the classic option price remains 60.9876, while the discrete strategy price increases from 17.7457 to 26.0187 and 35.2747, so the ratio rises from 0.2910 to 0.4266 and 0.5784. Throughout this range, the ratio is below 1, indicating a consistent price advantage of the discrete strategy over the classic option.

4.2. Sensitivity Analysis of the Option Price to Key Parameters

Figure 6, Figure 7, Figure 8 and Figure 9 illustrate how the option price varies with the underlying price S, exercise price $X_e$, investment interval $\mathsf{\delta }$, and interest rate r under different values of the maximum capital utilization ratio $\mathsf{\beta }$. Holding all other variables fixed and varying only one at a time, we observe that the option price increases with S, $\mathsf{\delta }$, and r, but decreases with $X_e$. Moreover, the higher the capital utilization ratio $\mathsf{\beta }$, the lower the option price.

5. Empirical Test

5.1. Data Source and Description

This study examines two European call options: the STAR Market 50 ETF call option expiring on 23 April 2025 (KeChuang 50 Call April 2025, strike 1.050; contract code: 10009005) and the E-mini $\mathrm{S}\&\mathrm{P}$ 500 futures call option on the ESH26 contract with strike 6300 (hereafter, $\mathrm{ESH}26\text{\_}6300\mathrm{C}$). The trading period for contract 10009005 runs from 27 February 2025 to 23 April 2025, covering 39 trading days. The trading period for contract $\mathrm{ESH}26\text{\_}6300\mathrm{C}$ runs from 19 March 2024 to 20 March 2026, covering 505 trading days. The remaining data analysis is presented below.

(1)

The underlying asset price S: The asset price for the current day is determined by the closing price of the previous day.

(2)

The validity period T: The validity period of an option is generally expressed in terms of trading days, with the standard convention in the stock market being that there are 252 trading days in a year. Thus, T is determined by dividing the number of trading days remaining until the option’s expiration by 252.

(3)

The risk-free interest rate r: For the SSE STAR Market 50 Index call option, we use the SHIBOR overnight rate as the risk-free interest rate, whereas for the E-mini $S\&P$ 500 futures call option on the ESH26 contract we use the Secured Overnight Financing Rate (SOFR) published by the Federal Reserve Bank of New York.

(4)

The volatility of the underlying asset prices $\mathsf{\sigma }$: Assuming that the future is a continuation of the past, we use historical volatility to measure the underlying asset’s volatility, which is obtained by calculating the standard deviation of the underlying asset’s daily logarithmic returns.

5.2. Model Parameter Calibration

To more accurately evaluate the performance of the option pricing model based on the discrete investment strategy, this study calibrates the model parameters $\mathsf{\beta }$, $\mathsf{\delta }$, and $\mathsf{ϵ}$ using option data. The parameters are estimated by minimizing the pricing error between the model-implied prices and observed market prices.

In the parameter estimation process, the optimal parameters are determined by minimizing the discrepancy between model-implied prices and observed market prices. Commonly used error metrics include the mean absolute error (MAE) and the root mean squared error (RMSE), which are defined as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{\mathrm{N}}}\sum _{\mathrm{i}=1}^{\mathrm{N}}\left|{\widehat{\mathrm{Y}}}_{\mathrm{i}}-{\mathrm{Y}}_{\mathrm{i}}\right|,$$

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{N}}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\widehat{\mathrm{Y}}}_{\mathrm{i}}-{\mathrm{Y}}_{\mathrm{i}}\right)}^2},$$

where $\widehat{\mathrm{Y}}$ denotes the model-implied price, $\mathrm{Y}$ denotes the market price, and $\mathrm{N}$ is the sample size.

Figure 10 compares the market price of a call option (10009005) with a strike price of 1.050 and an April maturity against the theoretical price from the classic Black–Scholes model and the option price implied by the discrete investment strategy. The green line represents the observed market prices, the orange line denotes the option prices under the discrete investment strategy, and the blue line indicates the prices given by the classical Black–Scholes model. After parameter calibration, the estimated parameters are $\mathsf{\beta }=0.75$, $\mathsf{\delta }=e^{-6}$ and $\mathsf{ϵ}=0.21$. The simulation results indicate that the option price under the discrete strategy is closer to the observed market price. Specifically, relative to the market data, the discrete strategy model achieves an MAE of 0.004 and an RMSE of 0.005, whereas the classic Black–Scholes model yields an MAE of 0.19 and an RMSE of 0.15, confirming that the discrete strategy pricing errors are smaller.

Figure 11 compares the market prices of the 6300-strike call option maturing in March ($\mathrm{ESH}26\text{\_}6300\mathrm{C}$) with the theoretical prices generated by the classic Black–Scholes model and by the option pricing formula under a discrete investment strategy. Using market data from 19 March 2024 to 11 June 2025, we calibrate the model parameters and obtain $\mathsf{\beta }=0.30$, $\mathsf{\delta }=978.91$, and $\mathsf{ϵ}=0.67$. The simulation results indicate that the option prices generated by the discrete strategy are closer to the observed market prices. In addition, the mean absolute error and root mean squared error between the discrete strategy prices and the market data are 17.71 and 21.17, respectively, whereas the MAE and RMSE of the Black–Scholes model relative to the market data are 133.20 and 137.30, confirming that the discrete strategy pricing exhibits smaller deviations from the observed prices.

6. Summary

This paper constructs a discrete investment strategy and introduces an early action mechanism: when the underlying asset price successively reaches the thresholds ${\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}$, ${\mathrm{X}}_{\mathrm{e}}$, ${\mathrm{S}}_1$, ${\mathrm{S}}_2$, …, ${\mathrm{S}}_{\mathrm{m}}$, …, ${\mathrm{S}}_{\mathrm{N}}$ $\left(({\mathrm{X}}_{\mathrm{e}}-\mathsf{ϵ}<{\mathrm{X}}_{\mathrm{e}}<{\mathrm{S}}_1<{\mathrm{S}}_2<{\mathrm{S}}_3,<\dots ,<{\mathrm{S}}_{\mathrm{m}},<\dots ,<{\mathrm{S}}_{\mathrm{N}})\right)$, the underlying asset is purchased in batches at predetermined proportions. On this basis, this paper derives a closed-form pricing formula for European options under a discrete investment strategy. A two-factor sensitivity analysis shows that, compared with the classical option pricing framework, option values under the discrete strategy are systematically lower. Moreover, the one-factor sensitivity analysis indicates that the option price decreases as the maximum capital allocation ratio $\mathsf{\beta }$ increases. Using market data on the SSE STAR 50 ETF call option and the $\mathrm{ESH}26\text{\_}6300\mathrm{C}$ call option, we further calibrate the model parameters and find that the pricing errors between the theoretical values implied by the discrete strategy and the observed option prices are significantly smaller than those of the classical model. This confirms the effectiveness of the proposed strategy. Overall, the findings of this paper provide useful implications for option pricing and risk management practice. First, the lower option prices generated by the discrete strategy offer investors and risk managers a practical tool for controlling option costs and building in a safety margin. Second, by dynamically adjusting entry timing and the position size according to pre-specified price thresholds, investors can strengthen protection against downside risk while preserving exposure to market upside, thus achieving more resilient portfolio allocation and decision-making in highly volatile markets.

This study still has several limitations, as investor behavior is shaped by many factors that are not yet fully captured in our framework. Future research will proceed along two main directions. First, we will enrich the discrete investment strategy to incorporate more realistic trading rules, such as take-profit and stop-loss orders, risk-management constraints, and subjective judgments that may lead investors to reduce positions, lock in gains, or even short the underlying, so that position dynamics need not be monotonic. Second, we will relax the basic modeling assumptions. In particular, instead of assuming constant volatility, we plan to model volatility as a stochastic process; the Heston framework, which extends the Black–Scholes equation by allowing volatility to follow Cox–Ingersoll–Ross dynamics, offers a natural starting point [22,23]. We also intend to incorporate transaction costs—e.g., along the lines of Leland’s scheme that distributes costs over an infinite sequence of trading intervals so that cumulative costs are proportional to the asset value [24]—as well as dividends, taxes, and other market frictions, to make the model more consistent with actual market conditions.