An Option Pricing Formula for Active Hedging Under Logarithmic Investment Strategy

Abstract

Classic options can no longer meet the diversified needs of investors; thus, it is of great significance to construct and price new options for enriching the financial market. This paper proposes a new option pricing model that integrates the logarithmic investment strategy with the classic Black–Scholes theory. Specifically, this paper focus on put options, introducing a threshold-based strategy whereby investors sell stocks when prices fall to a certain value. This approach mitigates losses from adverse price movements, enhancing risk management capabilities. After deriving an analytical solution, we utilized mathematical software to visualize the factors influencing new option prices in three-dimensional space. The findings suggest that the pricing of these new options is influenced not only by standard factors such as the underlying asset price, volatility, risk-free rate of interest, and time to expiration, but also by investment strategy parameters such as the investment strategy index, investment sensitivity, and holding ratios. Most importantly, the pricing of new put options is generally lower than that of classic options, with numerical simulations demonstrating that under optimal parameters the new options can achieve similar hedging effectiveness at approximately three-quarters the cost of standard options. These findings highlight the potential of logarithmic investment strategies as effective tools for risk management in volatile markets. To validate our theoretical model, numerical simulations using data from Shanghai 50 ETF options were used to confirm its accuracy, aligning well with theoretical predictions. The new option model proposed in this paper contributes to enhancing the efficiency of resource allocation in capital markets at a macro level, while at a micro level, it helps investors to apply investment strategies more flexibly and reduce decision-making errors.

1. Introduction

The acceleration of financial globalization has introduced significant volatility into commodity markets, driving enterprises and investors to adopt innovative financial instruments for risk management and return optimization. Options are a prominent class of financial derivatives that have gained widespread popularity among market participants for their utility in both hedging and arbitrage strategies. Over the past five decades, options trading has expanded to account for more than two-thirds of the global derivatives market, reflecting its effectiveness and increasing sophistication. As technological advancements and market complexities evolve, financial institutions are compelled to continually develop new products, giving rise to exotic options. Unlike classic options, exotic options are non-standardized instruments, characterized by unique cash flow structures and payment methods tailored to specific needs, and are often referred to as special-purpose or customized options. This study focuses on exotic options by integrating discrete investment strategies with classic options to construct and price a novel exotic option.

The option pricing formula was first derived by Black and Scholes in 1973 and was subsequently extended by Merton [1,2]. In their framework, a portfolio comprising an option and its underlying asset is constructed, and the positions dynamically adjusted to ensure that the portfolio’s return matches the risk-free rate of interest over an infinitesimally short time interval. This approach leads to the derivation of a partial differential equation that characterizes the behavior of the option’s price:

(1)$V_2=rV-rS{V}_1-{\displaystyle \frac{1}{2}}{\nu }^2{S}^2{V}_{11}$

where $V_1,V_2$ represent the first-order partial derivatives of the option price V on stock price S and time t, respectively, $V_{11}$ denotes the second-order partial derivative of the option price V on stock price S, $\nu$ signifies the volatility of stock returns, and r denotes the risk-free rate of interest. By defining $t^{*}$ as the time to maturity and c as the exercise price, we arrive at the following equation:

(2)$V(S,t^{*})=\left\{\begin{array}{cc}{\displaystyle S-c,}\hfill & {\displaystyle S\ge c,}\hfill \\ {\displaystyle 0,}\hfill & {\displaystyle S<c.}\hfill \end{array}\right.$

This equation has a unique solution $V(S,t)$ that satisfies the differential Equation (8) under the boundary condition (2). By transforming the equation into the form of the heat conduction equation and solving it, we present the pricing formula for the European call option, as shown below:

(3)$V(S,t)=SN\left(d_1\right)-c{e}^{-r(t^{*}-t)}N\left(d_2\right),$

where $d_1={\displaystyle \frac{ln\frac{S}{c}+(t^{*}-t)(r+\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}},d_2=d_1-\nu \sqrt{t^{*}-t}={\displaystyle \frac{ln\frac{S}{c}+(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}$, and $N(·)$ denotes the cumulative normal density function.

The Black–Scholes model has become an established foundational framework for the rational pricing of financial derivatives, and is extensively utilized across various financial domains. Concurrently, researchers have actively sought to explore and enhance option pricing methodologies. The Black–Scholes model is predicated on an idealized market that assumes the absence of taxes and transaction costs, which deviates significantly from real-world financial markets. As a result, scholars have endeavored to relax these assumptions in order to broaden the applicability of the classical model. Leland (1985) incorporated the magnitude and frequency of transaction costs into an option pricing model that includes proportional transaction costs [3]. Shokrollahi (2016) derived an option pricing formula accounting for transaction costs under the assumption that the price of the underlying asset follows a mixed fractional Brownian motion [4]. Merton (1978, 1982) formulated option pricing models that account for situations involving known dividend payments on stocks and stochastic interest rates [5,6]. Schroder (1999) simplified the option pricing framework through numerical transformation and formulated an option price model in random dividend scenarios [7]. Krausz (1985) and Shackleton and Wojakowski (2001) constructed pricing models for American and European options, respectively, that accommodate continuous dividend payments [8,9]. Heston (1993) employed a mean reversion process to characterize volatility randomness, thereby enriching option pricing methodologies [10]. Cox (1996) introduced the constant elasticity of variance (CEV) model, wherein variance is contingent on stock price while its elasticity remains independent of stock price dynamics [11]. Hull and White (1987) subsequently extended this model to consider scenarios with nonzero correlations between stock prices and volatility [12]. Furthermore, Merton (1976) and Madan (1998) proposed the classical hybrid jump-diffusion and variance gamma models, respectively, by incorporating jump components into the underlying asset price dynamics to mitigate pricing biases [13,14]; Researchers such as Ernesto Mordecki (2002), Jong Jun Park (2020), and Brignone Riccardo (2023) have also investigated option pricing challenges under conditions involving jumps in the underlying asset price changes [15,16,17].

Additionally, numerous scholars have investigated alternative methodologies for developing option pricing models. Cox et al. (1979) introduced the binomial tree method for pricing American options, which segments the option’s validity period into discrete intervals. This method assumes that the stock price can either increase or decrease at each step, creating a binomial tree structure. By working backward from the option’s strike price and stock price, they derived the expected option price [18]. Breen (1991) further enhanced this approach with the “accelerated binomial option pricing model”, which offers improved speed and efficiency compared to the traditional model [19]. Longstaff et al. (2001) applied the Monte Carlo method for calculating American option prices [20]. With advancements in technology, nonparametric machine learning algorithms have also been applied to asset pricing. Hutchinson et al. (1994) were pioneers in utilizing radial basis networks and backpropagation neural networks for option pricing, achieving accuracy on par with classic parametric models [21].

Meanwhile, a variety of exotic options have emerged, evolving from classic options and typically traded over the counter. These exotic options primarily include barrier options, Asian options, lookback options, and others. They offer flexibility and diversity, catering to the specific needs of investors and providing a broader range of investment choices; however, their inherent complexity poses challenges for pricing. Ito’s lemma serves as a foundational framework for pricing exotic options, allowing for the analysis of various forms of these derivatives. A partial differential equation analogous to Equation (1) can be derived for each type of exotic option; however, the corresponding boundary conditions differ among them. Therefore, identifying the boundary conditions that affect the final value is crucial in exotic option pricing.

While the rise of diverse exotic options has significantly advanced the options market, many studies overlook the proactive role of investors, who often adjust their positions in response to fluctuations in the underlying asset’s price. In particular, an investor might purchase a call option while simultaneously shorting the underlying stock to manage short-selling risks. Short-term investors may choose to cover their short positions to mitigate losses or even realize gains when the stock price rises. In contrast, long-term investors are typically advised to avoid covering short positions. Similarly, purchasing a put option while holding the underlying stock can help limit potential losses in the event of a price decline. Short-term investors might opt to sell the stock if its price falls, thereby reducing losses or securing gains, whereas long-term investors are likely to maintain their positions. Investors face numerous possibilities and scenarios for buying or selling stocks throughout the life of the option. Key factors to consider include price movements, the range of prices at which investors choose to act, the allocation of capital for stock purchases, and whether investors adhere to a discrete or continuous investment strategy.

Wang et al. (2009) were the first to derive a call option pricing model incorporating a linear investment strategy [22]. In this model, an investor follows a predetermined strategy that leads them to initiate buying or selling the stock upon reaching the exercise price,. Specifically, the option holder buys the stock when the stock price hits the exercise price for a call option, and sells it when the price hits the exercise price for a put option. However, Wang et al.’s assumption that the proportion of stock varies linearly and continuously with the stock price poses limitations, as continuous trading in the stock market incurs significant costs. To address this issue, Li et al. (2019) investigated option pricing in a discrete investment strategy as a way of enhancing the model’s practical applicability [23]. Additionally, prospect theory suggests that investors experience decreasing marginal utility in response to gains, while their marginal aversion increases in the face of losses, indicating that stock proportions should not correlate linearly with stock prices [24]. Acknowledging this, Qiao et al. (2021) proposed a logarithmic investment strategy [25]. Wu et al. (2023) further refined this approach by advancing the threshold for purchasing stock, leading to an exotic option pricing model grounded in a proactive investment strategy [26]. In summary, prior research has predominantly focused on European call options through specific investment strategies, raising the pertinent question of whether the same investment strategy could be applied to put options in order to effectively reduce their pricing.

Motivated by this question, the aim of the current research is to present a put option pricing model incorporating a specific investment strategy. In contrast to the classic option pricing model, a central assumption of this paper is that investors will adopt a logarithmic investment strategy to adjust their position proportions based on stock price trends after acquiring the option, thereby modifying the option’s intrinsic value function.

The rest of this paper is structured as follows: Section 2 reviews the principles of option pricing; Section 3 outlines the fundamental assumptions of the new pricing model, establishes the logarithmic investment strategy, and calculates the intrinsic value function of the new put option; Section 4 demonstrates the analytical solution of the new put option using the Black–Scholes partial differential equation; Section 5 presents a sensitivity analysis of the factors affecting the new option’s price; in Section 6, data from the Shanghai 50ETF are used for empirical testing to further validate the effectiveness of the new model; finally, Section 7 summarizes the main conclusions and outlines future research directions.

2. New Option Pricing Formula with Brownian Motion

2.1. The Law of Stock Price Change and Brownian Motion

Bachelier first applied Brownian motion to model stock price fluctuations in 1900 [27]. Brownian motion, also known as the Wiener process, is a continuous-time stochastic process that captures random motion. Mathematically, it can be defined as follows:

(1) $B\left(t\right)$ is a random process with $B\left(0\right)=0$.

(2) For any $\Delta t>0$, the increments $B(t+\Delta t)-B\left(t\right)$ are independent and normally distributed with mean 0 and variance $\Delta t$.

(3) The path of $B\left(t\right)$ is continuous but not differentiable at any point, representing erratic changes.

A standard Brownian motion $B\left(t\right)$ satisfies the above properties, where t represents time and the increments $B\left(t\right)-B\left(s\right)$ for $s<t$ are normally distributed with mean 0 and variance $t-s$.

In financial markets, the evolution of stock prices can be expressed as follows:

(4)$ln{\displaystyle \frac{S\left(t\right)}{S_0}}=(r-{\displaystyle \frac{{\nu }^2}{2}})t+\nu B\left(t\right),$

which can be rearranged to the following:

(5)$S\left(t\right)=S_0\times exp\left((r-{\displaystyle \frac{{\nu }^2}{2}})t+\nu B\left(t\right)\right).$

This formulation demonstrates that the price of the underlying asset evolves as a stochastic process. Specifically, the logarithm of the stock price follows the Brownian motion, leading to the conclusion that asset prices follow a geometric Brownian motion.

2.2. The Ito Formula

From the last section, it can be seen that the price change of the underlying asset in a risk-neutral world can be expressed by the Equation (5).

Because $S\left(t\right)=f(X_t,t),X_t=B_t$, where $f(x,t)=S_0\times exp\left((r-{\displaystyle \frac{{\nu }^2}{2}})t+\nu x\right)$. Thus,

(6)$\begin{array}{cc}{\displaystyle dS\left(t\right)}\hfill & =S_0[(r-{\displaystyle \frac{{\nu }^2}{2}})+{\displaystyle \frac{{\nu }^2}{2}}]\times exp\left((r-{\displaystyle \frac{{\nu }^2}{2}})t+\nu B\left(t\right)\right)dt+S_0\nu \times exp\left((r-{\displaystyle \frac{{\nu }^2}{2}})t+\nu B\left(t\right)\right)d{B}_t\hfill \\ {\displaystyle }\hfill & =rS\left(t\right)dt+\nu S\left(t\right)d{B}_t.\hfill \end{array}$

Therefore, the stock price satisfies the following differential equation:

(7)${\displaystyle \frac{d{S}_t}{S_t}}=rdt+\nu d{B}_t$

where ${\displaystyle \frac{d{S}_t}{S_t}}$ represents the return of $S_t$ in time period $dt$, $rdt$ is the risk-free rate of $S_t$ in the expected growth rate form, $\nu d{B}_t$ denotes the volatility of $S_t$’s return, and $\nu$ is referred to as the volatility.

2.3. European Call Option Pricing Formula

In 1973, Black and Scholes demonstrated that, under correct option pricing conditions, no arbitrage opportunities are formed by constructing portfolios with both long and short positions in options and their underlying assets. This insight formed the foundation of a theoretical framework for valuing options, encapsulated in the Black–Scholes PDE (partial differential equation):

(8)$\left\{\begin{array}{c}{\displaystyle V_2=rV-rS{V}_1-\frac{1}{2}{\nu }^2{S}^2{V}_{11},}\hfill \\ {\displaystyle V(S,t^{*})=\left\{\begin{array}{c}{\displaystyle {(S-c)}^{+},\left(\mathit{Call}\right)}\hfill \\ {\displaystyle {(c-S)}^{-},\left(\mathit{Put}\right)}\hfill \end{array}\right.}\hfill \end{array}\right.$

where V is the option’s price, t is the current time, r is the risk-free rate of interest, $\nu$ is the volatility of stock price, and S is the current stock price.

Lemma 1. 

Given a function f with finite expectation $E\left[f\left[W\left[t^{*}\right]\right]\right]<\infty$, for any $t<t^{*}$, one has

(9) ${\tilde{E}}_t\left[f\left[B\left[t^{*}\right]\right]\right]={\int }_{\mathbb{R}}{\displaystyle \frac{1}{\sqrt{2\pi \left(t^{*}-t\right)}}}exp\left(-{\displaystyle \frac{{(S-B\left[t\right])}^2}{2(t^{*}-t)}}\right).$

3. European Put Option with the Logarithmic Investment Strategy

3.1. Basic Assumptions

The basic assumptions of the new option pricing model are broadly consistent with those of the Black–Scholes model, as follows:

• (1). Asset prices follow a geometric Brownian motion described by

  (10)$d{S}_t=\mu Sdt+\nu Sd{B}_t,$

  where $\mu$ denotes the asset’s expected rate of return, $\nu$ represents the volatility of the asset price, and $d{B}_t$ corresponds to the increment of a Wiener process (Brownian motion).

• (2). The risk-free interest rate, represented by r, remains constant over the investment period.

• (3). No dividends are paid.

• (4). Neither transaction costs nor taxes are considered in the model.

• (5). No arbitrage opportunities exist.

• (6). Investors adjust their positions dynamically in response to changes in asset prices.

Assumption (6) is the key assumption that distinguishes the new option model from the classic option model.

3.2. Establishment of the Logarithmic Investment Strategy

Assume that the total investment amount is A with a holding ratio b, which is defined as the proportion of total assets allocated to stocks. Let S denote the stock price, b represent the initial holding ratio, and c be the exercise price of the put option. A threshold, expressed as $(c-\xi )$ (where $c>\xi >0$), triggers the implementation of a nonlinear investment strategy. The parameter $\xi$ reflects the investor’s sensitivity to market movement or tolerance for risk, a concept we refer to as ‘investment lag’. A higher value of $\xi$ indicates a greater investment lag, signifying stronger risk tolerance and lower sensitivity to price changes. The range of $\xi$ extends from $[0,+\infty )$. The investor is assumed to reduce stock holdings as the stock price declines from $(c-\xi )$ to $(1-a)(c-\xi )$, where a represents the investment strategy index. A higher a ($0<a<1$) implies that the stock-selling process occurs over a broader range of price fluctuations. The specific strategy employed by the investor is expressed as shown in Figure 1.

(11)$\beta \left[S\right]=\left\{\begin{array}{cc}{\displaystyle b,}\hfill & {\displaystyle 0\le S<(1-a)(c-\xi ),}\hfill \\ {\displaystyle b+\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}ln\left[\frac{S}{(1-a)(c-\xi )}\right],}\hfill & {\displaystyle (1-a)(c-\xi )\le S<(c-\xi ),}\hfill \\ {\displaystyle b_0,}\hfill & {\displaystyle c-\xi \le S}\hfill \end{array}\right.$

3.3. Defining the Loss Function

When the stock price is S, investors can purchase ${\displaystyle \frac{A}{S}}$ shares by allocating all available funds to the stock. If investors anticipate a decline in the stock price from S to $S-\Delta S$, the reduction in the portfolio’s value due to the sale of shares is provided by $\Delta q=\Delta QA=A{Q}^{\prime }\left[S\right]\Delta S$, where $\Delta Q$ is the change in the holding proportion. If the stock price decreases from S to $S_0$, then the expected return from selling the stock is as follows: $R={\displaystyle \frac{\Delta q}{S}}(S-S_0)={\displaystyle \frac{A{Q}^{\prime }\left[S\right]\Delta S}{S}}(S-S_0)$, where ${\displaystyle \frac{\Delta q}{S}}$ is the number of shares sold. Therefore, if the stock price falls from c to $S_0$, the expected return is provided by the following:

(12)$\begin{array}{cc}{\displaystyle R\left(S_0\right)}\hfill & =\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle S\ge c-\xi ,}\hfill \\ {\displaystyle {\int }_{S_0}^{c-\xi }\frac{Q^{\prime }\left[S\right]A}{c}(c-\xi -S)dS+\frac{Q\left[S_0\right]A}{c}(c-S_0),}\hfill & {\displaystyle (1-a)(c-\xi )\le S<c-\xi ,}\hfill \\ {\displaystyle {\int }_{(1-a)(c-\xi )}^{c-\xi }\frac{Q^{\prime }\left[S\right]A}{c}(c-\xi -S)dS+\frac{\beta A}{c}(c-S_0),}\hfill & {\displaystyle 0\le S<(1-a)(c-\xi )}\hfill \end{array}\right.\hfill \\ {\displaystyle }\hfill & =\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle S\ge c-\xi ,}\hfill \\ {\displaystyle \frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left((c-\xi )ln\left[\frac{1}{1-a}\right]-a(c-\xi )\right),}\hfill & {\displaystyle (1-a)(c-\xi )\le S<c-\xi ,}\hfill \\ {\displaystyle \frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left((c-\xi )ln\left[\frac{c-\xi }{S_0}\right]-(c-\xi -S_0)\right),}\hfill & {\displaystyle 0\le S<(1-a)(c-\xi ).}\hfill \end{array}\right.\hfill \end{array}$

If the investor fails to reduce their position as the stock price falls from c to $S_0$, the resulting loss can be expressed as

$L={\displaystyle \frac{A}{c}}(c-S_0).$

If the investor reduces their position in accordance with the aforementioned strategy as the stock price declines from c to $S_0$, then a portion of the potential loss will be offset. Consequently, the total loss incurred by the investor can be expressed as follows:

(13)$L=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle c\le S_0,}\hfill \\ {\displaystyle \frac{A}{c}(c-S_0),}\hfill & {\displaystyle c-\xi \le S_0<c,}\hfill \\ {\displaystyle {\int }_{S_0}^{c-\xi }\frac{Q^{\prime }\left[S\right]A}{c}(c-\xi -S)dS+\frac{Q\left[S_0\right]A}{c}(c-S_0),}\hfill & {\displaystyle (1-a)(c-\xi )\le S_0<c-\xi ,}\hfill \\ {\displaystyle {\int }_{(1-a)(c-\xi )}^{c-\xi }\frac{Q^{\prime }\left[S\right]A}{c}(c-\xi -S)dS+\frac{bA}{c}(c-S_0),}\hfill & {\displaystyle 0\le S_0<(1-a)(c-\xi ),}\hfill \end{array}\right.$

Therefore, the loss per share is as

(14)$L^{\prime }={\displaystyle \frac{L}{\frac{A}{c}}}$

$=\left\{\begin{array}{cc}{\displaystyle V_{t^{*}1}\left[S_0\right],}\hfill & {\displaystyle c\le S_0,}\hfill \\ {\displaystyle V_{t^{*}2}\left[S_0\right],}\hfill & {\displaystyle c-\xi \le S_0<c,}\hfill \\ {\displaystyle V_{t^{*}3}\left[S_0\right],}\hfill & {\displaystyle (1-a)(c-\xi )\le S_0<c-\xi ,}\hfill \\ {\displaystyle V_{t^{*}4}\left[S_0\right],}\hfill & {\displaystyle 0\le S_0<(1-a)(c-\xi ),}\hfill \end{array}\right.$

where

(15)$\begin{array}{cc}\hfill & {\displaystyle V_{t^{*}1}\left[S_0\right]=0,}\hfill \\ \hfill & {\displaystyle V_{t^{*}2}\left[S_0\right]=c-S_0,}\hfill \\ \hfill & {\displaystyle V_{t^{*}3}\left[S_0\right]=\left(b+\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}ln\left[\frac{S_0}{(1-a)(c-\xi )}\right]\right)(c-\xi -S_0)}\hfill \\ \hfill & {\displaystyle +\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left((c-\xi )ln\left[\frac{c-\xi }{S_0}\right]-(c-\xi -S_0)\right)}\hfill \\ {\displaystyle }\hfill & V_{t^{*}4}\left[S_0\right]={\displaystyle \frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}}\left((c-\xi )ln\left[{\displaystyle \frac{1}{1-a}}\right]-a(c-\xi )\right)+b(c-\xi -S_0).\hfill \end{array}$

Therefore, the intrinsic value of a put option can be expressed as follows:

(16)$V_t^{*}(S\left(t\right),t)=\left\{\begin{array}{cc}{\displaystyle V_{t^{*}1}\left[S_0\right],}\hfill & {\displaystyle c\le S_0,}\hfill \\ {\displaystyle V_{t^{*}2}\left[S_0\right],}\hfill & {\displaystyle c-\xi \le S_0<c,}\hfill \\ {\displaystyle V_{t^{*}3}\left[S_0\right],}\hfill & {\displaystyle (1-a)(c-\xi )\le S_0<c-\xi ,}\hfill \\ {\displaystyle V_{t^{*}4}\left[S_0\right],}\hfill & {\displaystyle 0\le S_0<(1-a)(c-\xi ).}\hfill \end{array}\right.$

4. Derivation Process

According to the classical Black–Scholes PDE,

${\displaystyle \frac{\partial f}{\partial t}}+rS{\displaystyle \frac{\partial f}{\partial S}}+{\displaystyle \frac{{\nu }^2}{2}}{S}^2{\displaystyle \frac{{\partial }^{2}f}{\partial S^2}}=rf,$

where f is the price of the derivative, t represents time, r is the risk-free interest rate, $\nu$ is the volatility of the stock price, and S is the stock price. The analytical solution to the Black–Scholes PDE is as follows:

(17)$V[S,t]={\displaystyle \frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}}{\int }_{-\infty }^{+\infty }f(S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-{\displaystyle \frac{{\nu }^2}{2}})\right))e^{-{\displaystyle \frac{X^2}{2}}}dX$

where $t^{*}$ is expiration date and $f(S,t)$ is the intrinsic value function of the option. At maturity ($t=t^{*}$), the intrinsic value function $f(S,t)$ corresponds to the option’s payoff. For a standard European put option, the intrinsic value function can be expressed as follows:

(18)$f(S,t^{*})=\left\{\begin{array}{cc}{\displaystyle 0,}\hfill & {\displaystyle S\left(t^{*}\right)<c,}\hfill \\ {\displaystyle S\left(t^{*}\right)-c.}\hfill & {\displaystyle S\left(t^{*}\right)\ge c.}\hfill \end{array}\right.$

The intrinsic value function of a put option reflects the per-share loss, and is expressed by $f(S,t)$, i.e.

(19)$f(S,t)=\left\{\begin{array}{cc}{\displaystyle V_{t^{*}1}\left[S_0\right],}\hfill & {\displaystyle c\le S_0,}\hfill \\ {\displaystyle V_{t^{*}2}\left[S_0\right],}\hfill & {\displaystyle c-\xi \le S_0<c,}\hfill \\ {\displaystyle V_{t^{*}3}\left[S_0\right],}\hfill & {\displaystyle (1-a)(c-\xi )\le S_0<c-\xi ,}\hfill \\ {\displaystyle V_{t^{*}4}\left[S_0\right].}\hfill & {\displaystyle 0\le S_0<(1-a)(c-\xi ).}\hfill \end{array}\right.$

Pricing Formula for the European Put Option

For a European put option, the intrinsic value function corresponds to the previously defined loss function, and is provided by

(20)$V[S,t]=V_1[S,t]+V_2[S,t]+V_3[S,t]+V_4[S,t],$

where

$\begin{array}{cc}\hfill & {\displaystyle V_1[S,t]=\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{S\ge c}{V}_{t^{*}1}[S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-\frac{{\nu }^2}{2})\right),t]e^{-\frac{X^2}{2}}dX,}\hfill \\ \hfill & {\displaystyle V_2[S,t]=\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{c-\xi \le S\le c}{V}_{t^{*}2}[S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-\frac{{\nu }^2}{2})\right),t]e^{-\frac{X^2}{2}}dX,}\hfill \\ \hfill & {\displaystyle V_3[S,t]=\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{(1-a)(c-\xi )\le S<c-\xi }{V}_{t^{*}3}[S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-\frac{{\nu }^2}{2})\right),t]e^{-\frac{X^2}{2}}dX,}\hfill \\ \hfill & {\displaystyle V_4[S,t]=\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{0\le S<(1-a)(c-\xi )}{V}_{t^{*}4}[S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-\frac{{\nu }^2}{2})\right),t]e^{-\frac{X^2}{2}}dX.}\hfill \end{array}$

Because $V_{t^{*}1}=0$, it follows that $V_1[S,t]=0$. Let $\Pi =S\times exp\left(\nu \sqrt{t^{*}-t}+(t^{*}-t)(r-{\displaystyle \frac{{\nu }^2}{2}})\right)$; then, the integral boundaries can be reformulated.

When $c-\xi \le \Pi <c$, the bounds on X are as follows:

(21)${\displaystyle \frac{ln\left[\frac{c-\xi }{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}\le X<{\displaystyle \frac{ln\left[\frac{c}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}.$

When $(1-a)(c-\xi )\le \Pi <(c-\xi )$, the corresponding bounds on X are as follows:

(22)${\displaystyle \frac{ln\left[\frac{(1-a)(c-\xi )}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}\le X<{\displaystyle \frac{ln\left[\frac{c-\xi }{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}.$

When $\Pi <(1-a)(c-\xi )$, then

$X<{\displaystyle \frac{ln\left[\frac{(1-a)(c-\xi )}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}}.$

To facilitate simplification, we let

(23)$\begin{array}{cc}\hfill & {\displaystyle d_1=\frac{ln\left[\frac{(1-a)(c-\xi )}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}},}\hfill \\ \hfill & {\displaystyle d_2=\frac{ln\left[\frac{c-\xi }{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}},}\hfill \\ \hfill & {\displaystyle d_3=\frac{ln\left[\frac{c}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}},}\hfill \\ \hfill & {\displaystyle d_4=d_1-\nu \sqrt{t^{*}-t},}\hfill \\ \hfill & {\displaystyle d_5=d_2-\nu \sqrt{t^{*}-t},}\hfill \\ \hfill & {\displaystyle d_6=d_3-\nu \sqrt{t^{*}-t}.}\hfill \end{array}$

We further introduce the term $\Sigma =\nu \sqrt{t^{*}-t}X+(t^{*}-t)(r-{\displaystyle \frac{{\nu }^2}{2}})$ to streamline the expression of the option’s value. Accordingly, the value function $V_2[S,t]$ is provided by the following:

$\begin{array}{cc}{V}_2[S,t]\hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_2}^{d_3}(c-S)e^{-\frac{X^2}{2}}dX=(c-S)e^{-r(t^{*}-t)}(N\left[d_3\right]-N\left[d_2\right])}\hfill \end{array}$

where $N[·]$ denotes the cumulative distribution function of the standard normal distribution:

(24)$\begin{array}{cc}{\displaystyle V_3[S,t]}\hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}(-M{e}^{\Sigma }-H{e}^{\Sigma }X+c+PX)e^{-\frac{X^2}{2}}dX}\hfill \\ \hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}(-M{e}^{\Sigma }-H{e}^{\Sigma }X+c)e^{-\frac{X^2}{2}}dX+P\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(e^{\frac{d_1^2}{2}-\frac{d_2^2}{2}}\right)}\hfill \\ \hfill & {\displaystyle =-\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}M{e}^{\Sigma }{e}^{-\frac{X^2}{2}}dX-\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}H{e}^{\Sigma }X{e}^{-\frac{X^2}{2}}dX+\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}c{e}^{-\frac{X^2}{2}}dX+P\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(e^{\frac{d_1^2}{2}-\frac{d_2^2}{2}}\right)}\hfill \\ \hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}c{e}^{-\frac{X^2}{2}}dX+P\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(e^{\frac{d_1^2}{2}-\frac{d_2^2}{2}}\right)-\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{d_1}^{d_2}H{e}^{\sum }X{e}^{-\frac{X^2}{2}}dX}\hfill \\ \hfill & {\displaystyle -\frac{M}{\sqrt{2\pi }}{\int }_{d_4}^{d_5}{e}^{-\frac{{(X-\nu \sqrt{t^{*}-t})}^2}{2}}d(X-\nu \sqrt{t^{*}-t})}\hfill \\ \hfill & {\displaystyle =-M(N\left[d_5\right]-N\left[d_4\right])-\frac{H}{\sqrt{2\pi }}\left((e^{-\frac{d_4^2}{2}}-e^{-\frac{d_5^2}{2}})+\nu \sqrt{t^{*}-t}\sqrt{2\pi }(N\left[d_5\right]-N\left[d_4\right])\right)}\hfill \end{array}$

(25)$\begin{array}{cc}{\displaystyle V_4[S,t]}\hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_0^{d_1}\left[\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left((c-\xi )ln\left[\frac{1}{1-a}\right]-a(c-\xi )\right)+b(c-\xi -\Pi )\right]e^{-\frac{X^2}{2}}dX}\hfill \\ \hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_{-\infty }^{d_1}\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left((c-\xi )ln\left[\frac{1}{1-a}\right]-a(c-\xi )\right)+b(c-\xi )-b\Pi )\right)e^{-\frac{X^2}{2}}dX}\hfill \\ \hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left(ln\left[\frac{1}{1-a}\right]-a\right)(c-\xi )+b(c-\xi )\right){\int }_0^{d_1}{e}^{-\frac{X^2}{2}}dX-b\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}{\int }_0^{d_1}S{e}^{\Sigma }{e}^{-\frac{X^2}{2}}dX}\hfill \\ \hfill & {\displaystyle =\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left(ln\left[\frac{1}{1-a}\right]-a\right)(c-\xi )+b(c-\xi )\right)\sqrt{2\pi }N\left[d_1\right]-\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}bS{e}^{r(t^{*}-t)}\sqrt{2\pi }N\left[d_4\right]}\hfill \\ \hfill & {\displaystyle =e^{-r(t^{*}-t)}\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left(ln\left[\frac{1}{1-a}\right]-a\right)(c-\xi )+b(c-\xi )\right)N\left[d_1\right]-bSN\left[d_4\right].}\hfill \end{array}$

(26)$\begin{array}{cc}V[S,t]\hfill & {\displaystyle =V_1[S,t]+V_2[S,t]+V_3[S,t]+V_4[S,t]}\hfill \\ \hfill & {\displaystyle =\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\left(ln\left[\frac{1}{1-a}\right]-a\right)(c-\xi )+b(c-\xi )\right)e^{-r(t^{*}-t)}N\left[d_1\right]-bSN\left[d_4\right]-\frac{H}{\sqrt{2\pi }}(e^{-\frac{d_4^2}{2}}-e^{-\frac{d_5^2}{2}})}\hfill \\ \hfill & {\displaystyle + P\frac{e^{-r(t^{*}-t)}}{\sqrt{2\pi }}\left(e^{\frac{d_1^2}{2}-\frac{d_2^2}{2}}\right)+(c-S)e^{-r(t^{*}-t)}(N\left[d_3\right]-N\left[d_2\right])+(-M-H\nu \sqrt{t^{*}-t})(N\left[d_5\right]-N\left[d_4\right])}\hfill \\ \hfill & {\displaystyle + L{e}^{-r(t^{*}-t)}(N\left[d_2\right]-N\left[d_1\right]),}\hfill \end{array}$

where

$\begin{array}{cc}M\hfill & {\displaystyle =\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}ln\left[\frac{S}{(1-a)(c-\xi )}\right]-\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]-b}\right)+ln\left[\frac{1}{1-a}\right](t^{*}-t)(r-\frac{{\nu }^2}{2})\right)S,}\hfill \\ H\hfill & {\displaystyle =\left(\frac{b_0-b}{ln\left[\frac{1}{1-a}\right]}\nu \sqrt{t^{*}-t}\right)S,}\hfill \\ L\hfill & {\displaystyle =\left(\frac{(b_0-b)(c-\xi )}{ln\left[\frac{1}{1-a}\right]}ln\left[\frac{c-\xi }{S_0}\right]-\frac{(b_0-b)(c-\xi )}{ln\left[\frac{1}{1-a}\right]}+b(c-\xi )+\frac{(b_0-b)(c-\xi )}{ln\left[\frac{1}{1-a}\right]}ln\left[\frac{S}{(1-a)(c-\xi )}\right]\right.}\hfill \\ \hfill & {\displaystyle \left.+ \frac{(b_0-b)(c-\xi )}{ln\left[\frac{1}{1-a}\right]}(t^{*}-t)(r-\frac{{\nu }^2}{2})\right)S,}\hfill \\ P\hfill & {\displaystyle =\frac{(b_0-b)(c-\xi )}{ln\left[\frac{1}{1-a}\right]}\nu \sqrt{t^{*}-t}.}\hfill \end{array}$

We discuss a special case to preliminarily verify the correctness of the model. When $\xi =0$ and $b_0=b=1$, the investor refrains from selling the stock, and Equation (26) should equal the classic European put option; in this case, the option value is provided by the following:

(27)$V[S,t]=c{e}^{-r(t^{*}-t)}N\left[d_6\right]-SN\left[d_3\right]$

where

(28)$d_3={\displaystyle \frac{ln\left[\frac{c}{S}\right]-(t^{*}-t)(r-\frac{{\nu }^2}{2})}{\nu \sqrt{t^{*}-t}}},d_6=d_3+\nu \sqrt{t^{*}-t}.$

5. Numerical Simulation

It is important to highlight that as the newly proposed exotic option structure is quite intricate, it is necessary to comprehensively evaluate its price influencing factors in order to better understand its risk and return characteristics and help investors to effectively manage and control various market risks. This research analyzes how varying parameter values affect the pricing behavior of the exotic option in comparison to a standard option. The values of the other parameters are as follows: $t=0, c=30, S=30, t^{*}=0.5, r=0.025$.

The key results that can be drawn from Table 1 are as follows:

• (1). Consistent Pricing Difference across Strategies: The price of the new option remains consistently lower than that of the classic option across all parameter configurations, confirming that active investment strategies contribute to lowering the new option’s value by mitigating risks. This aligns with the idea that incorporating subjective initiative into trading practices reduces the option’s intrinsic value.

• (2). Effect of Parameter a on Option Prices: Increasing a (from 0.5 to 0.8) generally results in a lower price for the new option. This suggests that as the investor’s confidence in their strategy increases (or as more weight is placed on certain signals), the option becomes less valuable, reflecting reduced uncertainty; for example, with $b=0.5, \xi =0, b_0=0.6$, the new option price decreases from 0.4156 to 0.4070 when a rises from 0.5 to 0.8.

• (3). Impact of Volatility on Option Prices: As the volatility $\nu$ increases from 0.1 to 0.5, the prices of both the classic and new options rise. However, the relative difference between the two narrows with higher volatility, indicating that the new option becomes more valuable relative to the classic option in more volatile markets. For instance, with $a=0.5, b=0.3, b_0=0.6, \xi =0$, the price ratio of the new option to the classic option increases from $66.96\%$ ($\nu =0.1$) to $77.72\%$ ($\nu =0.5$).

• (4). Effect of Strategy Parameters $b_0$ and $\xi$: A higher $b_0$ (moving from 0.6 to 1) tends to increase the price of the new option, narrowing the gap with the classic option. This suggests that as the weight on active trading grows, the impact of strategic adjustments on reducing option prices becomes less pronounced. Similarly, increasing $\xi$ (which could reflect the magnitude of position adjustments) reduces the price of the new option. For example, with $a=0.5, b=0.3, b_0=0.6$, the price drops from 0.4456 when $\xi =0$ to 0.0031 when $\xi =6$, highlighting that larger strategic adjustments significantly diminish option value.

• (5). Interaction between Parameters: The interplay between $a, b, b_0$ influences the degree of price reduction; for instance, with $a=0.5, b=0.3, \xi =6$, the new option price drops to as low as 0.0031, showing that higher parameter values can compound the effect of strategy on option prices.

These results demonstrate that regardless of the investment strategy employed, the new option consistently trades at a lower price than the traditional option, confirming the effectiveness of the strategy. We present a three-dimensional visualization to illustrate the interaction between key factors and their impact on option pricing.

As shown in Figure 2, the classic and new option pricing formulas are expressed as functions of b and c. The values of the other parameters are as follows: $a=0.5, r=0.05, S=30, t=0, t^{*}=0.5, \nu =0.3, b_0=1, \xi =2$. In particular, we only need to observe the region where the new option price is lower than that of the classic option, because the scenario in which the new option price exceeds the classic option price contradicts our assumptions, despite being mathematically plausible. From a mathematical perspective, when the stock price falls from c to $c-\xi$, the delay in investor response can result in losses. Although investors may recover some of their losses by selling a portion of their holdings as the price rebounds from $c-\xi$ to $(1-a)(c-\xi )$, the initial loss incurred may surpass the recovered amount, leading to a total loss that could exceed that associated with classic options. This phenomenon is evident in the graph, where the price of the new option appears higher than the price of the classic option. In practice, however, the most detrimental outcome for investors is to refrain from taking any action throughout the entire price fluctuation range, which effectively mimics the behavior of classic options. Consequently, the price of the new option should be less than or at least equivalent to that of the classic option. We introduce two additional constraints to account for this: (1) $0\le b\le b_0\le 1$, and (2) $0\le$ the price of new option ≤ the price of the classic option. The pricing formula of classic options exclusively depends on the exercise price c, indicating that the price of classic options is directly related to c. Thus, a higher strike price results in a higher classic option price. In contrast, it can be observed that the price of options priced with nonlinear strategies is proportional to both the strike price and the minimum holding ratio. Specifically, in the case of put options governed by nonlinear investment strategies, an increased minimum holding ratio b results in fewer shares being sold by the investor. When the stock price declines, the investor consequently faces heightened losses, which enhances the intrinsic value of the put option, and consequently its price. Similarly, a higher initial holding ratio $b_0$ amplifies the losses incurred by the investor when the stock price falls, leading to a corresponding increase in the intrinsic value and price of the put option. As b approaches 0, c approaches 30, indicating that when the stock price declines to $c-\xi$, the investor liquidates all shares. When the stock price continues to decrease, the investor’s losses are minimized, resulting in the lowest option price in the nonlinear strategy, which equals 1.8608. Conversely, as b approaches 1 and c approaches 50, the investor refrains from selling any held stocks regardless of price fluctuations. In this scenario, the total loss aligns closely with that of classic options. The price of the new option is 15.5327 at this point, which remains lower than the classic option price of 18.7959.

When the option price is expressed as a function of b and $\xi$, with the parameters set to $c=45, a=0.5, r=0.05, S=30, t=0, T=0.5, \nu =0.3, b_0=1$, we impose the following constraints: (1) $0\le b\le b_0\le 1$, and (2) $0\le$ the price of new option ≤ the price of the classic option. Under these conditions, we focus on the scenario in which the price of the new option with the logarithmic strategy is lower than that of the classical option. As shown in Figure 3, the price of the classical option remains constant at 14.0002, independent of b and $\xi$; in contrast, the price of the new option varies proportionally with b and $\xi$. Specifically, an increased investment lag results in a lower threshold price for investors to sell stocks, thereby amplifying potential losses from holding these assets. This dynamic leads to an increase in the intrinsic value of the new options, consequently driving up their prices. Conversely, a higher minimum holding ratio results in fewer stocks being sold, which also leads to increased losses and a corresponding rise in the intrinsic value of put options, further elevating their prices. As b approaches 0 and $\xi$ approaches 0, it indicates that when the stock price reaches $c-\xi$, investors will sell all of their stocks. In this scenario, investor losses are minimized and the price of the new option reaches its lowest point, calculated at 9.9741, which is lower than that of the classical option; conversely, when b approaches 1 and $\xi$ approaches 5, the new option attains a maximum value of 12.5906, which remains lower than the price of the classical option.

When the option prices are expressed as functions of b and S, with parameters set to $c=45, a=0.5, r=0.05, t=0, t^{*}=0.5, \nu =0.3, \xi =2, b_0=1$, we observe the following constraints: (1) $0\le b\le b_0\le 1$, and (2) $0\le$ the price of the new option ≤ the price of the classic option. It can be seen from Figure 4 that we focus on the region where the option price in the nonlinear strategy remains below that of the classic option. The prices of classic options are independent of b and inversely proportional to S; specifically, as the stock price increases, the potential loss faced by investors is diminished, leading to a lower option price. Notably, when b approaches 0.5 and S approaches 50, the minimum option price is 1.6319; conversely, when b approaches 0.8 and S approaches 30, the maximum option price reaches 14.0002. In contrast, the price of the new options is directly proportional to b and inversely proportional to S. Thus, a higher minimum holding ratio results in fewer shares being sold by investors, which increases potential losses and consequently raises the intrinsic value of the put options, leading to higher prices. For example, when b approaches 0 and S approaches 50, the minimum option price is 1.1020. As b approaches 1 and S approaches 30, the maximum option price is 13.0432. Overall, under the specified constraints, the price of the new option remains lower than that of the classic option.

When the option prices are expressed as functions of b and $b_0$ with the parameters set to $c=45, a=0.5, r=0.05, S=30, t=0, t^{*}=0.5, \nu =0.3, \xi =2$, the following constraints apply: (1) $0\le b\le b_0\le 1$, and (2) $0\le$ the price of the new option ≤ the price of the classic option. As shown in the Figure 5, we focus on the region where the price of the new option is lower than that of the classical option. In classical options, the pricing does not incorporate b and $b_0$, resulting in a constant price of 14.0002. In contrast, the pricing of new options is directly influenced by b and $b_0$; specifically, in the case of put options within the nonlinear investment strategy, a higher minimum holding ratio b leads to a reduction in the quantity of stocks that investors are willing to sell. Consequently, when stock prices decline, investors face greater losses, which increases the intrinsic value of the put options, thereby elevating their prices. When b approaches 0 and $b_0$ approaches 1, investors liquidate all their stocks when the stock price falls to $c-\xi$. As the stock price continues to decrease, investor losses are minimized, resulting in the lowest price for the new option being 2.1129. Conversely, when both b approaches 1 and $b_0$ approaches 1, the maximum value within the defined constraints is 13.0432.

6. Empirical Test

The simulation results presented in the previous section demonstrate that the put option employing a dynamic investment strategy exhibits a considerable pricing advantage. However, certain parameter selections may be subjective and lack a robust empirical basis. Consequently, we sought to further assess the model’s effectiveness by utilizing real market data, as described in this section.

6.1. Data Source and Description

The Shanghai 50ETF option, introduced in February 2015, was the first exchange-traded option launched on the Shanghai Stock Exchange. This option has a substantial operational history, an active trading volume, and a wealth of high-quality data, making it an ideal candidate for empirical testing. Therefore, we selected the Shanghai 50ETF option as the focal point of our research. Specifically, we consider the 50EFT option contract (1005647) with a strike price of 2.55 expiring in August 2024 as an example. The data sources are detailed below:

• (1). Underlying asset price S: The underlying asset for the Shanghai 50ETF option contract is the Shanghai 50 Exchange-Traded Fund (code: 510050), which operates as an index fund. Market prices are available on the official website of the Shanghai Stock Exchange or through various stock quotation platforms.

• (2). Exercise price c: According to the option contract data provided in the Wind database, the contract’s exercise price is established at 2.55.

• (3). Time to expiration $t^{*}$: The expiration of an option is typically measured in trading days. The stock market generally has 252 trading days per year, meaning that T is calculated as the number of trading days remaining until expiration divided by 252. Assuming that investors buy this option contract on 27 June 2024 (the issuance date) and that there are 45 trading days until expiration on 28 August, we have $t^{*}=45/252$.

• (4). Risk-free rate of interest r: The Shanghai Interbank Offered Rate (Shibor) is commonly used to represent the risk-free rate in the Chinese financial market. Various Shibor rates are published, including overnight, one week, two weeks, one month, three months, six months, nine months, and one year. Because the option expires in 45 days and no corresponding Shibor rate exists, we use the arithmetic average of the monthly Shibor rate sourced from the China Currency Network, which is 2.25%.

• (5). Volatility of the underlying asset returns $\xi$: In this analysis, the volatility is derived from historical volatility calculations. Given the significant fluctuations in the stock market, insufficient data may introduce considerable errors. Therefore, we calculated the volatility using the underlying asset prices within the T-period preceding the option purchase (from 22 April to 27 June), resulting in an estimated volatility of $\xi =0.1850$.

6.2. Determination and Implementation of Investment Strategy

Investors typically purchase put options for two primary reasons: first, to hedge against potential losses from stock price declines when holding a substantial number of shares; and second, to speculate on anticipated market downturns based on prevailing market conditions. In the first half of 2024, the A-share market experienced significant volatility and transformation, with the Shanghai Stock Exchange Index fluctuating by 18.12%, moving from 2974.93 at the beginning of the year to 2967.40 on June 28. Although the overall decline was modest, the high market volatility underscored rapid shifts in investor sentiment. This instability has prompted some investors to withdraw from the stock market, further intensifying the downward pressure on stock prices. This research posits that investors hold a substantial number of Shanghai 50ETF shares and opt to purchase put options expiring in August 2024 on 27 June 2024, as a risk management strategy.

Simultaneously, the investor develops an investment strategy to adjust the proportions of their position based on stock price trends, with the following parameter values: the minimum capital utilization rate is established at $b=0.2$, the initial capital occupancy rate is $b_0=0.8$, and the investment strategy index is $a=0.8$. Determining the investment sensitivity $\xi$ is critical to the feasibility of the investment strategy; if sensitivity is too high, it may surpass the fluctuation range of the underlying asset’s price, while if it is too low it can lead to frequent trading and substantial transaction costs. Consequently, it is imperative to establish a reasonable investment spacing, denoted by $\delta$. Given the limited fluctuation range of stock prices in the short term, the following formula was employed in this study to define the investment spacing $\delta$:

(29)${\displaystyle \delta =\frac{p_{max,i}-p_{min,j}}{N},i,j\in [t_0-t^{*},t_0]}$

where T denotes the time to expiration, $p_{max,i}$ and $p_{min,j}$ respectively represent the peak and trough prices of the underlying asset, and the variable N refers to the total number of stock transactions. Notably, from 27 June 2024, to 27 August 2024, the Shanghai 50ETF reached a maximum price of 2.539 and a minimum of 2.390. If we assume two transactions, the investment spacing derived from Formula (29) is calculated as 0.08. Thus, when the stock price hits $S_1=c-\delta =2.47$ and $S_2=c-2\delta =2.39$, the investor will sell the corresponding proportion of shares. Furthermore, to ensure the operability of the investment strategy, the investor is are required to conduct no more than one transaction per day. The investment process is illustrated in Figure 6.

6.3. Comparative Analysis

Based on the earlier assumptions, we computed the theoretical prices of options using a put option pricing model that incorporates a nonlinear investment strategy and using the classic Black–Scholes model. These two theoretical values were then compared with actual prices to assess the strategy’s effectiveness. To evaluate the effect of investment spacing on option pricing, we selected three different spacings, $\delta =0.04, 0.08, 0.16$, which represent scenarios where the investor buys or sells the stock four times, two times, and one time. The results of these calculations are presented in Table 2.

Table 2 indicates that the theoretical price of the traditional option closely matches the actual market price, with fluctuations around it. The maximum observed deviation is 0.0071, while the minimum is only 0.0001. Importantly, in the case of investment spacings of $\delta =0.04$ and $\delta =0.08$, the price of the exotic option utilizing a nonlinear investment strategy consistently remains below both the theoretical price of the classic option and the actual market price. Conversely, when the investment spacing increases to 0.16, the price of the exotic option surpasses the theoretical price of the classic option within a few trading days. Additionally, an examination of the data in the last three columns reveals a progressive increase in the prices of exotic options as the investment spacing widens. This phenomenon occurs because larger investment spacings result in fewer transactions by the investor, or potentially no trading activity at all, limiting opportunities for active hedging. The investor is consequently unable to leverage stock gains to offset losses, leading to the largest total loss and subsequently the highest price for the exotic option.

Figure 7 provides an intuitive representation of the relationship between put option prices across various models. The trend of the theoretical put option price aligns with the actual price, confirming the validity of the proposed model. The price curve for the classic option closely follows the actual price, while the curve for the option using the investment strategy typically sits below that of the classic option. Moreover, an increase in investment spacing results in an elevated price curve, drawing it closer to that of the classic option. This phenomenon occurs because when the investment spacing is $\delta =0.16$, the offer price of the stock is $c-\delta =2.39$ and the actual stock price remains constant at this offer price throughout the option holding period; in other words, the investor refrains from trading any stocks during this period, resulting in the price being most closely aligned with that of the classic option. In contrast, with an investment spacing of $\delta =0.04$, the offer prices of the stock are $S_1=c-\delta =2.51, S_2=c-2\delta =2.47, S_3=c-3\delta =2.43$ and $S_4=c-4\delta =2.39$, respectively, enabling the investor to conduct multiple trades and offset more losses, resulting in a lower intrinsic value for the put option compared to the classic option.

Furthermore, we used copper options data (code: CU2501P82000.SHF) to compare the new option pricing model with the classic model, covering the period from 7 June 2024, to 9 August 2024. The parameters were set accordingly. The price of the underlying asset (CU2501P82000) was sourced from the Wind database. According to the option contract data from Wind, the exercise price is set at 82,000. Assuming that investors purchase this option contract on 7 June 2024, with 45 trading days until expiration on 9 August, the time to expiration is $t^{*}$ = 45/252. The average Shibor rate from 7 June 2024 to 9 August 2024 is used as the risk-free rate ($r=1.89\%$). The volatility $\xi$ is the average volatility of the underlying asset (CU2501P82000) for the same period (rounded to two decimal places as 0.18). The minimum capital utilization rate is set at $b=0.2$, the initial capital occupancy rate is $b_0=0.8$, and the investment strategy index is $a=0.8$. The variable N represents the total number of stock transactions. Notably, during the period from 7 June 2024 to 9 August 2024, the highest price of the copper option’s underlying asset was 81,530 and the lowest price was 70,680. Assuming two transactions, the investment spacing derived from Formula (29) is calculated as 5425.

The results of the numerical simulation shown in Figure 8 indicate that the price of the new option is lower than that of the classic option. Additionally, the new pricing model aligns closely with the real market data trends.

7. Conclusions

This paper proposes a new option pricing model that incorporates a nonlinear investment strategy building on the classic Black–Scholes framework. We argue that for put options, when the stock price falls to a certain threshold ($c-\xi$), investors using the logarithmic investment strategy are likely to sell the stock. This strategic move allows investors to mitigate some of the losses incurred due to the price drop, effectively enhancing their risk management capabilities. Through the derivation of an analytical solution, we utilize mathematical analysis software to visualize the factors influencing the pricing of new options in three-dimensional space. The following conclusions can be drawn:

(1) The price of new options is not only related to fundamental factors such as the underlying asset price, volatility, and risk-free interest rate, but is also influenced by the selection of investment strategy parameters. After constructing the pricing model for new options, a sensitivity analysis of the factors affecting their prices was conducted in this study, with the results presented in graphical form. Our findings indicate that the same factors affecting the price of standard options, such as stock price, price volatility, time to expiration, and risk-free interest rate, also influence the price of new options, with the same directional effect. Additionally, investment strategy parameters also impact the price of new options.

(2) Regardless of the chosen investment strategy, the price of new options is consistently lower than that of traditional standard options. Numerical simulations of the prices of new options compared to classic options indicate that the investment strategy coefficients a, investment sensitivity $\xi$, and both the minimum and initial capital utilization ratios b and $b_0$ significantly impact the pricing of new options. More importantly, with appropriate parameters, new options exhibit a notable price advantage compared to classic options, with prices being approximately three-quarters of those of classic options. In other words, investors can achieve a comparable level of hedging effectiveness to that of classic options for only three-quarters of the classic option price by setting appropriate investment strategies.

These findings underscore the potential of nonlinear strategies as an effective tool for investors seeking to navigate the complexities of financial risk. The model not only offers a practical approach for option pricing but also highlights a strategic pathway for optimizing investment decisions under adverse market conditions. To empirically validate our theoretical framework, we conducted numerical simulations utilizing data from the Shanghai 50 ETF options. The results from these simulations align closely with our theoretical predictions, thereby reinforcing the robustness and reliability of our proposed model. This alignment between theory and empirical data lends credence to the applicability of our findings in real-world trading scenarios.

Despite these promising outcomes, we acknowledge the inherent complexity of financial markets, which necessitates further refinement of our model. Future research will focus on relaxing certain assumptions within the model as well as on incorporating a more nuanced understanding of investor behavior. By addressing these areas, we aim to enhance the model’s applicability and predictive power in diverse market environments.

Author Contributions

Software, writing—review and editing, M.Z.; formal analysis, M.W.; investigation, J.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. The investment strategy.

Figure 2. Option to b and c.

Figure 3. Option price to b and [Forumla omitted. See PDF.].

Figure 4. Option price to b and S.

Figure 5. Option price to b and [Forumla omitted. See PDF.].

Figure 6. Investment process.

Figure 7. Comparison of put option price.

Figure 8. Comparison of copper option prices.

Table 1

Comparison of two option prices with different investment strategies.

Parameters				Classic Option			New Option
$\mathit{a}$	$\mathit{b}$	${\mathit{b}}_{\mathbf{0}}$	$\mathit{\xi }$	$\mathit{\nu }\mathbf{=}\mathbf{0.1}$	$\mathit{\nu }\mathbf{=}\mathbf{0.3}$	$\mathit{\nu }\mathbf{=}\mathbf{0.5}$	$\mathit{\nu }\mathbf{=}\mathbf{0.1}$	$\mathit{\nu }\mathbf{=}\mathbf{0.3}$	$\mathit{\nu }\mathbf{=}\mathbf{0.5}$
0.5	0.3	0.6	0	0.6655	2.2802	3.7468	0.4456	1.6140	2.9121
0.5	0.3	0.6	3	0.6655	2.2802	3.7468	0.1815	1.5312	2.6737
0.5	0.3	0.6	6	0.6655	2.2802	3.7468	0.0031	1.1497	2.6112
0.5	0.3	1	0	0.6655	2.2802	3.7468	0.6428	2.3418	3.4657
0.5	0.3	1	3	0.6655	2.2802	3.7468	0.2047	1.9498	3.2243
0.5	0.3	1	6	0.6655	2.2802	3.7468	0.0032	1.6402	2.9873
0.5	0.5	0.6	0	0.6655	2.2802	3.7468	0.4156	1.4354	2.6331
0.5	0.5	0.6	3	0.6655	2.2802	3.7468	0.1779	1.2883	2.4076
0.5	0.5	0.6	6	0.6655	2.2802	3.7468	0.0030	1.1301	2.3908
0.5	0.5	1	0	0.6655	2.2802	3.7468	0.6427	2.3233	3.2868
0.5	0.5	1	3	0.6655	2.2802	3.7468	0.2010	1.8870	3.1582
0.5	0.5	1	6	0.6655	2.2802	3.7468	0.0032	1.4606	2.8988
0.8	0.3	0.6	0	0.6655	2.2802	3.7468	0.4200	1.3195	2.8974
0.8	0.3	0.6	3	0.6655	2.2802	3.7468	0.1784	1.2802	2.6171
0.8	0.3	0.6	6	0.6655	2.2802	3.7468	0.0030	1.1472	2.5852
0.8	0.3	1	0	0.6655	2.2802	3.7468	0.6129	1.8443	3.2941
0.8	0.3	1	3	0.6655	2.2802	3.7468	0.1974	1.5638	3.0676
0.8	0.3	1	6	0.6655	2.2802	3.7468	0.0032	1.1809	2.7661
0.8	0.5	0.6	0	0.6655	2.2802	3.7468	0.4070	1.2740	2.2989
0.8	0.5	0.6	3	0.6655	2.2802	3.7468	0.1768	1.2052	2.1467
0.8	0.5	0.6	6	0.6655	2.2802	3.7468	0.0030	1.1060	2.0560
0.8	0.5	1	0	0.6655	2.2802	3.7468	0.6000	1.7991	3.1956
0.8	0.5	1	3	0.6655	2.2802	3.7468	0.1958	1.4818	3.0072
0.8	0.5	1	6	0.6655	2.2802	3.7468	0.0031	1.1797	2.7454

Table 2

Comparison of put option price.

Date	Stock Price	Actual Price	Classic Option	New Option
				$\mathit{\delta }\mathbf{=}\mathbf{0.04}$	$\mathit{\delta }\mathbf{=}\mathbf{0.08}$	$\mathit{\delta }\mathbf{=}\mathbf{0.16}$
27/6/2024	2.434	0.1221	0.1206	0.0574	0.0847	0.1137
28/6/2024	2.441	0.1144	0.1148	0.0555	0.0815	0.1074
1/7/2024	2.450	0.1052	0.1075	0.0529	0.0772	0.0990
2/7/2024	2.462	0.0966	0.0982	0.0492	0.0710	0.0875
3/7/2024	2.462	0.0947	0.0979	0.0493	0.0712	0.0875
4/7/2024	2.454	0.1009	0.1036	0.0519	0.0757	0.0954
5/7/2024	2.434	0.1238	0.1192	0.0581	0.0862	0.1151
8/7/2024	2.422	0.1308	0.1290	0.0615	0.0919	0.1266
9/7/2024	2.444	0.1110	0.1106	0.0554	0.0817	0.1058
10/7/2024	2.440	0.1135	0.1136	0.0567	0.0840	0.1100
11/7/2024	2.466	0.0934	0.0929	0.0486	0.0701	0.0837
12/7/2024	2.481	0.0830	0.0816	0.0434	0.0612	0.0684
15/7/2024	2.491	0.0753	0.0742	0.0399	0.0551	0.0582
16/7/2024	2.500	0.0723	0.0720	0.0365	0.0494	0.0531
17/7/2024	2.510	0.0658	0.0653	0.0327	0.0429	0.0525
18/7/2024	2.522	0.0628	0.0613	0.0280	0.0359	0.0411
19/7/2024	2.539	0.0555	0.0523	0.0214	0.0244	0.0301
22/7/2024	2.515	0.0620	0.0591	0.0307	0.0393	0.0463
23/7/2024	2.483	0.0798	0.0771	0.0435	0.0610	0.0652
24/7/2024	2.474	0.0879	0.0834	0.0470	0.0672	0.0749
25/7/2024	2.450	0.1090	0.1024	0.0553	0.0823	0.1018
26/7/2024	2.444	0.1144	0.1073	0.0572	0.0859	0.1087
29/7/2024	2.435	0.1182	0.1150	0.0598	0.0908	0.1189
30/7/2024	2.422	0.1313	0.1266	0.0630	0.0967	0.1328
31/7/2024	2.463	0.0917	0.0903	0.0518	0.0761	0.0875
1/8/2024	2.448	0.1039	0.1028	0.0567	0.0855	0.1052
2/8/2024	2.433	0.1188	0.1162	0.0607	0.0933	0.1225
5/8/2024	2.417	0.1355	0.1311	0.0640	0.0994	0.1395
6/8/2024	2.401	0.1504	0.1465	0.0666	0.1035	0.1542
7/8/2024	2.399	0.1511	0.1485	0.0667	0.1037	0.1561
8/8/2024	2.406	0.1436	0.1417	0.0654	0.1023	0.1507
9/8/2024	2.403	0.1468	0.1448	0.0656	0.1027	0.1536
12/8/2024	2.404	0.1448	0.1439	0.0652	0.1022	0.1530
13/8/2024	2.403	0.1456	0.1450	0.0650	0.1020	0.1541
14/8/2024	2.390	0.1589	0.1580	0.0665	0.1027	0.1636
15/8/2024	2.422	0.1280	0.1264	0.0616	0.0985	0.1368
16/8/2024	2.434	0.1159	0.1147	0.0596	0.0957	0.1236
19/8/2024	2.445	0.1080	0.1039	0.0578	0.0925	0.1104
20/8/2024	2.438	0.1158	0.1109	0.0581	0.0944	0.1182
21/8/2024	2.426	0.1233	0.1230	0.0588	0.0955	0.1311
22/8/2024	2.430	0.1234	0.1192	0.0575	0.0941	0.1257
23/8/2024	2.445	0.1062	0.1043	0.0546	0.0912	0.1072
26/8/2024	2.442	0.1090	0.1075	0.0542	0.0900	0.1091
27/8/2024	2.444	0.1074	0.1058	0.0535	0.0875	0.1061

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