Asian option pricing under sub-fractional vasicek model

1.   Introduction

Since the emerge of Black-Scholes framework in the 1970s, option pricing has always been one of the hot issues in financial research. Classic Black-Scholes framework heavily relies on unrealistic assumptions, and only works for plain vanilla options, like European options. However, plain vanilla options are susceptible to possible spot manipulation at maturity. As a natural and reasonable extension, exotic options, especially the path-dependence options, attract the attention of both partitioner and researchers. Asian options, generally described as options whose payoff depends on the average price of the underlying asset during a pre-specified period within an option's lifetime and pre-specified observation frequency, is one of the most-used hedging tools in the over-the-counter (OTC) marketplace (Zhang, 1997).

Theoretically, there are two types of Asian options, arithmetic Asian options and geometric Asian options. The key difference between them is that when the underlying asset prices are log-normally distributed, the geometric Asian options follow the same distribution while the arithmetic Asian options do not. In real-world practice, the arithmetic Asian option is the only type traded in the market, whose closed-form expression is very difficult to obtain. The pricing process of arithmetic Asian options involves the price of geometric Asian options as a control variate, due to the fact that the closed-form solution of geometric Asian options is relatively easy to obtain. Therefore, it is worth studying the pricing dynamics of geometric Asian options. In recent years, there are some research results regarding geometric Asian options, such as Sander (2019), Yao and Li (2018) and Gen and Zhou (2018). However, up to the authors' best knowledge, for most existing literature, the underlying asset price is modelled by a stochastic process driven by standard Brownian motion, which is a martingale with the Markov property. The Markovian setting is equivalent to the weak form of efficient market hypothesis (Hull, 2003), which is very ideal and unrealistic. According to Greene and Fielitz (1977) and Lo (1991), there is empirical evidence showing that the asset pricing dynamics involve past information. As a result, it is natural to introduce fractional Brownian motion (fBm), a stochastic process which preserves long-dependence and self-similarity properties. Mandelbrot and Van Ness (1968) firstly used fBm to model financial asset pricing dynamics, Xiao and Zhou (2008) and Liu et al. (2008) use fractional Brownian motion to describe the evolution process of financial asset prices, and more research results can be found in Duncan et al. (2000) and Hu and Øksendal (2003) and references there in. Since fractional Brownian motion has stationary increment, which can't match the virtue of financial data, a more general Gaussian process, the sub-fractional Brownian motion (sfBm), has been introduced to model the dynamics of financial assets. According to Bojdecki et al. (2004), the sub-fractional Brownian motion not only preserves the long-dependence and self-similarity properties, but also has a faster degeneration. In Tudor (2008), it has been shown that the sub-fractional Brownian motion can be a good choice to describe the volatility. Although, both fractional Brownian motion and sub-fractional Brownian motion are neither Markovian nor martingale, and there are arbitrages in such market models, which was first proved by Rogers (1997). However, it still worth discussing in the pricing formula of financial derivatives under such environment, since such arbitrages can be excluded from these models by restricting the class of trading strategies, more details can be found in Cheridito (2003) and Xiao et al. (2021).

The classical B-S model assumes that the interest rate is a constant, but in the real financial market, the interest rate preserves the term structure. Especially in the long run, the interest rate fluctuates within a certain range under the influence of both macro and micro economic factors, that is, the interest rate has the phenomenon of "mean reversion". To characterize the stochastic behavior of interest rates, the Vasicek model has been proposed and widely used in the study of stochastic interest rates. Cajueiro and Tabak (2007) found that there is a long-term dependence of stochastic interest rates. Zhou and Li (2014), Huang et al. (2012) studied the option pricing formulae under the fractional Vasicek rate model. Tudor (2008) found that sub-fractional Brownian motion had stronger memory than fractional Brownian motion and could better describe the long-term dependence of financial assets. Guo and Zhang (2017) used the sub-fractional Vasicek model to price European options and obtained the pricing formula of European options. See Ji et al. (2022), Yang et al. (2022) and Wang et al. (2021) for more recent progress.

Enlightened by these existing results, in this paper underlying asset is assumed to follow the sub-fractional Brownian motion and the interest rates satisfy the sub-fractional Vasicek model, then the B-S type PDE corresponding to Asian geometric average option pricing formula is derived, and a close-form of such formula can be obtained. Such a formula will play an essential role when a more precise control variate price is required in the pricing of arithmetic Asian options.

The rest of this paper is organized as follows: In Section 2, we introduce some necessary preliminary knowledge about option pricing and the zero-coupon bond pricing model. In Section 3, we investigate the Asian geometric average option pricing model, and give the analytical solution and the pricing formula for Asian geometric average options where the underlying asset is driven by the sub-fractional Brownian motion with interest rate satisfying the sub-fractional Vasicek model. In Section 4, we discusses the influence of Hurst index, stock price and zero-coupon bond price on Asian option price. Some further implications will be added in the last section as well.

2.   Preliminaries

Let $(\Omega, {\cal F}_t, P)$ be a complete probability space with a filtration $\{{\cal F}_t\}_{t\geq 0}$ satisfying usual conditions.

Definition 2.1. Let $H \in (0, 1)$ be the Hurst index. The sub-fractional Brownian motion ${\xi ^H} = \left\{ {\xi _t^H:t \in \mathbb{R}} \right\}$ with the Hurst index $H$ is a continuous Gaussian process satisfying that

1. ${\rm{E}}[\xi _t^H] = 0$,

2. ${\rm{E}}[\xi _t^H\xi _s^H] = {t^{2H}} + {s^{2H}} - \frac{1}{2}\left({{{\left| {t + s} \right|}^{2H}} + {{\left| {t - s} \right|}^{2H}}} \right), s, t \in \mathbb{R}$,

where $E = {E_p}$ denotes the expectation under the probability measure $P$.

In case $H = 1/2$, $\xi _t^H$ is a standard Brownian motion. The sub-fractional Brownian motion is a modified fractional Brownian motion, which has the properties of self-similarity and long memory. More details can be found in Tudor (2008), (Guo and Zhang, 2017), Bojdecki et al. (2004) and references therein.

Definition 2.2. Rao (2016): Assumes that ${A_t}$ is the path-dependent variable of Asian options, which represents the average value of stock price on $[0, t]$. It is a geometric average, that is,

The Asian option corresponding to this path is called the Asian geometric average option. If at the maturity time $t$, the Asian option is executed at the fixed strike price $K$, the return of Asian geometric average call option can be expressed as ${\left({{A_t} - K} \right)^ + }$ and the return of Asian geometric average put option can be expressed as ${\left({K - {A_t}} \right)^ + }$.

Lemma 2.3. Yan et al. (2011): Suppose $\{ Y(t)\}$ is a sub-fractional ${\rm{It\hat o}}$ process on probability space $(\Omega, {\cal F}, P)$, which is given by ${\rm{d}}Y(t) = r{\rm{d}}t + \sigma {\rm{d}}\xi _t^H$. Let $f(t, Y(t)) \in {C^{1, 2}}({\mathbb{R}_ + } \times \mathbb{R} \to \mathbb{R})$, and that $f(t, Y(t))$, $\int_0^t {\frac{{\partial f}}{{\partial \tau }}} (\tau, Y(\tau)){\rm{d}}\tau$, $\int_0^t {\frac{{{\partial ^2}f}}{{\partial {Y^2}}}} (\tau, Y(\tau))d\tau$, $\int_0^t {\frac{{{\partial ^2}f}}{{\partial {Y^2}}}} (\tau, Y(\tau)){\tau ^{2H - 1}}d\tau$ belong to the ${L^2}(P)$ space, then we have the following sub-fractional ${\rm{It\hat o}}$ formula,

The proof of Lemma 2.3 can be found in Yan et al. (2011).

Before proceeding to our main results, the term structure of interest rate should be discussed. Suppose that interest rate ${r_t}$ satisfies the following stochastic differential equation under risk neutral measure Q (Guo and Zhang, 2017)

where $a, \theta, \sigma$ are constants, $\left\{ {\xi _t^H; t > 0} \right\}$ is sub-fractional Brownian motion, $a$ is the speed of interest rate adjustment, $\theta$ is the long-term interest rate, $\sigma$ is the coefficient of influence on interest rates. This equation model is the so-called sub-fractional Vasicek model.

The sub-fractional Vasicek model is a modified form and an extension of the fractional Vasicek model. In this paper, we use the sub-fractional Brownian motion in place of the fractional Brownian motion, simultaneously, we use the sub-fractional Vasicek model instead of the fractional Vasicek model.

We give the corresponding assumptions as follows.

1. Financial markets are complete and frictionless;

2. Short selling is allowed;

3. Underlying assets pay no dividend;

4. There is no transaction cost and it is tax-free;

5. The expected return rate of the risk-free portfolio is equal to the risk-free interest rate;

Suppose there are two kinds of continuous free-trade assets in the market, one is the risk-free asset, such as treasury bills, and the other is the risky asset, for instant stocks. The stock price, ${S_t}$, satisfies the following equation

while ${r_t}$, the risk-free interest rate, satisfies the following sub-fractional Vasicek model

where ${\sigma _1}$ and $\sigma_2$ are constants, and $\left\{ {\xi _1^H(t); t > 0} \right\}$ and $\left\{ {\xi _2^H(t); t > 0} \right\}$ are two correlated sub-fractional Brownian motions such that

Since the interest rate is non-exchangeable, a zero-coupon bond, as its carrier, plays a unique role in the study of option pricing when the interest rate is of stochastic nature. In this paper, we denote the face value of the zero-coupon bond at time $t$ as $F(r_t, t; T),$ and the face value of the zero-coupon rate at maturity is $1$ dollar, i.e. $F(r_T, T; T) = 1$.

An application of Delta hedging principle, together with Lemma 2.3, yields that price of zero-coupon bond $F({r_t}, t; T)$ satisfies the following partial differential equation:

where $\hat \theta = \theta - \frac{\lambda }{a}{\sigma _2}$, $\lambda$ is the price of the financial market with interest rate risk. According to (Guo and Zhang, 2017), Equation (5) has unique explicit solution

where

When $T - t \ll 1$, we can get $A(t, T) \sim 0$, $B(t, T) \sim T - t$, and the zero-coupon bond price is $F({r_t}, t; T) = {e^{ - {r_t}(T - t)}}$.

3.   Main result

Theorem 3.1. Assume that stock price ${S_t}$ satisfies Equation (3), interest rate ${r_t}$ satisfies Equation (4), denote $V(t, {r_t}, {S_t}, {A_t})$ as the Asian geometric average call option at time $t$ ($0 \le t \le T$), with fixed strike pricing $K$ and maturity $T$, then $V(t, {r_t}, {S_t}, {A_t})$ satisfies the partial differential equation

with terminal condition

where ${A_t} = \exp \left\{ {\frac{1}{t}\int_0^t {\ln {S_\tau }} d\tau } \right\}$ denotes the geometric mean of stock price on $[0, t]$ which is equivalent to the path of Asian geometric options.

Proof. According to Delta hedging principle, a portfolio is constructed with a long position of one unit Asian option $V$, and a short position of ${\Delta _1}$ unit ${S_t}$ and ${\Delta _2}$ zero-coupon bonds $F$. Define the value of portfolio at the t as

then the change of the portfolio in the time interval $d t$ is

Differentiating ${A_t}$ with respect to $t$ yields

According to Lemma 2.3, we can get

and

Applying Equations (10) and (11) over the time interval $d t$, we can get

Let ${\Delta _1} = \frac{{\partial V}}{{\partial {S_t}}}$ and ${\Delta _2} = \frac{{\partial V/\partial {r_t}}}{{\partial F/\partial {r_t}}}$, we have

A simple application of Delta hedging principle yields

By combining (12) and (13), we obtain that

with $V(T, {r_T}, {S_T}, {A_T}) = {({A_T} - K)^ + }$. □

Theorem 3.2. Assume that stock price ${S_t}$ satisfies Equation (3), interest rate ${r_t}$ satisfies Equation (4), then for fixed strike pricing $K$ and maturity date $T$, the value of Asian geometric average call option $V(t, {r_t}, {S_t}, {A_t})$ is given by

where

Proof. Consider

with

Let $E = \exp \left({\frac{1}{a}\left({{e^{ - a(T - t)}} - 1} \right){r_t}} \right)$, and recall the price solutions of zero-coupon bonds (5), we get

A substitution of ${r_t}, \, {A_t}, \, {S_t}$ yields that

A direct calculation yields that

Substituting the previous Equation into (14), we obtain that

where

At maturity date $T$, $V(T, {r_T}, {S_T}, {A_T}) = {({A_T} - K)^ + }$ can be rewritten as

For Equation (15), it can be transformed into a heat equation by substitution

where $\alpha (t)$, $\beta (t)$, $\gamma (t)$ are undetermined functions.

By calculation,

Substituting the previous Equations into (15) yields

By letting

According to the terminal conditions $\alpha (T) = \beta (T) = \gamma (T) = 0$, we can get

Substituting them back into (16), one gets

while the terminal condition $\hat V(T, y) = {({e^y} - K)^ + }$ can be converted into

According to the theory of heat Equation Yao and Li (2018), a solution to Equation (17) can be obtained as

For ${I_1}$, let $t = \frac{{z - \eta - 2\tau }}{{\sqrt {2\tau } }}$, then

For ${I_2}$, let $\omega = \frac{{z - \eta }}{{\sqrt {2\tau } }}$, then

By transformation $U(\eta, \tau) = \hat V(y, t){e^{\beta (t)}}$,

By substituting the expression of $\eta, \tau, \beta (t)$ into (18), we can get

where,

By using substitution of variable conditions, together with Equation (19), we can get

where

□

Corollary 3.3. Under the same assumptions in Theorem 3.2. Assume that stock price ${S_t}$ satisfies Equation (3), interest rate ${r_t}$ Equation formula (4), denote $V_p(t, {r_t}, {S_t}, {A_t})$ as the value of Asian geometric average put option at time $t$ ($0 \le t \le T$) with fixed strike pricing $K$ and maturity $T$, then

Proof. According to the terminal condition $V_p(T, {r_T}, {S_T}, {A_T}) = {(K - {A_T})^ + }$, the value of put option ${V_p}(t, {r_t}, {S_t}, {A_t})$ at time $t$ ($0 \le t \le T$) can be obtained by solving Equation (14) with the solution method in Theorem 3.2. □

4.   Simulations

In this chapter, the impacts of Hurst index $H$, stock price ${S_t}$, interest rate ${r_t}$ and zero-coupon bond price $F$ on Asian geometric average call option prices are discussed. According to the Asian geometric average call option pricing formula from Theorem 3.2, the constant parameters in the Asian geometric average call option formula of Theorem 3.2 are assumed to be as follows:

Under such setting of parameters, Table 1 and Table 2 illustrate that the Asian geometric average call option prices at fixed strike prices under different Hurst indices, stock prices and zero-coupon bond prices.

From Table 1, and Table 2, the interaction of Hurst index, initial price of the stock, face value of the zero-coupon bond and the price of Asian geometric average call options are illustrated. It is natural to see that for a call-type option, the price of the Asian geometric average call option is positive-related with the initial price of the stock. The same pricing dynamic applies for the relationship between the price of the Asian geometric average call option and face value of the zero-coupon. Keeping the initial price of the stock and the face value of the zero-coupon unchanged, the data from the Table 1 and Table 2 demonstrates that the prices of Asian geometric average call options are negatively-related to the Hurst Index.

For more values of Hurst index, this negative-relation stays consistent, as shown in the Figure 1, where the asset driven by a geometric Brownian motion (GBM) has been employed as a benchmark. Such convergence results validate our model.

Figure 2 illustrates the prices of Asian geometric average call options when the underlying assets are modelled by geometric Brownian motion (GBM), which indicates that when $H = 0.5$, prices of Asian geometric average call options driven by sfBm will converge to the prices of Asian geometric average call option driven by GBM.

Figure 3 illustrates the Delta of Asian geometric average call option with respect to time to maturity as well as the price of underlying asset. The Delta is most sensitive when the price of underlying asset is close to the strike price $K = 66.$ However, such measurement of Delta is not an optional indicator for the Asian geometric average call option, since it is simultaneous, which is inconsistent with the path-dependent property of Asian geometric average call options.

Figure 4 and 5 illustrate the Gamma and Vega of Asian geometric average call option with respect to time to maturity as well as the price of underlying asset. Both Greeks preserve the properties that they are very sensitive when the price of underlying asset is close to the strike price. Such observations will lead to a natural conclusion that the existing technique of calculation of Greeks does not apply, and modification of Greeks' calculation should be taken into consideration in the future research.

5.   Conclusions

In this paper, the underlying asset is modelled by sfBm and the interest rate follows the sub-fractional Vasicek model. The explicit pricing formula for the Asian geometric average option has been derived. According to Theorem 3.2, such a result can be seen as an natural extension of the fractional Vasicek model Zhou and Li (2014). The numerical simulation also suggests that when $H = 0.5,$ the price of the Asian geometric average call option driven by sfBm converges to the the Asian geometric average call option driven by GBM.

The results of this paper have more implications, in both theoretical and industrial aspects. For the theoretical aspect, Theorem 3.2 can be extended easily to the case where all parameters $a, \theta, \sigma$ are time-dependent, i.e., the corresponding sub-fractional Hull-White model. However, for the sub-fractional CIR model, such result is no longer applicable. For the industrial aspects, our simulation results suggest that the existing calculation technique for Greeks does not work for Asian-type options. More generally speaking, for the path-dependent option, the Greeks could be redefined, i.e., an "average version" rather than a "simultaneous" version. In the future, some restriction will be implemented to remove some trading strategy with arbitrages.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 12271483). This work is partially supported by Zhejiang Research Base Key Project of Philosophy, and Social Sciences of Zhejiang Modern Service Industry Research Center (No. 20JDZD071). We are very grateful to the reviewers for their valuable comments. We also appreciate Ms. Run Wen and Mr. Jiajie Qiu for their valuable advice in the simulation and programming.

Conflict of interest

All authors declare no conflicts of interest in this paper.