Asian option pricing under sub-fractional vasicek model

: This paper investigates the pricing formula for geometric Asian options where the underlying asset is driven by the sub-fractional Brownian motion with interest rate satisfying the sub-fractional Vasicek model. By applying the sub-fractional Itˆo formula, the Black-Scholes (B-S) type Partial Di ff erential Equations (PDE) to Asian geometric average option is derived by Delta hedging principle. Moreover, the explicit pricing formula for Asian options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the stock price, the Hurst index, the speed of interest rate adjustment


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 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.

Preliminaries
Let (Ω, F t , P) be a complete probability space with a filtration {F t } t≥0 satisfying usual conditions. Definition 2.1. Let H ∈ (0, 1) be the Hurst index. The sub-fractional Brownian motion ξ H = ξ H t : t ∈ R with the Hurst index H is a continuous Gaussian process satisfying that where E = E p denotes the expectation under the probability measure P .
In case H = 1/2 , ξ H t 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 (A t − K) + and the return of Asian geometric average put option can be expressed as (K − A t ) + . Lemma 2.3. Yan et al. (2011): Suppose {Y(t)} is a sub-fractional Itô process on probability space (Ω, F , P), which is given by ∂Y 2 (τ, Y(τ))τ 2H−1 dτ belong to the L 2 (P) space, then we have the following sub-fractional Itô 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, θ, σ are constants, ξ H t ; t > 0 is sub-fractional Brownian motion, a is the speed of interest rate adjustment, θ is the long-term interest rate, σ 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 σ 1 and σ 2 are constants, and ξ H 1 (t); t > 0 and ξ H 2 (t); t > 0 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 zerocoupon bond F(r t , t; T ) satisfies the following partial differential equation: Quantitative Finance and Economics Volume 7, Issue 3, 403-419. whereθ = θ − λ a σ 2 , λ is the price of the financial market with interest rate risk. According to Guo and Zhang (2017), Equation (5) has unique explicit solution where

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 ≤ t ≤ T ), with fixed strike pricing K and maturity T , then V(t, r t , S t , A t ) satisfies the partial differential equation where A t = exp 1 t t 0 ln S τ dτ 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 ∆ 1 unit S t and ∆ 2 zero-coupon bonds F. Define the value of portfolio at the t as then the change of the portfolio in the time interval dt 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 dt , we can get 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 Let E = exp 1 a e −a(T −t) − 1 r t , and recall the price solutions of zero-coupon bonds (5), we get A substitution of r t , A t , S t yields that V(r t , A t , S t , t) = EV(y, t) = Fe A(t,T )V (y, t).
A direct calculation yields that Substituting the previous Equation into (14), we obtain that where For Equation (15), it can be transformed into a heat equation by substitution τ = γ(t), η = y + α(t), U(η, τ) =V(y, t)e β(t) , where α(t), β(t), γ(t) are undetermined functions. By calculation, Substituting the previous Equations into (15) yields By letting According to the terminal conditions α(T ) = β(T ) = γ(T ) = 0, we can get Substituting them back into (16), one gets while the terminal conditionV(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 2 , let ω = z − η √ 2τ , then By transformation U(η, τ) =V(y, t)e β(t) , By substituting the expression of η, τ, β(t) into (18), we can get V(y, t) = e y+ T t [α 1 (s)+α 2 (s)+α 3 (s)]ds N (d 1 ) − Ke 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 ≤ t ≤ 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 ≤ t ≤ T ) can be obtained by solving Equation (14) with the solution method in Theorem 3.2.

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.

Conclusions
In this paper, the underlying asset is modelled by sfBm and the interest rate follows the subfractional 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, θ, σ are timedependent, 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.