Asian Option Pricing under Uncertain Volatility Model

In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.


Introduction
An option on a traded account is a financial contract that allows the buyer of the contract the right to trade an underlying asset for a specified price, called the strike price, during the lifetime of the option. ere are various options, such as European options, American options, Asian options and barrier options. e foundation for the modern analysis of options, the Black-Scholes-Merton pricing formula for European options, was introduced by Black and Scholes [1] and Merton [2]. e Black-Scholes-Merton model assumes constant volatility. However, constant volatility cannot explain the observed market prices for options. After Black, Scholes and Merton's work, some scholars studied option pricing models with stochastic volatility. A series of papers introduced several models for stochastic volatility, such as the Hull-White stochastic volatility model [3] and the Heston stochastic volatility model [4]. e uncertain volatility model is another approach to describe nonconstant volatility. In 1995, Lyons [5] and Avellaneda et al. [6] introduced uncertain volatility models. In these models, volatility is assumed to lie within a range of values, so prices are no longer unique. We can only get the best-case and worst-case scenario prices. Several studies investigate problems with uncertain volatility. We can see these results in Lyons [5], Avellaneda et al. [6], Dokuchaev and Savkin [7], Zhou and Li [8], and Forsyth and Vetzal [9]. ese papers show pricing in uncertain volatility models involving nonlinear partial differential equations. Vorbrink [10] and Epstein and Ji [11] generalized the no-arbitrage theory to financial markets with ambiguous volatility in the mathematically rigorous framework of G-Brownian motion. Method of approximating the valuation equations and the latest research on Fourier transform was given by Zhang et al. [12] and Yu et al. [13]. Pooley et al. [14] and Avellaneda et al. [6] propose some numerical methods.
In 2014, Fouque and Ren [15] studied the price of European derivatives in the worst-case scenario with the uncertain volatility model. ey provide an approximate method of pricing the derivatives with a small volatility interval. In addition, the paper also shows that the solution reduces to a constant volatility problem for simple options with convex payoffs.
is study examines the pricing problem of Asian options. e payoff function is path dependent on risky asset price processes with the addition of another variable to solve the problem. e first problem in estimating the worst-case scenario Asian option prices is obtaining the Hamilton-Jacobi-Bellman (HJB) equation for the prices. e HJB equation is called the Black-Scholes-Barenblatt (BSB) equation in financial mathematics. We can obtain the BSB equation using stochastic control theory. e next difficulty is to prove the convergence of the estimation. To control the error term, we obtain its expectation form using the Dynkin's formula and determine the conditions to impose on the payoff function through proof and deduction. Finally, we obtain the approximation procedure for the prices. Compared to Fouque and Ren's paper [15], we add an equation to the stochastic control system, which we can also reflect in the BSB equation. In terms of the dynamics of the risky asset price process, we provide an equation to describe the path dependence. When estimating the expectation form, we use the relationship between the two processes, in Section 4.4, we fix one of the two variables first to simplify the problem. We manage the two variables using another method that changes the form of the BSB equation. e paper is organized as follows. In Section 2, we briefly describe Asian options under the uncertain volatility model and give the BSB equations for the option prices. In Section 3, we estimate the Asian option prices in the worst-case scenario, where the estimation relies on two Black-Scholeslike PDEs. Next, we propose the main result of this study, which shows the rationality of the estimation. In Section 4, we give the proof of the main result. rough the conditions imposed on the payoff function, we obtain the convergence of the error term. In the process, we obtain the expectation form of the error term, which we divide into three parts. We derive the control for each part using stochastic control theory and the properties of the worst-case scenario Asian option price process. Finally, we conclude the paper.

Asian Options under Uncertain Volatility Model
In this section, we introduce Asian options under the uncertain volatility model. en, we provide the BSB equation of Asian option prices. Suppose that X is an Asian option written on a risky asset with maturity Tand payoff φ(·). φ(·) is a nonconvex function and the result is identical to the Black-Scholes result under convex conditions. at is to say, this study results cover generalized Asian options. Here, generality means that the payoff function φ(·) can be in different forms, as long as it is nonconvex. Assume that the price process of the risky asset X t solves the stochastic differential equation: where r is the constant risk-free interest rate and W t is a standard Brownian motion on the probability space (Ω, F, P). Let σ and σ are two constants and there is σ ≤ σ. e volatility process σ t ∈ A[σ, σ] for each t ∈ [0, T], which is a family of progressively measurable and [σ, σ]-valued processes. By the abovementioned definition, we know that volatility in an uncertain volatility model is not a stochastic process with a probability distribution, but a family of stochastic processes with unknown prior information. us, we can use model ambiguity to distinguish between uncertain volatility models. Due to the path dependence of risky asset price processes, we assume that Y t,T satisfies where Y t � t 0 X u du. en, we can obtain Asian option prices in the worst-case scenario at time t < T as follows: where esssup is the essential supremum. By the ambiguity of the uncertain volatility model, we obtain the definition of price as equation (3). Obviously, the worst-case scenario price is for the option seller and is related to the coherent risk measure that quantifies the model risk induced by volatility uncertainty (see [16]). Moreover, model ambiguity in mathematical finance has captured the attention of many. erefore, we should pay attention to the importance of the worst-case prices.

Lemma 1. V(t, X t , Y t ) satisfies the following BSB equation:
Proof. Note that the stochastic control system is en, for all (s, x, y) ∈ [0, T] × R + × R + , we first establish the dynamic program frame: e cost function is where en, we obtain 2 Mathematical Problems in Engineering Dividing both sides of the inequality by s − s, we have Here, we assume that φ is Lipschitz continuous. en, according to It o's formula and equation (6), we obtain which is In contrast, for any From the argument above, we obtain Combining (14) with (17), we have Here, adding variable Y into the dynamic system leads to a more complex stochastic control system, which adds the dimensionality of the BSB equation.
Remark 2. Note that (4) is a fully nonlinear PDE which has no solution, unlike the Black-Scholes equation. us, we solve the problem by reducing it to two Black-Scholes-like PDEs.

Black-Scholes-Like PDEs and Main Result
In this section, we first reparameterize the uncertain volatility model to study prices in the worst-case scenario. Assume that the risky asset price process satisfies the following SDE: where e cost function is where E txy [·] refers to the conditional expectation taken with respect to By Lemma 1, we obtain the following BSB equation for V ε : which is equivalent to where It is obvious that Mathematical Problems in Engineering the worst-case scenario price is higher than any Black--Scholes price with a constant volatility of σ 0 ∈ [σ, σ]. In the following section, we will show that the worst-case scenario price of Asian options will converge to its Black-Scholes price with constant volatility σ 0 . In addition, we can obtain the rate of convergence of the Asian option prices as the volatility interval shrinks to a single point. en, we can estimate prices through this result when the interval is sufficiently small.
Let V 0 be the Black-Scholes prices, . Now, we suppose that V ε is continuous with respect to ε. en, by the continuity of V ε and equation (3), It is well known that V 0 satisfies the following partial differential equation: In contrast, we have V 1 � z ε V ε | ε�0 , which is the rate of convergence of the Asian option prices as ε approaches 0. To obtain the equation characterizing V 1 , we differentiate both sides of equation (23) with respect to ε and let ε � 0, then we have We now have two Black-Scholes-like PDEs. Next, we want to find the connection between V ε and V 0 , V 1 . en, we try to prove whether it is possible to impose additional conditions on the payoff function to make the error term V ε − (V 0 + εV 1 ) be of order o(ε). at is to say, the estimation of the worst-case scenario Asian option prices will approach the truth-value as the model ambiguity decreases.
is will also provide a method to estimate the worst-case Asian option prices. By the deduction in Section 4, we obtain the following theorem, which is the main result of this study. Theorem 1. Assume that φ ∈ C 2 p (R + ) is Lipschitz continuous, the fourth derivative of φ exists and the second derivative of φ is continuous. en, Here, φ ∈ C 2 p (R + ) means that its derivatives up to order 2 have polynomial growth.

Remark 3.
ere are some difficulties in proving eorem 1. e first is how to convert the error term into an estimable form. Here, we obtain its expectation form and divide it into three parts in Section 4. e second difficulty is how to estimate the three parts. Here, we will use stochastic control theory, the zero set property of equation (33), the properties of sublinear expectation in [18], and the properties of the worst-case scenario Asian option price processes.
Remark 4. By eorem 1, we can compute Asian option price is the Black-Scholes price of the Asian option and we can compute numerically by a simple difference scheme according to (25) (see [14]).
us, when we compute V ε with different ε, we only need to compute V 0 and V 1 once for all small values of ε by eorem 1.

Proof of the Main Result
In this section, we try to control the error term to prove that we can compute V ε with its estimation V 0 + εV 1 . Additionally, from the conditions imposed on φ mentioned in eorem 1, we have the following process of proof. e following parts also reflect our thought process.

e Lipschitz Continuity of the Payoff Function.
From Section 3, we know that only with the continuity of V ε can we obtain the PDEs of V 0 � (V ε | ε�0 ) and V 1 � (z ε V ε | ε�0 ).
us, to obtain the continuity of V ε , we suppose that φ is Lipschitz continuous. en, there exists a constant K 1 such that us, we have the following Lemma.

Lemma 2.
Assume that φ is Lipschitz continuous. en, V ε is continuous with respect to ε.
We have By the Lipschitz continuity of φ and equation (1), there is a constant K 1 such that 4 Mathematical Problems in Engineering With the estimates of the moments of solutions of the stochastic differential equations ( eorem 9 in Section 2.9 and Corollary 12 in Section 2.5 of [19]), we have the constants N � N(q, r, σ 0 us, we have where e continuity of V ε with respect to ε can be proven similarly when ε ≤ ε 0 .

Expectation Form of the Error Term.
In this section, we analyze the error term and give its expectation form as preparation work before proving the convergence of V 0 + εV 1 .

e Polynomial Growth Condition of the Payoff Function.
From Section 4.2, we know that to control the error term, we need to analyze the three parts. By (41), we have erefore, it is sufficient to prove Obviously, it is necessary to obtain controls of the terms |I 2 | and |I 3 |. For |I 1 |, we need to prove its convergence. We first consider controls of the terms |I 2 | and |I 3 |.
By the expressions of I 2 and I 3 , we can see that the partial derivatives of V 0 and V 1 are involved.
us, we should consider estimating them before controlling I 2 and I 3 . Next, we can obtain the expectation form of V 0 and V ε by the classical result. When ε � 0, we have where H(� (1/T) T 0 exp (r − (σ 2 0 /2))(u − t) + σ 0 (W u − W t )}du) is a random variable for fixed t ∈ [0, T]. Similarly, we have where By equations (45) and (46), we note that it is necessary to impose polynomial growth conditions on φ to control z 2 xx V 0 and z 2 xx V ε . en, we estimate z 2 xx V 0 (t, x, y) and z 2 xx V ε (t, x, y) in the following Lemma.