Asian Option Pricing under an Uncertain Volatility Model

Han, Yuecai; Liu, Chunyang

doi:https://doi.org/10.1155/2020/4758052

Mathematical Problems in Engineering

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Stochastic Process Theory and Its Applications

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Research Article | Open Access

Volume 2020 | Article ID 4758052 | https://doi.org/10.1155/2020/4758052

Asian Option Pricing under an Uncertain Volatility Model

Yuecai Han¹and Chunyang Liu¹

Guest Editor: Zhimin Zhang

Received23 Jan 2020

Accepted05 Mar 2020

Published21 Apr 2020

Abstract

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

1. 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. There are various options, such as European options, American options, Asian options and barrier options. The 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]. The 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].

The 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]. These 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. They 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.

This study examines the pricing problem of Asian options. The payoff function is path dependent on risky asset price processes with the addition of another variable to solve the problem. The first problem in estimating the worst-case scenario Asian option prices is obtaining the Hamilton–Jacobi–Bellman (HJB) equation for the prices. The HJB equation is called the Black–Scholes–Barenblatt (BSB) equation in financial mathematics. We can obtain the BSB equation using stochastic control theory. The 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.

The 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–Scholes-like 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. Through 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.

2. Asian Options under Uncertain Volatility Model

In this section, we introduce Asian options under the uncertain volatility model. Then, we provide the BSB equation of Asian option prices. Suppose that is an Asian option written on a risky asset with maturity T and payoff . is a nonconvex function and the result is identical to the Black–Scholes result under convex conditions. That is to say, this study results cover generalized Asian options. Here, generality means that the payoff function can be in different forms, as long as it is nonconvex. Assume that the price process of the risky asset solves the stochastic differential equation:where is the constant risk-free interest rate and is a standard Brownian motion on the probability space . Let and are two constants and there is . The volatility process for each , which is a family of progressively measurable and -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. Thus, we can use model ambiguity to distinguish between uncertain volatility models.

Due to the path dependence of risky asset price processes, we assume that satisfieswhere . Then, we can obtain Asian option prices in the worst-case scenario at time 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. Therefore, we should pay attention to the importance of the worst-case prices.

Through stochastic control theory (see [17]), satisfies the HJB (BSB) equation.

Lemma 1. satisfies the following BSB equation:

Proof. Note that the stochastic control system isThen, for all , we first establish the dynamic program frame:The cost function iswhere . The value function isFor all , , we haveThen, we obtainDividing both sides of the inequality by , we haveHere, we assume that is Lipschitz continuous. Then, according to It ’s formula and equation (6), we obtainLet . For all , we havewhich isIn contrast, for any , there is a such thatThus, we haveFrom the argument above, we obtainCombining (14) with (17), we have

Remark 1. Here, adding variable 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. Thus, we solve the problem by reducing it to two Black–Scholes-like PDEs.

3. 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 and . The cost function iswhere refers to the conditional expectation taken with respect to , . The value function is

By Lemma 1, we obtain the following BSB equation for :which is equivalent towhere . It is obvious that the worst-case scenario price is higher than any Black–Scholes price with a constant volatility of . 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 . In addition, we can obtain the rate of convergence of the Asian option prices as the volatility interval shrinks to a single point. Then, we can estimate prices through this result when the interval is sufficiently small.

Let be the Black–Scholes prices, , . Now, we suppose that is continuous with respect to . Then, by the continuity of and equation (3), we have . It is well known that satisfies the following partial differential equation:

In contrast, we have , which is the rate of convergence of the Asian option prices as approaches 0. To obtain the equation characterizing , we differentiate both sides of equation (23) with respect to and let , then we have

We now have two Black–Scholes-like PDEs. Next, we want to find the connection between and . Then, we try to prove whether it is possible to impose additional conditions on the payoff function to make the error term be of order . That is to say, the estimation of the worst-case scenario Asian option prices will approach the truth-value as the model ambiguity decreases. This 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 is Lipschitz continuous, the fourth derivative of exists and the second derivative of is continuous. Then,

Here, means that its derivatives up to order 2 have polynomial growth.

Remark 3. There are some difficulties in proving Theorem 1. The 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. The 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 Theorem 1, we can compute Asian option price with its approximation, , where is the Black–Scholes price of the Asian option and we can compute numerically by a simple difference scheme according to (25) (see [14]).

Remark 5. Note that (24) and (25) are independent of . Thus, when we compute with different , we only need to compute and once for all small values of by Theorem 1.

4. Proof of the Main Result

In this section, we try to control the error term to prove that we can compute with its estimation . Additionally, from the conditions imposed on mentioned in Theorem 1, we have the following process of proof. The following parts also reflect our thought process.

4.1. The Lipschitz Continuity of the Payoff Function

From Section 3, we know that only with the continuity of can we obtain the PDEs of and . Thus, to obtain the continuity of , we suppose that is Lipschitz continuous. Then, there exists a constant such that

Thus, we have the following Lemma.

Lemma 2. Assume that is Lipschitz continuous. Then, is continuous with respect to .

Proof. Let . Note thatWe haveBy the Lipschitz continuity of and equation (1), there is a constant such thatWith the estimates of the moments of solutions of the stochastic differential equations (Theorem 9 in Section 2.9 and Corollary 12 in Section 2.5 of [19]), we have the constants such thatThus, we havewhere .
Let . We have .
The continuity of with respect to can be proven similarly when .

4.2. 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 .

Let be the worst-case scenario volatility process and be the worst-case scenario risky asset process. Then, we can rewrite equation (19) as follows:

We can obtain the expression of by equation (23), and we have , where

Similarly, by solving equation (25) for , we have the volatility process, , where

Here, we use the short notation and for and , respectively. Let . To estimate the error term , we define the operator . According to partial differential equations (22), (24) and (25), we havewith the boundary condition . We have the following expectation form of by the Dynkin’s formula:where

We can therefore estimate by controlling , , and .

4.3. The 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

Therefore, it is sufficient to prove

Obviously, it is necessary to obtain controls of the terms and . For , we need to prove its convergence. We first consider controls of the terms and .

By the expressions of and , we can see that the partial derivatives of and are involved. Thus, we should consider estimating them before controlling and . Next, we can obtain the expectation form of and by the classical result. When , we have

Thus,where is a random variable for fixed . Similarly, we havewhere is a random variable for a fixed .

By equations (45) and (46), we note that it is necessary to impose polynomial growth conditions on to control and . Then, we estimate and in the following Lemma.

Lemma 3. Suppose that the second derivative of the payoff function satisfies the polynomial growth condition, that is, there are constants and such that . Then, we have constant such thatwhere depends on , , , and . Moreover, there is a constant such thatwhere depends on , , , and .

Proof. As the assumption of in the lemma, we haveHere, depends on , , , and .
Indeed, for a constant , we haveWe obtain the control similarly. Then, there is a constant that depends on , , , and such thatNow, by the following proposition, we can obtain controls of the terms and .

Proposition 1. Assume that and the Lipschitz continuity condition holds. Then, there exist constants and such that in equations (39) and (40) satisfy

Proof. By Lemma 3, we have the following inequality from (23) and (48):By the expression of , it is true thatBy (25) and (47), we get the controls for :where depends on , , and . We can obtain the existence and uniqueness of from Theorem 5.2.1 in [20]. Then, by the estimates of the moments of solutions of the stochastic differential equations (Corollary 12 in Section 2.5 of [19]), there is a constant for a fixed such thatBy (40), (55), and (56), we have the following inequality:Here, depends on , , , and .
By (39), (47), (55), and (56), we obtain the control of term .where depends on , , , , , and , .Combining (57) and (58), we have a constant such that

4.4. The Continuity of the Second Derivative of the Payoff Function

By Proposition 1, we obtain controls of the terms and . Next, for the fixed point , it suffices to prove that

Note that if (i.e., its derivatives up to order 2 have polynomial growth), we can obtain the following inequality by (38), (47), (56), and Hölder inequality:

Here, depends on , and . Moreover, is independent of .

Let . By (34) and (35), we have

Thus, to prove as , it suffices to prove that

By the expression of , we should analyze the derivatives of and . Here, we find that the continuity of is necessary.

Lemma 4. Assume that is continuous. Then, and are continuous with respect to .

Proof. By (45), we have and . If is continuous, then for all , , there is a constant such thatfor all . So, for all , , and , we haveThus, we obtainSimilarly, we can obtain the continuity of .

Remark 6. Rationally, and its derivatives converge to and its corresponding derivatives as approaches 0 by Lemma 2.

Remark 7. To simplify the complexity brought by the variable , which is called path dependence, and to study the behavior of , we defineLet . Then, we can obtain the following equation when is continuous:

Remark 8. To control , we divide into two parts. Let . We will discuss the characteristics of and .

Lemma 5. Assume that is continuous and the fourth derivative of exists. Then, we have

Here, refers to the conditional probability with respect to and .

Proof. By (33) and (24), we can obtain the equationLet . Then, by equation (70) and the existence of the fourth derivative of , we haveLet . Then, we haveNote that the coefficients in equation (72) are constants and Q is bounded on by the continuity of and Lemma 4. Moreover, by equation (72), we find that has no relationship with the equations. Then, by Theorem A of [21] and the remark below it, we find that the number of zero points of is only countable for all . Thus, has only countable zero points. Hence, we have by Lemma 4.10 of [15] and then the proof of Lemma 5 is complete.
Based on the previous analysis, we will now prove (63). We split the expectation into two parts. By proving the convergence of each part, we can show the convergence of the expectation.

Proposition 2. Assume that , is continuous, and the fourth derivative of exists. Then, we obtain equation (63).

Proof. Let be the closure of , , and .
By the definition of , we haveNow, we consider the second part of (73) first. By (56) and Chebyshev’s inequality,Thus, we haveFor the first part, we note thatLet ; then,Note that . Then, is a sequence of decreasing closed sets as . Obviously, converges weakly to . Thus, is weakly compact. By Lemma 8 in [18], we can see thatBy Lemma 4, there is . Then, by Lemma 5, we haveNext, by the definition of , we haveThus,Then, we obtainBy equations (75) and (82), for any , there is such thatNext, for a given and , there is such thatTherefore, for any , there is such thati.e.,

4.5. Proof of the Main Result

Now, from the analysis above, we can give the brief proof of Theorem 1.

By inequality (61) and Proposition 2, we have

By inequality (41), we have

By Proposition 1 and equation (87), we obtain the theorem.

5. Conclusion

In this study, we analyze the behavior of Asian option prices in the worst-case scenario using an uncertain volatility model with volatility interval . As approaches 0, the ambiguity of the model vanishes. We can also see that the worst-case scenario prices of Asian options converge to its Black–Scholes prices with constant volatility as the interval shrinks. Additionally, this study provides an approach to estimate the worst-case scenario Asian option prices. At the same time, we also provide an estimation method to solve a fully nonlinear PDE (22) by imposing additional conditions on the boundary condition and splitting it into two Black–Scholes-like equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was partially supported by NSFC (Grant nos. 11871244 and 11901233).

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Copyright © 2020 Yuecai Han and Chunyang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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