# Asian Option Pricing under an Uncertain Volatility Model

Authors: Yuecai Han; Chunyang Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 Section2, 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 thatX is an Asian option written on a risky asset with maturity T and payoff φ·. φ· 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 φ· can be in different forms, as long as it is nonconvex. Assume that the price process of the risky asset Xt solves the stochastic differential equation:(1)dXt=rXtdt+σtXtdWt,where r is the constant risk-free interest rate and Wt is a standard Brownian motion on the probability space Ω,F,ℙ. Let σ¯ and σ¯ are two constants and there is σ¯≤σ¯. The volatility process σt∈Aσ¯,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. Thus, we can use model ambiguity to distinguish between uncertain volatility models.Due to the path dependence of risky asset price processes, we assume thatYt,T satisfies(2)Yt,T=YT−YtT−t,where Yt=∫0tXudu. Then, we can obtain Asian option prices in the worst-case scenario at time t<T as follows:(3)Vt,Xt,Yt=exp−rT−tesssupσ∈Aσ¯,σ¯EφY0,TFt,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]), Vt,Xt,Yt satisfies the HJB (BSB) equation.Lemma 1.
V
t
,
X
t
,
Y
t satisfies the following BSB equation:(4)∂tV+rx∂xV−V+x∂yV+supσ∈Aσ¯,σ¯12x2σ2∂xx2V=0,0≤t≤T,x≥0,y≥0,VT,x,y=φyT,x≥0,y≥0.Proof.
Note that the stochastic control system is(5)dXt=rXtdt+σtXtdWt,σt∈Aσ¯,σ¯,dYt=Xtdt.
Then, for alls,x,y∈0,T×R+×R+, we first establish the dynamic program frame:(6)dXt=rXtdt+σtXtdWt,dYt=Xtdt,Xs=x,Ys=y.
The cost function is(7)Js,x,y;σ=Ese−rT−sφY0,T,where Es·=E⋅Fs. The value function is(8)Vs,x,y=esssupσ∈Aσ¯,σ¯Js,x,y;σ.
For all0≤s≤s^≤T, σ∈Aσ¯,σ¯, we have(9)Vs,x,y≥Ese−rT−sφY0,T=Es∫ss^−re−rT−tφdt+e−rT−s^φ.
Then, we obtain(10)0≥Es∫ss^−re−rT−tφdt+Vs^,x,y−Vs,x,y.
Dividing both sides of the inequality bys^−s, we have(11)0≥Es∫ss^−re−rT−tφdts^−s+Vs^,x,y−Vs,x,ys^−s.
Here, we assume thatφ is Lipschitz continuous. Then, according to It o^’s formula and equation (6), we obtain(12)dV=Vtdt+VxdXt+VydYt+12VxxdXtdXt+12VyydYtdYt+12VxydXtdYt=Vt+rXtVx+XtVy+12σt2Xt2Vxxdt+σtXtVxdWt.
Lets^⟶s. For all σ∈Aσ¯,σ¯, we have(13)0≥−rEse−rT−sφ+Vt+rXsVx+XsVy+12σs2Xs2Vxx≥−rVs,x,y+Vts,x,y+rxVxs,x,y+xVys,x,y+12σs2Xs2Vxxs,x,y,which is(14)0≥−rV+Vt+rxVx+xVy+supσ∈Aσ¯,σ¯12σ2x2Vxx.
In contrast, for anyε>0, there is a σε∈Aσ¯,σ¯ such that(15)Vs,x,y−εs^−s≤Ese−rT−sφ=Es∫ss^−re−rT−tφdt+Ese−rT−s^φ.
Thus, we have(16)−ε≤Es∫ss^−re−rT−tφdts^−s+Vs^,x,y−Vs,x,ys^−s.
From the argument above, we obtain(17)0≤−rV+Vt+rxVx+xVy+supσ∈Aσ¯,σ¯12σ2x2Vxx.
Combining (14) with (17), we have(18)0=−rV+Vt+rxVx+xVy+supσ∈Aσ¯,σ¯12σ2x2Vxx.Remark 1.
Here, adding variableY 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:(19)dXtε=rXtεdt+σtXtεdWt,dYtε=Xtεdt,where σt∈Aε=σtσtis aσ0,σ0+ε−valued progressively measurable process and σ0∈σ¯,σ¯. The cost function is(20)Jεt,x,y;σ=e−rT−tEtxyφY0,Tε,where Etxy· refers to the conditional expectation taken with respect to Xtε=x, Ytε=y. The value function is(21)Vεt,x,y;σ=esssupσ∈AεJεt,x,y;σ.By Lemma1, we obtain the following BSB equation for Vε:(22)∂tVε+rx∂xVε−Vε+x∂yVε+supσ∈Aε12x2σ2∂xx2Vε=0,0≤t≤T,x≥0,y≥0,VεT,x,y=φyT,x≥0,y≥0,which is equivalent to(23)∂tVε+rx∂xVε−Vε+x∂yVε+supγ∈A0,112x2σ0+εγ2∂xx2Vε=0,0≤t≤T,x≥0,y≥0,VεT,x,y=φyT,x≥0,y≥0,where A0,1=γtγtis a0,1−valued progressively measurable process. It is obvious that the worst-case scenario price is higher than any Black–Scholes price with a constant volatility of σ0∈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 σ0. 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.LetV0 be the Black–Scholes prices, V0=Vεε=0, V1=∂εVεε=0. Now, we suppose that Vε is continuous with respect to ε. Then, by the continuity of Vε and equation (3), we have V0=V0=Vεε=0. It is well known that V0 satisfies the following partial differential equation:(24)∂tV0+rx∂xV0−V0+x∂yV0+12σ02x2∂xx2V0=0,0≤t≤T,x≥0,y≥0,V0T,x,y=φyT,x≥0,y≥0.In contrast, we haveV1=∂εVεε=0, which is the rate of convergence of the Asian option prices as ε approaches 0. To obtain the equation characterizing V1, we differentiate both sides of equation (23) with respect to ε and let ε=0, then we have(25)∂tV1+rx∂xV1−V1+x∂yV1+12σ02x2∂xx2V1+supγ∈A0,1γσ0x2∂xx2V0=0,0≤t≤T,x≥0,y≥0,V1T,x,y=0,x≥0,y≥0.We now have two Black–Scholes-like PDEs. Next, we want to find the connection betweenVε and V0,V1. Then, we try to prove whether it is possible to impose additional conditions on the payoff function to make the error term Vε−V0+εV1 be of order oε. 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 thatφ∈Cp2R+ is Lipschitz continuous, the fourth derivative of φ exists and the second derivative of φ is continuous. Then,(26)limε↓0Vε−V0+εV1ε=0.Here,φ∈Cp2R+ means that its derivatives up to order 2 have polynomial growth.Remark 3.
There are some difficulties in proving Theorem1. 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 Theorem1, we can compute Asian option price Vεt,Xtε,Ytε with its approximation, V0t,Xtε,Ytε+εV1t,Xtε,Ytε, where V0t,Xtε,Ytε is the Black–Scholes price of the Asian option and we can compute V1t,Xtε,Ytε numerically by a simple difference scheme according to (25) (see [14]).Remark 5.
Note that (24) and (25) are independent of ε. Thus, when we compute Vε with different ε, we only need to compute V0 and V1 once for all small values of ε by Theorem 1.


## 4. Proof of the Main Result

In this section, we try to control the error term to prove that we can computeVε with its estimation V0+εV1. Additionally, from the conditions imposed on φ 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 Section3, we know that only with the continuity of Vε can we obtain the PDEs of V0=Vεε=0 and V1=∂εVεε=0. Thus, to obtain the continuity of Vε, we suppose that φ is Lipschitz continuous. Then, there exists a constant K1 such that(27)φx−φy≤K1x−y,for allx≠y,x,y∈R+.Thus, we have the following Lemma.Lemma 2.
Assume thatφ is Lipschitz continuous. Then, Vε is continuous with respect to ε.Proof.
Let0≤ε0≤ε<1. Note that(28)Vεt,x,y;σ=esssupσ∈Aεe−rT−tEtxyφY0,Tε.
We have(29)erT−tVε0t,x,y;σ=esssupσ∈Aε0EtxyφY0,Tε0σ=esssupσ∈AεEtxyφY0,Tεσ∧σ0+ε0.
By the Lipschitz continuity ofφ and equation (1), there is a constant K1 such that(30)erT−tVεt,x,y;σ−Vε0t,x,y;σ≤esssupσ∈AεEtxyφY0,Tεσ−EtxyφY0,Tεσ∧σ0+ε0≤K1esssupσ∈AεEtxyY0,Tεσ−Y0,Tεσ∧σ0+ε021/2≤K1/Tesssupσ∈AεEtxy∫0TXuεσ−Xuεσ∧σ0+ε02du1/2.
With 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 N=Nq,r,σ0,N′=N′q,r,σ0,andC=maxNN′,N+N′ such that(31)Etxysups≤uXsεσ−Xsεσ∧σ0+ε02q≤Nuq−1eNuEtxy∫0uXsεσ2q·σs−σs∧σs+ε02qds≤Nuq−1eNuN′eN′uu1+x2qε−ε02q=CuqeCu1+x2qε−ε02q.
Thus, we have(32)erT−tVεt,x,y−Vε0t,x,y≤K1Tesssupσ∈Aε∫0TEtxysups∈0,uXuεσ−Xuεσ∧σ0+ε02du1/2≤K1Tesssupσ∈Aε∫0TCueCu1+x2ε−ε02du1/2≤K1′1+x21/2ε−ε0,where K1′=K1′K1,C,T.
Letε⟶ε0. We have Vεt,x,y−Vε0t,x,y⟶0.
The continuity ofVε with respect to ε can be proven similarly when ε≤ε0.


### 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 ofV0+εV1.Letσ^t be the worst-case scenario volatility process and X^tε be the worst-case scenario risky asset process. Then, we can rewrite equation (19) as follows:(33)dX^tε=rX^tεdt+σ^tX^tεdWt,dY^tε=X^tεdt.We can obtain the expression ofσ^ by equation (23), and we have σ^ε=σ0+εγ^, where(34)γ^t,x,y;ε=1,∂xx2Vεt,x,y≥0,0,∂xx2Vεt,x,y<0.Similarly, by solving equation (25) for V1, we have the volatility process, σ¯ε=σ0+εγ¯, where(35)γ¯t,x,y=1,∂xx2V0t,x,y≥0,0,∂xx2V0t,x,y<0.Here, we use the short notationγ^t and γ¯t for γ^t,x,y;ε and γ¯t,x,y, respectively. Let Zε=Vε−V0+εV1. To estimate the error term Zε, we define the operator Lσ=∂t+rx∂x−r+1/2σ2x2∂xx2+x∂y. According to partial differential equations (22), (24) and (25), we have(36)Lσ^tZε=Lσ^tVε−V0+εV1=0−Lσ^tV0+εV1=−Lσ^t−Lσ0V0−Lσ0V0−εLσ^t−Lσ0V1−εLσ0V1=εγ¯t−γ^tσ0x2∂xx2V0−ε22γ^t2x2∂xx2V0+2σ0γ^tx2∂xx2V1−ε32γ^t2x2∂xx2V1=−fεt,x,y,with the boundary condition ZεT=VεT−V0T−εV1T=0. We have the following expectation form of Zε by the Dynkin’s formula:(37)Zε=Etxy∫tTfεs,x,yds=εEtxy∫tTγ^s−γ¯s⋅σ0⋅X^sε2∂xx2V0s,X^sε,Y^sεds+ε2Etxy∫tT12γ^s2X^sε2∂xx2V0s,X^sε,Y^sε+σ0γ^sX^sε2℘xx2V1s,X^sε,Y^sεds+ε3Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds=εI1+ε2I2+ε3I3,where(38)I1=Etxy∫tTγ^s−γ¯s⋅σ0⋅X^sε2∂xx2V0s,X^sε,Y^sεds,(39)I2=Etxy∫tT12γ^s2X^sε2∂xx2V0s,X^sε,Y^sε+σ0γ^sX^sε2∂xx2V1s,X^sε,Y^sεds,(40)I3=Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds.(41)Zε≤εI1+ε2I2+ε3I3.We can therefore estimateZε by controlling I1, I2, and I3.


### 4.3. The Polynomial Growth Condition of the Payoff Function

From Section4.2, we know that to control the error term, we need to analyze the three parts. By (41), we have(42)Zεε≤I1+εI2+εI3.Therefore, it is sufficient to prove(43)limε↓0I1+εI2+εI3=0.Obviously, it is necessary to obtain controls of the termsI2 and I3. For I1, we need to prove its convergence. We first consider controls of the terms I2 and I3.By the expressions ofI2 and I3, we can see that the partial derivatives of V0 and V1 are involved. Thus, we should consider estimating them before controlling I2 and I3. Next, we can obtain the expectation form of V0 and Vε by the classical result. When ε=0, we have(44)Xu=xexpr−σ022u−t+σ0Wu−Wt.Thus,(45)V0t,x,y=e−rT−tEtxyφY0,T=e−rT−tEtxyφ1T∫0TXudu=e−rT−tEtxyφ1T⋅x⋅∫0Ter−σ02/2u−t+σ0Wu−Wtdu=e−rT−tEtxyφx·H,where H=1/T∫0Texpr−σ02/2u−t+σ0Wu−Wtdu is a random variable for fixed t∈0,T. Similarly, we have(46)Vεt,x,y=e−rT−tesssupσ∈AεEtxyφY0,Tε=e−rT−tEtxyφx⋅G,where G=1/T∫0Texpr−σ^u2/2u−t−σ^uWu−Wtdu is a random variable for a fixed t∈0,T.By equations (45) and (46), we note that it is necessary to impose polynomial growth conditions on φ to control ∂xx2V0 and ∂xx2Vε. Then, we estimate ∂xx2V0t,x,y and ∂xx2Vεt,x,y in the following Lemma.Lemma 3.
Suppose that the second derivative of the payoff function satisfies the polynomial growth condition, that is, there are constantsK2 and m such that φ″x≤K21+xm. Then, we have constant K3 such that(47)∂xx2V0t,x,y≤K31+xm,where K3 depends on T,t, EtxyH2, EtxyHm+2, and K2. Moreover, there is a constant K4 such that(48)∂xx2Vεt,x,y≤K41+xm,where K4 depends on T,t, EtxyG2, EtxyGm+2, and K2.Proof.
As the assumption ofφ in the lemma, we have(49)∂xx2V0t,x,y=e−rT−tEtxyφ″xHH2≤e−rT−tEtxyK21+xHmH2≤K31+xm.
Here,K3 depends on T,t, EtxyH2, EtxyHm+2, and K2.
Indeed, for a constantm>0, we have(50)EHm=1TmE∫−tT−texpr−σ022u+σ0Wudum≤1TmE∫−tT−ter−σ02/2T−t+σ0Wudum≤1Tmemr−σ02/2T−tEsups∈−t,T−teσ0Wsm<+∞.
We obtain the controltialxx2Vε similarly. Then, there is a constant K4 that depends on T,t, EtxyG2, EtxyGm+2, and K2 such that(51)∂xx2Vεt,x,y≤K41+xm.
Now, by the following proposition, we can obtain controls of the termsI2 and I3.Proposition 1.
Assume thatφ∈Cp2R+ and the Lipschitz continuity condition holds. Then, there exist constants C1 and p1 such that I2andI3 in equations (39) and (40) satisfy(52)I2+I3≤C11+xp1.Proof.
By Lemma3, we have the following inequality from (23) and (48):(53)∂tVε+rx∂xVε−Vε+x∂yVε≤12σ0+ε2x2∂xx2Vε≤K42σ0+ε2x2+xm+2.
By the expression ofV1, it is true that(54)∂tV1+rx∂xV1−V1+x∂yV1≤K4σ0x2+xm+2.
By (25) and (47), we get the controls for x2∂xx2V1:(55)x2∂xx2V1=∂tV1+rx∂xV1−V1+x∂yV1+g¯tσ0x2∂xx2V0·2σ02≤∂tV1+rx∂xV1−V1+x∂yV1+σ0x2∂xx2V0·2σ02≤M1x2+xm+2,where M1 depends on K3, K4, and σ0. We can obtain the existence and uniqueness of X^tε 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 N1q for a fixed q>0 such that(56)Etxysups∈t,TX^sεq≤N1qeN1qT−t1+xq.
By (40), (55), and (56), we have the following inequality:(57)I3=Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds≤M12Etxy∫tTX^sε2+X^sεm+2ds≤M1′1+xm+2.
Here,M1′ depends on T,t, N12, N1m+2, and M1.
By (39), (47), (55), and (56), we obtain the control of term I2.(58)I2=Etxy∫tT12γ^s2X^sε2∂xx2V0+σ0γ^sX^sε2∂xx2V1ds≤K32Etxy∫tTX^sε2+X^sεm+2ds+M1Etxy∫tTX^sε2+X^sεm+2ds≤M21+xp1,where M2 depends on T, t, M1, K3, N12, and N1m+2, p1≥m+2.Combining (57) and (58), we have a constant C1 such that(59)I2+I3≤C11+xp1.


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

By Proposition1, we obtain controls of the terms I2 and I3. Next, for the fixed point t,x,y∈0,T×R+×R+, it suffices to prove that(60)limε↓0I1=0.Note that ifφ∈Cp2R+ (i.e., its derivatives up to order 2 have polynomial growth), we can obtain the following inequality by (38), (47), (56), and Hölder inequality:(61)I1≤Etxy∫tTσ0X^sε2∂xx2V02ds1/2Etxy∫tTγ^s−γ¯s2ds1/2≤M31+xp21/2Etxy∫tTγ^s−γ¯sds1/2.Here,M3 depends on K3,T,t,σ0, and p2≥4+2m. Moreover, M3 is independent of ε.Lethεt,x,y=γ^t,x,y;ε−γ¯t,x,y. By (34) and (35), we have(62)hεt,x,y=1,∂xx2Vε∂xx2V0<0,0,∂xx2Vε∂xx2V0≥0.Thus, to proveI1⟶0 as ε⟶0, it suffices to prove that(63)limε↓0Etxy∫tThεs,X^sε,Y^sεds=0.By the expression ofhε, we should analyze the derivatives of V0 and Vε. Here, we find that the continuity of φ″ is necessary.Lemma 4.
Assume thatφ″ is continuous. Then, ∂xx2V0 and ∂xx2Vε are continuous with respect to x,y.Proof.
By (45), we have V0t,x,y=e−rT−tEtxyφxH and ∂xx2V0t,x,y=e−rT−tEtxyφ″xHH2. If φ″ is continuous, then for all x0∈R+, δ>0, there is a constant ξ=ξδ,x0 such that(64)φ″xH−φ″x0H≤δ,for all xH∈x0H−ξ,x0H+ξ. So, for all x0,y0∈R+×R+, xH∈x0H−ξ,x0H+ξ, and y∈y0−ξ,y0+ξ, we have(65)∂xx2V0t,x,y−∂xx2V0t,x0,y0=e−rT−tEtxyφ″xHH2−φ″x0HH2≤e−rT−tEtxyH2φ″xH−φ″x0H≤e−rT−tδEtxyH2.
Thus, we obtain(66)limx,y⟶x0,y0∂xx2V0t,x,y=∂xx2V0t,x0,y0.
Similarly, we can obtain the continuity of∂xx2Vε.Remark 6.
Rationally,Vε and its derivatives converge to V0 and its corresponding derivatives as ε approaches 0 by Lemma 2.Remark 7.
To simplify the complexity brought by the variableY, which is called path dependence, and to study the behavior of hε, we define(67)Dtyλ=x∈R+∂xx2Vε0∂xx2V0≤0,∃ε0>λ.
LetDty0=limλ↓0Dtyλ. Then, we can obtain the following equation when ∂xx2Vε is continuous:(68)Dty0=x∈R+∂xx2V0t,x,y=0.Remark 8.
To controlhε, we divide Dtyλ into two parts. Let αρ=−ρ,ρ. We will discuss the characteristics of Dtyλ∩αρ and Dtyλ∩αρc.Lemma 5.
Assume thatφ″ is continuous and the fourth derivative of φ exists. Then, we have(69)PtxyDsys0∩αρ=0,fors∈t,T.Here,Ptxy⋅ refers to the conditional probability with respect to Xtε=x and Ytε=y.Proof.
By (33) and (24), we can obtain the equation(70)2∂tV0+rx∂xV0−V0+12σ02x2∂xx2V0=0,V0T=φxH.
LetQ=∂xx2V0. Then, by equation (70) and the existence of the fourth derivative of φ, we have(71)2∂tQ+r+σ02Q+r+2σ02x∂xQ+12σ02x2∂xx2Q=0,QT=φ″xHH2.
Letx=logk. Then, we have(72)2∂tQ+r+σ02Q+r+2σ02∂kQ+12σ02∂k2Q=0,QT=φ″logkHH2.
Note that the coefficients in equation (72) are constants and Q is bounded on Dsys0∩αρ by the continuity of φ″ and Lemma 4. Moreover, by equation (72), we find that y 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 Q is only countable for all s,ys∈t,T×R. Thus, ∂xx2V0 has only countable zero points. Hence, we have PtxyDsys0∩αρ=0 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 thatφ∈Cp2R+, φ″ is continuous, and the fourth derivative of φ exists. Then, we obtain equation (63).Proof.
LetD¯tyλ be the closure of Dtyλ, D¯ty0=limλ↓0D¯tyλ, and 0≤λ<ε<1.
By the definition ofDtyλ, we have(73)Etxy∫tThεs,X^sε,Y^sεds≤Etxy∫tTID¯sY^sελX^sεds=Etxy∫tTID¯sY^sελ∩αρX^sεds+Etxy∫tTID¯sY^sελ∩αρcX^sεds=Φ1+Φ2.
Now, we consider the second part of (73) first. By (56) and Chebyshev’s inequality,(74)Φ2≤Etxy∫tTIαρcX^sεds≤∫tTPtxysups∈t,TX^sε≥ρds≤T−tρEtxysups∈t,TX^sε≤T−tN11ρeN11T−t1+x.
Thus, we have(75)limρ⟶∞Φ2=0.
For the first part, we note that(76)Φ1=∫tTPtxyX^sε∈D¯sY^sελ∩αρds.
LetθΩ=supλ∈0,1PtxyΩ; then,(77)PtxyX^sε∈D¯sY^sελ∩αρ≤θD¯sY^sελ∩αρ.
Note thatλ<ε. Then, D¯syλ is a sequence of decreasing closed sets as ε↓0. Obviously, X^sε converges weakly to Xs. Thus, Xs is weakly compact. By Lemma 8 in [18], we can see that(78)limε↓0θD¯sY^sελ∩αρ=θD¯sY^s00∩αρ.
By Lemma4, there is D¯sy0=Dsy0. Then, by Lemma 5, we have(79)PtxyX^sε∈D¯sY^sε0∩αρ=0,forε≥0.
Next, by the definition ofθΩ, we have(80)θD¯sY^s00∩αρ=0.
Thus,(81)limε↓0θD¯sY^sελ∩αρ=0.
Then, we obtain(82)limε↓0Φ1=limε↓0PtxyX^sε∈D¯sY^sελ∩αρ=0.
By equations (75) and (82), for any δ>0, there is ρ0=ρ0t,x,y,δ>0 such that(83)Φ2<δ2,for  allρ>ρ0.
Next, for a givenρ0 and δ, there is ε0=ε0t,x,y,δ,ρ0t,x,y,δ such that(84)Φ1<δ2,for allε<ε0.
Therefore, for anyδ>0, there is ε0=ε0t,x,y,δ such that(85)Φ1+Φ2<δ,for allε<ε0,i.e.,(86)limε↓0Etxy∫tThεs,X^sε,Y^sεds=0.


### 4.5. Proof of the Main Result

Now, from the analysis above, we can give the brief proof of Theorem1.By inequality (61) and Proposition 2, we have(87)limε↓0I1=0.By inequality (41), we have(88)Vε−V0+εV1ε≤I1+εI2+εI3=0.By Proposition1 and equation (87), we obtain the theorem.


## 4.1. The Lipschitz Continuity of the Payoff Function

From Section3, we know that only with the continuity of Vε can we obtain the PDEs of V0=Vεε=0 and V1=∂εVεε=0. Thus, to obtain the continuity of Vε, we suppose that φ is Lipschitz continuous. Then, there exists a constant K1 such that(27)φx−φy≤K1x−y,for allx≠y,x,y∈R+.Thus, we have the following Lemma.Lemma 2.
Assume thatφ is Lipschitz continuous. Then, Vε is continuous with respect to ε.Proof.
Let0≤ε0≤ε<1. Note that(28)Vεt,x,y;σ=esssupσ∈Aεe−rT−tEtxyφY0,Tε.
We have(29)erT−tVε0t,x,y;σ=esssupσ∈Aε0EtxyφY0,Tε0σ=esssupσ∈AεEtxyφY0,Tεσ∧σ0+ε0.
By the Lipschitz continuity ofφ and equation (1), there is a constant K1 such that(30)erT−tVεt,x,y;σ−Vε0t,x,y;σ≤esssupσ∈AεEtxyφY0,Tεσ−EtxyφY0,Tεσ∧σ0+ε0≤K1esssupσ∈AεEtxyY0,Tεσ−Y0,Tεσ∧σ0+ε021/2≤K1/Tesssupσ∈AεEtxy∫0TXuεσ−Xuεσ∧σ0+ε02du1/2.
With 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 N=Nq,r,σ0,N′=N′q,r,σ0,andC=maxNN′,N+N′ such that(31)Etxysups≤uXsεσ−Xsεσ∧σ0+ε02q≤Nuq−1eNuEtxy∫0uXsεσ2q·σs−σs∧σs+ε02qds≤Nuq−1eNuN′eN′uu1+x2qε−ε02q=CuqeCu1+x2qε−ε02q.
Thus, we have(32)erT−tVεt,x,y−Vε0t,x,y≤K1Tesssupσ∈Aε∫0TEtxysups∈0,uXuεσ−Xuεσ∧σ0+ε02du1/2≤K1Tesssupσ∈Aε∫0TCueCu1+x2ε−ε02du1/2≤K1′1+x21/2ε−ε0,where K1′=K1′K1,C,T.
Letε⟶ε0. We have Vεt,x,y−Vε0t,x,y⟶0.
The continuity ofVε with respect to ε can be proven similarly when ε≤ε0.


## 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 ofV0+εV1.Letσ^t be the worst-case scenario volatility process and X^tε be the worst-case scenario risky asset process. Then, we can rewrite equation (19) as follows:(33)dX^tε=rX^tεdt+σ^tX^tεdWt,dY^tε=X^tεdt.We can obtain the expression ofσ^ by equation (23), and we have σ^ε=σ0+εγ^, where(34)γ^t,x,y;ε=1,∂xx2Vεt,x,y≥0,0,∂xx2Vεt,x,y<0.Similarly, by solving equation (25) for V1, we have the volatility process, σ¯ε=σ0+εγ¯, where(35)γ¯t,x,y=1,∂xx2V0t,x,y≥0,0,∂xx2V0t,x,y<0.Here, we use the short notationγ^t and γ¯t for γ^t,x,y;ε and γ¯t,x,y, respectively. Let Zε=Vε−V0+εV1. To estimate the error term Zε, we define the operator Lσ=∂t+rx∂x−r+1/2σ2x2∂xx2+x∂y. According to partial differential equations (22), (24) and (25), we have(36)Lσ^tZε=Lσ^tVε−V0+εV1=0−Lσ^tV0+εV1=−Lσ^t−Lσ0V0−Lσ0V0−εLσ^t−Lσ0V1−εLσ0V1=εγ¯t−γ^tσ0x2∂xx2V0−ε22γ^t2x2∂xx2V0+2σ0γ^tx2∂xx2V1−ε32γ^t2x2∂xx2V1=−fεt,x,y,with the boundary condition ZεT=VεT−V0T−εV1T=0. We have the following expectation form of Zε by the Dynkin’s formula:(37)Zε=Etxy∫tTfεs,x,yds=εEtxy∫tTγ^s−γ¯s⋅σ0⋅X^sε2∂xx2V0s,X^sε,Y^sεds+ε2Etxy∫tT12γ^s2X^sε2∂xx2V0s,X^sε,Y^sε+σ0γ^sX^sε2℘xx2V1s,X^sε,Y^sεds+ε3Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds=εI1+ε2I2+ε3I3,where(38)I1=Etxy∫tTγ^s−γ¯s⋅σ0⋅X^sε2∂xx2V0s,X^sε,Y^sεds,(39)I2=Etxy∫tT12γ^s2X^sε2∂xx2V0s,X^sε,Y^sε+σ0γ^sX^sε2∂xx2V1s,X^sε,Y^sεds,(40)I3=Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds.(41)Zε≤εI1+ε2I2+ε3I3.We can therefore estimateZε by controlling I1, I2, and I3.


## 4.3. The Polynomial Growth Condition of the Payoff Function

From Section4.2, we know that to control the error term, we need to analyze the three parts. By (41), we have(42)Zεε≤I1+εI2+εI3.Therefore, it is sufficient to prove(43)limε↓0I1+εI2+εI3=0.Obviously, it is necessary to obtain controls of the termsI2 and I3. For I1, we need to prove its convergence. We first consider controls of the terms I2 and I3.By the expressions ofI2 and I3, we can see that the partial derivatives of V0 and V1 are involved. Thus, we should consider estimating them before controlling I2 and I3. Next, we can obtain the expectation form of V0 and Vε by the classical result. When ε=0, we have(44)Xu=xexpr−σ022u−t+σ0Wu−Wt.Thus,(45)V0t,x,y=e−rT−tEtxyφY0,T=e−rT−tEtxyφ1T∫0TXudu=e−rT−tEtxyφ1T⋅x⋅∫0Ter−σ02/2u−t+σ0Wu−Wtdu=e−rT−tEtxyφx·H,where H=1/T∫0Texpr−σ02/2u−t+σ0Wu−Wtdu is a random variable for fixed t∈0,T. Similarly, we have(46)Vεt,x,y=e−rT−tesssupσ∈AεEtxyφY0,Tε=e−rT−tEtxyφx⋅G,where G=1/T∫0Texpr−σ^u2/2u−t−σ^uWu−Wtdu is a random variable for a fixed t∈0,T.By equations (45) and (46), we note that it is necessary to impose polynomial growth conditions on φ to control ∂xx2V0 and ∂xx2Vε. Then, we estimate ∂xx2V0t,x,y and ∂xx2Vεt,x,y in the following Lemma.Lemma 3.
Suppose that the second derivative of the payoff function satisfies the polynomial growth condition, that is, there are constantsK2 and m such that φ″x≤K21+xm. Then, we have constant K3 such that(47)∂xx2V0t,x,y≤K31+xm,where K3 depends on T,t, EtxyH2, EtxyHm+2, and K2. Moreover, there is a constant K4 such that(48)∂xx2Vεt,x,y≤K41+xm,where K4 depends on T,t, EtxyG2, EtxyGm+2, and K2.Proof.
As the assumption ofφ in the lemma, we have(49)∂xx2V0t,x,y=e−rT−tEtxyφ″xHH2≤e−rT−tEtxyK21+xHmH2≤K31+xm.
Here,K3 depends on T,t, EtxyH2, EtxyHm+2, and K2.
Indeed, for a constantm>0, we have(50)EHm=1TmE∫−tT−texpr−σ022u+σ0Wudum≤1TmE∫−tT−ter−σ02/2T−t+σ0Wudum≤1Tmemr−σ02/2T−tEsups∈−t,T−teσ0Wsm<+∞.
We obtain the controltialxx2Vε similarly. Then, there is a constant K4 that depends on T,t, EtxyG2, EtxyGm+2, and K2 such that(51)∂xx2Vεt,x,y≤K41+xm.
Now, by the following proposition, we can obtain controls of the termsI2 and I3.Proposition 1.
Assume thatφ∈Cp2R+ and the Lipschitz continuity condition holds. Then, there exist constants C1 and p1 such that I2andI3 in equations (39) and (40) satisfy(52)I2+I3≤C11+xp1.Proof.
By Lemma3, we have the following inequality from (23) and (48):(53)∂tVε+rx∂xVε−Vε+x∂yVε≤12σ0+ε2x2∂xx2Vε≤K42σ0+ε2x2+xm+2.
By the expression ofV1, it is true that(54)∂tV1+rx∂xV1−V1+x∂yV1≤K4σ0x2+xm+2.
By (25) and (47), we get the controls for x2∂xx2V1:(55)x2∂xx2V1=∂tV1+rx∂xV1−V1+x∂yV1+g¯tσ0x2∂xx2V0·2σ02≤∂tV1+rx∂xV1−V1+x∂yV1+σ0x2∂xx2V0·2σ02≤M1x2+xm+2,where M1 depends on K3, K4, and σ0. We can obtain the existence and uniqueness of X^tε 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 N1q for a fixed q>0 such that(56)Etxysups∈t,TX^sεq≤N1qeN1qT−t1+xq.
By (40), (55), and (56), we have the following inequality:(57)I3=Etxy∫tT12γ^s2X^sε2∂xx2V1s,X^sε,Y^sεds≤M12Etxy∫tTX^sε2+X^sεm+2ds≤M1′1+xm+2.
Here,M1′ depends on T,t, N12, N1m+2, and M1.
By (39), (47), (55), and (56), we obtain the control of term I2.(58)I2=Etxy∫tT12γ^s2X^sε2∂xx2V0+σ0γ^sX^sε2∂xx2V1ds≤K32Etxy∫tTX^sε2+X^sεm+2ds+M1Etxy∫tTX^sε2+X^sεm+2ds≤M21+xp1,where M2 depends on T, t, M1, K3, N12, and N1m+2, p1≥m+2.Combining (57) and (58), we have a constant C1 such that(59)I2+I3≤C11+xp1.


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

By Proposition1, we obtain controls of the terms I2 and I3. Next, for the fixed point t,x,y∈0,T×R+×R+, it suffices to prove that(60)limε↓0I1=0.Note that ifφ∈Cp2R+ (i.e., its derivatives up to order 2 have polynomial growth), we can obtain the following inequality by (38), (47), (56), and Hölder inequality:(61)I1≤Etxy∫tTσ0X^sε2∂xx2V02ds1/2Etxy∫tTγ^s−γ¯s2ds1/2≤M31+xp21/2Etxy∫tTγ^s−γ¯sds1/2.Here,M3 depends on K3,T,t,σ0, and p2≥4+2m. Moreover, M3 is independent of ε.Lethεt,x,y=γ^t,x,y;ε−γ¯t,x,y. By (34) and (35), we have(62)hεt,x,y=1,∂xx2Vε∂xx2V0<0,0,∂xx2Vε∂xx2V0≥0.Thus, to proveI1⟶0 as ε⟶0, it suffices to prove that(63)limε↓0Etxy∫tThεs,X^sε,Y^sεds=0.By the expression ofhε, we should analyze the derivatives of V0 and Vε. Here, we find that the continuity of φ″ is necessary.Lemma 4.
Assume thatφ″ is continuous. Then, ∂xx2V0 and ∂xx2Vε are continuous with respect to x,y.Proof.
By (45), we have V0t,x,y=e−rT−tEtxyφxH and ∂xx2V0t,x,y=e−rT−tEtxyφ″xHH2. If φ″ is continuous, then for all x0∈R+, δ>0, there is a constant ξ=ξδ,x0 such that(64)φ″xH−φ″x0H≤δ,for all xH∈x0H−ξ,x0H+ξ. So, for all x0,y0∈R+×R+, xH∈x0H−ξ,x0H+ξ, and y∈y0−ξ,y0+ξ, we have(65)∂xx2V0t,x,y−∂xx2V0t,x0,y0=e−rT−tEtxyφ″xHH2−φ″x0HH2≤e−rT−tEtxyH2φ″xH−φ″x0H≤e−rT−tδEtxyH2.
Thus, we obtain(66)limx,y⟶x0,y0∂xx2V0t,x,y=∂xx2V0t,x0,y0.
Similarly, we can obtain the continuity of∂xx2Vε.Remark 6.
Rationally,Vε and its derivatives converge to V0 and its corresponding derivatives as ε approaches 0 by Lemma 2.Remark 7.
To simplify the complexity brought by the variableY, which is called path dependence, and to study the behavior of hε, we define(67)Dtyλ=x∈R+∂xx2Vε0∂xx2V0≤0,∃ε0>λ.
LetDty0=limλ↓0Dtyλ. Then, we can obtain the following equation when ∂xx2Vε is continuous:(68)Dty0=x∈R+∂xx2V0t,x,y=0.Remark 8.
To controlhε, we divide Dtyλ into two parts. Let αρ=−ρ,ρ. We will discuss the characteristics of Dtyλ∩αρ and Dtyλ∩αρc.Lemma 5.
Assume thatφ″ is continuous and the fourth derivative of φ exists. Then, we have(69)PtxyDsys0∩αρ=0,fors∈t,T.Here,Ptxy⋅ refers to the conditional probability with respect to Xtε=x and Ytε=y.Proof.
By (33) and (24), we can obtain the equation(70)2∂tV0+rx∂xV0−V0+12σ02x2∂xx2V0=0,V0T=φxH.
LetQ=∂xx2V0. Then, by equation (70) and the existence of the fourth derivative of φ, we have(71)2∂tQ+r+σ02Q+r+2σ02x∂xQ+12σ02x2∂xx2Q=0,QT=φ″xHH2.
Letx=logk. Then, we have(72)2∂tQ+r+σ02Q+r+2σ02∂kQ+12σ02∂k2Q=0,QT=φ″logkHH2.
Note that the coefficients in equation (72) are constants and Q is bounded on Dsys0∩αρ by the continuity of φ″ and Lemma 4. Moreover, by equation (72), we find that y 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 Q is only countable for all s,ys∈t,T×R. Thus, ∂xx2V0 has only countable zero points. Hence, we have PtxyDsys0∩αρ=0 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 thatφ∈Cp2R+, φ″ is continuous, and the fourth derivative of φ exists. Then, we obtain equation (63).Proof.
LetD¯tyλ be the closure of Dtyλ, D¯ty0=limλ↓0D¯tyλ, and 0≤λ<ε<1.
By the definition ofDtyλ, we have(73)Etxy∫tThεs,X^sε,Y^sεds≤Etxy∫tTID¯sY^sελX^sεds=Etxy∫tTID¯sY^sελ∩αρX^sεds+Etxy∫tTID¯sY^sελ∩αρcX^sεds=Φ1+Φ2.
Now, we consider the second part of (73) first. By (56) and Chebyshev’s inequality,(74)Φ2≤Etxy∫tTIαρcX^sεds≤∫tTPtxysups∈t,TX^sε≥ρds≤T−tρEtxysups∈t,TX^sε≤T−tN11ρeN11T−t1+x.
Thus, we have(75)limρ⟶∞Φ2=0.
For the first part, we note that(76)Φ1=∫tTPtxyX^sε∈D¯sY^sελ∩αρds.
LetθΩ=supλ∈0,1PtxyΩ; then,(77)PtxyX^sε∈D¯sY^sελ∩αρ≤θD¯sY^sελ∩αρ.
Note thatλ<ε. Then, D¯syλ is a sequence of decreasing closed sets as ε↓0. Obviously, X^sε converges weakly to Xs. Thus, Xs is weakly compact. By Lemma 8 in [18], we can see that(78)limε↓0θD¯sY^sελ∩αρ=θD¯sY^s00∩αρ.
By Lemma4, there is D¯sy0=Dsy0. Then, by Lemma 5, we have(79)PtxyX^sε∈D¯sY^sε0∩αρ=0,forε≥0.
Next, by the definition ofθΩ, we have(80)θD¯sY^s00∩αρ=0.
Thus,(81)limε↓0θD¯sY^sελ∩αρ=0.
Then, we obtain(82)limε↓0Φ1=limε↓0PtxyX^sε∈D¯sY^sελ∩αρ=0.
By equations (75) and (82), for any δ>0, there is ρ0=ρ0t,x,y,δ>0 such that(83)Φ2<δ2,for  allρ>ρ0.
Next, for a givenρ0 and δ, there is ε0=ε0t,x,y,δ,ρ0t,x,y,δ such that(84)Φ1<δ2,for allε<ε0.
Therefore, for anyδ>0, there is ε0=ε0t,x,y,δ such that(85)Φ1+Φ2<δ,for allε<ε0,i.e.,(86)limε↓0Etxy∫tThεs,X^sε,Y^sεds=0.


## 4.5. Proof of the Main Result

Now, from the analysis above, we can give the brief proof of Theorem1.By inequality (61) and Proposition 2, we have(87)limε↓0I1=0.By inequality (41), we have(88)Vε−V0+εV1ε≤I1+εI2+εI3=0.By Proposition1 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 intervalσ0,σ0+ε. As ε 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.



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