Approximation for Option Prices under Uncertain

Volatility

Jean-Pierre Fouque∗ Bin Ren†

February 1, 2013

Abstract

In this paper, we study the asymptotic behavior of the worst case scenario optionprices as the volatility interval in an uncertain volatility model (UVM) degenerates to asingle point, and then provide an approximation procedure for the worst case scenarioprices in a UVM with small volatility interval. Numerical experiments show that thisapproximation procedure performs well even as the size of the volatility band is not sosmall.

1 Introduction to UVM

Since their introduction in [3] and [13], uncertain volatility models have received intensiveattention in mathematical finance. In a simplest UVM, it is assumed that the market hastwo assets: one riskless asset and one risky asset. Their price processes are denoted as (Bt)and (Xt). It is also assumed that the price process of the riskless asset (Bt) has dynamics

dBt = rBtdt,

where r is a constant.The price process of the risky asset (Xt) solves the following stochastic differential equa-

tion (SDE)dXt = rXtdt + αtXtdWt,

where (Wt) is a Brownian motion on the probability space (Ω,F , F, Q) and the volatilityprocess (αt) ∈ A which is a family of progressively measurable and [σ, σ]-valued processes.For each stochastic volatility process α ∈ A, one has a general stochastic volatility modelfor (Xt). In a UVM, we only know that the true model lies in the above family of generalstochastic volatility models. Note that we do not have a prior belief (a probability distribu-tion) over the family of general stochastic volatility models. Therefore, we use “ambiguity”to distinguish this type of uncertainty. Intuitively, we can consider the size of the volatilityinterval as the degree of model ambiguity.

Due to the presence of model ambiguity (or absence of a prior distribution), the worstcase scenario analysis is applied in derivatives pricing under a UVM. Suppose that χ is aEuropean derivative written on the risky asset with maturity T and payoff ϕ(XT ). It isknown that its worst case scenario price at time t < T is given by

V (t, Xt) := supα∈A

Et [ϕ(XT )] ,

∗Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110,fouque@pstat.ucsb.edu. Work supported by NSF grant DMS-1107468.

†Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110,ren@pstat.ucsb.edu.

1

where Et[·] is the conditional expectation given Ft with respect to the measure Q, see [3][13]. It is proved in [8] that the seller of the derivative χ can super-replicate χ with initialwealth supα∈A E [ϕ(XT )] whatever the true volatility process is. The importance of theworst case scenario price is not only because of its super-replication property, but also dueto its relationship with coherent risk measures [2] [5]. Following the arguments in stochasticcontrol theory, V (t, x) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation (inmath-finance it is called Black-Scholes-Barenblatt (BSB) equation)

∂tV + r(x∂xV − V ) + supα∈[σ,σ]

[12x2α2∂2

xV

]= 0, (.)

V (T ) = ϕ.

If ϕ is convex (like the payoff of a European call), it is known that the worst case scenarioprice of χ is equal to its Black-Scholes price with constant volatility σ. For concave ϕ, wehave a similar result, see [17] for details. However, the fully nonlinear PDE (.) does nothave a closed form solution, like Black-Scholes formula, for a general terminal payoff functionϕ. In order to evaluate the worst case scenario price, we have to resort to numerical methods[3] and [15]. Similarly, the best case scenario price of χ can be defined as infα∈A E [χ] .Moreover, it is shown in [17] that given any price between infα∈A E [χ] and supα∈A E [χ], themarket is arbitrage free.

It is clear that the worst case scenario price is larger than any Black-Scholes price witha constant volatility σ ∈ [σ, σ]. In this paper, we shall consider how the worst case scenarioprice behaves as the volatility interval [σ, σ] degenerates to a single point σ ∈ [σ, σ]. In-tuitively, if the model ambiguity is reduced, then the extra price (which is included in theworst case scenario price) paid for that should be less. As it is shown in the sequel, the worstcase scenario price of χ will converges to its Black-Scholes price with constant volatility σ.Indeed, Cont suggested in [5] a measure of impact of model uncertainty on the worst casescenario price for any derivative χ:

µ(χ) = supα∈A

E [χ]− infα∈A

E [χ] ,

which vanishes as the volatility interval shrinks to a single point.In addition, our study partially answers the sensitivity problem of the worst case scenario

to the degree of model ambiguity which is proposed in [5]. In fact, we obtain the rate ofconvergence of the worst case scenario prices as volatility interval shrinks to a single point.Therefore, this result gives us an approximation of the worst case scenario price when theinterval is sufficiently small. Along the paper we denote the Black-Scholes price as V0 andthe rate of convergence as V1, which are the solutions to linear partial differential equations.Consequently, the first order approximation V0 + (σ − σ)V1 of the worst case scenario priceis achieved. Of course, the approximated price V0 + (σ − σ)V1 does not have the propertyof super-replication. What did we gain in the approximation procedure? First, the problemof solving a fully nonlinear BSB equation is reduced to solving two Black-Scholes like PDEs.The numerical examples also show that the approximation procedure is stable even withreasonably large volatility interval. Second, we are able to see how a linear expectation turninto a sublinear expectation.

In order to study the asymptotic behavior of worst case scenario prices, we re-parameterizeour uncertain volatility model and assume that the risky asset price process (Xα,ε

t ) has adynamic

dXα,εt = rXα,ε

t dt + αtXα,εt dWt, (.)

where α := (αt) ∈ Aε, the family of progressively measurable, [σ, σ + ε]-valued processes and(Wt) is a Brownian motion on the probability space (Ω,F , F, Q). If ε = 0 and no danger ofconfusion, we shall use (Xt) to denoted (Xα,0) which is indeed a geometric Brownian motionwith constant volatility σ,

dXt = rXtdt + σXtdWt.

2

We define the worst case scenario price as a value function of a stochastic control problem

Jε(t, x, α) := Etx[ϕ(Xα,εT )],

V ε(t, x) = supα∈Aε

[Jε(t, x, α)] ,

where the conditional expectation Etx[·] is taken with respect to the law of Xα,εT given

Xα,εt = x. The worst case scenario option price when ε = 0 is a Black-Scholes price. We also

represent it as a value function of a trivial stochastic control problem

V0(t, x) = J(t, x, σ) := Etx [ϕ(XT )] ,

where the subscripts in Etx[·] also mean that Xt = x.This paper is structured as follows. In Section 2, we briefly recall some well-known results

from stochastic control theory. The continuity of the worst case scenario price with respect tothe parameter ε is discussed in Section 3. In Section 4, we heuristically derive the equationfor the convergence rate of V ε as ε vanishes. Section 5 is devoted to studying the worstcase scenario asset price process for the clam χ. In Section 6, the analysis of error term isdeveloped. In Sections 7, 8 we perform a numerical experiment and conclude the paper.

2 Preliminary results from stochastic control

Because we are only interested in the case where ε is close to 0, it is legitimate to assumethat all ε ≤ 1. In particular, |αt| ≤ σ + 1. In order to introduce the results that are neededin this paper, we borrow the notations from [12].

Given a SDE with random coefficients:

dXt = bt(Xt)dt + σt(Xt)dWt, (.)

if there exists a constant K > 0 such that

|bt(x)− bt(y)| ≤ K |x− y| , |σt(x)− σt(y)| ≤ K |x− y|

for all t ∈ [0, T ], ω ∈ Ω, and x, y ∈ R, then we say that (L ) condition is satisfied. If for allt ∈ [0, T ], ω ∈ Ω, and x ∈ R there exist some K > 0 such that

|bt(x)| ≤ K |x|+ ht, |σt(x)|2 ≤ K |x|2 + r2t

for some stochastic processes (ht) and (rt), we say that (R) condition is satisfied. It can beseen that the condition (L ) implies the condition (R), by noticing that

|bt(x)| ≤ |bt(x)− bt(0)|+ |bt(0)||σt(x)| ≤ |σt(x)− σt(0)|+ |σt(0)| .

Note that

|rx− ry| ≤ r |x− y| , |αtx− αty| ≤ |αt| · |x− y| ≤ (σ + 1) |x− y| .

Therefore, it is clear that the SDE of (Xα,εt ) satisfies the condition (L ). According to

Corollary 12 in the section 2.5 [12], we have the following universal estimates of the momentsof (Xα,ε

t )

E

[sup

s∈[0,t]

|Xα,εs |q

]≤ NeNt (1 + |x0|)q

, (.)

for all α ∈ Aε, t ∈ [0, T ] and q > 0, where N = N(q, σ, r) (we assumed that ε < 1) andXα,ε

0 = x0.

3

For another ε′ ∈ (0, 1], it is assumed that ε′ < ε without losing generality. We considerthe process (Xα∧(σ+ε′),ε

t ) which satisfies the SDE (.) with volatility process (αt ∧ (σ + ε′))for some (αt) ∈ Aε. It is also assumed that Xα,ε′

0 = x0. By Theorem 9 in the section 2.9[12] and the estimates of the moments, we can conclude that

E

[sup

s∈[0,t]

∣∣∣Xα,εs −Xα∧(σ+ε′),ε

s

∣∣∣2q]

≤ N ′tq−1eN ′tE[∫ t

0

|Xα,ε|2q · |αs − αs ∧ (σ + ε′)|2q ds

]≤ N ′tq−1eN ′tN ′′eN ′′t(1 + |x0|2q)t(ε− ε′)

= N ′tqeN ′tN ′′eN ′′t(1 + |x0|2q)(ε− ε′)= NtqeNt(1 + |x0|2q)(ε− ε′), (.)

for any q ≥ 1, where N ′ = N ′(q, σ, r), N ′′ = N ′′(q, σ, r), and N = max N ′N ′′, N ′ + N ′′.

3 Continuity of V ε in ε

In this section, we mainly analyze the continuity of V ε with respect to ε and the main resultis:

Theorem 3.1. Given ϕ which is Lipschitz continuous with Lipschitz constant K1 and forany ε0 ∈ [0, 1)

limε→ε0

V ε(t, x) = V ε0(t, x)

for all (t, x) ∈ [0, T ]× R.

Proof. First, recall that Aε is the family of progressively measurable and [σ, σ + ε]-valuedprocesses. If 0 ≤ ε0 < ε < 1, we have that

V ε0(t, x) = supα∈Aε0

Etx [ϕ(Xα,ε0T )] = sup

α∈Aε

Etx

[ϕ(Xα∧(σ+ε0),ε

T )].

Therefore, by the estimate (.)

|V ε(t, x)− V ε0(t, x)| ≤ supα∈Aε

∣∣∣Etx[ϕ(Xα,εT )]− Etx[ϕ(Xα∧(σ+ε0),ε

T )]∣∣∣

≤ K1 supα∈Aε

(Etx

∣∣∣Xα,εT −X

α∧(σ+ε0),εT

∣∣∣2)1/2

≤ K1

(N(T − t)eN(T−t)(1 + x2) |ε− ε0|

)1/2

.

It can be seen that for any fixed point (t, x) ∈ [0, T ] × R, |V ε(t, x)− V ε0(t, x)| → 0 as εapproaches ε0 from above.

It can be proved similarly that limε↑ε0 |V ε(t, x)− V ε0(t, x)| = 0 for ε0 > 0.

In particular, when ε0 = 0 we have one-sided convergence of V ε(t, x)ε>0 which is statedin the following corollary.

Corollary 3.1. When the conditions in Theorem 3.1 are satisfied,

limε↓0

V ε(t, x) = V0(t, x),

for all (t, x) ∈ [0, T ]× R.

4

As the volatility interval [σ, σ + ε] becomes smaller, the above corollary tells us that theBlack-Scholes price V0 of χ with constant volatility σ is the main contributor to its worstcase scenario price relative to the extra price paid for the model ambiguity, i.e. V0 would bethe leading term in the approximation of V ε. We now investigate the first order correctionterm in this approximation, which will help us understand the sensitivity of the worst casescenario price to the degree of model ambiguity.

4 Heuristic derivation of the first order correction term

To simplify the notations, we assume r = 0 in the sequel, but all the results still hold whenr 6= 0. Recall that V ε solves the following BSB equation

∂tVε + sup

α∈[σ,σ+ε]

12α2x2∂2

xV ε = 0,

V ε(T ) = ϕ.

To study the asymptotic behavior of V ε, we also re-parameterize the BSB equation asfollows

∂tVε + sup

g∈[0,1]

12(σ + εg)2x2∂2

xV ε = 0, (.)

V ε(T ) = ϕ.

Note that V0 is the solution of the following Black-Scholes equation

∂tV0 +12σ2x2∂2

xV0 = 0, (.)

V0(T ) = ϕ.

In this heuristic derivation, we assume the differentiability of V ε with respect to ε andthe interchangeability of partial differential operators ∂ε, ∂t and ∂2

x. We differentiate theequation (.) with respect to ε

∂ε

∂tV

ε +12σ2x2∂2

xV ε + ε2 supg∈[0,1]

12g2x2∂2

xV ε + ε supg∈[0,1]

gσx2∂2xV ε

= 0.

Let V1 = ∂εVε|ε=0, and according to Corollary 3.1 we also note that V ε|ε=0 = V0. Therefore,

V1 is the unique solution of the following linear PDE with source and zero terminal condition

∂tV1 +12σ2x2∂2

xV1 + supg∈[0,1]

gσx2∂2xV0 = 0, (.)

V1(T ) = 0.

Note that the source term in the above equation is known from V0 and is in fact equal toσx2∂2

xV0I∂2xV0>0. It is nonlinear in V0 and can be seen as the first manifestation of the

nonlinearity of the full problem (.).In the above heuristic argument, we obtained the equation characterizing V1. It will be

verified that V1 which solves (.) is the first order derivative of V ε with respect to ε atε = 0. That is, we shall prove that the error term V ε − (V0 + εV1) is of order o(ε). Moreprecisely, the following theorem is achieved with more technical conditions imposed on ϕ.

Theorem 4.1. Assume that the payoff function ϕ ∈ C4p(R+) (p for polynomial growth), ϕ

is Lipschitz and its derivatives up to order 4 have polynomial growth. Moreover, we alsoassume that the second derivative of ϕ has a finite number of zero points. Then, pointwise

limε↓0

V ε − (V0 + εV1)ε

= 0.

5

5 Asset price process in the worst case scenario and es-timates of its moments

It is known from [18] and [6] that if ϕ is locally Lipschitz continuous, ϕ and ϕ′ have polynomialgrowth, then the viscosity solution V ε of (.) belongs to C1,2

p ([0, T ) × R) and there existsκ ∈ (0, 1] such that ∂2

xV ε is Holder-κ continuous.

5.1 Existence and uniqueness of (X∗,εt )

The equation (.) would produce the worst case scenario volatility process α∗,ε = σ + εg∗,ε

for the claim χ, where

g∗,ε(t, x) =

1 ∂2

xV ε(t, x) ≥ 0,

0 ∂2xV ε(t, x) < 0.

From the equation (.) of V1, we would have another choice of the volatility process forthe claim χ: α = σ + εg, where

g(t, x) =

1 ∂2

xV0(t, x) ≥ 0,

0 ∂2xV0(t, x) < 0.

Therefore, the asset price process in the worst case scenario for the claim χ is a stochasticprocess which satisfies the SDE (.) with α = α∗,ε and r = 0, i.e.

dX∗,εt = α∗,εt X∗,ε

t dWt. (.)

Define a transformationY ∗,ε

t := log X∗,εt ,

which is well-defined for any t < τ δ where

τ δ = inft > 0 | X∗,ε

t = δ or X∗,εt = 1/δ

= inf

t > 0 | Y ∗,ε

t = log δ or Y ∗,εt = − log δ

,

for any δ > 0. By Ito’s formula, the process (Y ∗,εt ) satisfies the following SDE

dY ∗,εt = −1

2(α∗,εt )2dt + α∗,εt dWt. (.)

It is noted that the coefficients in (.) are bounded and progressively measurable. More-over, the diffusion coefficient is bounded away from zero: α∗,εt ≥ σ > 0. Therefore, thanksto Theorem 1 in section 2.6 in [12] or the result 7.3.3 in [16], the SDE (.) has a uniqueweak solution. That is, we have a unique solution to the SDE (.) until τ δ for any δ > 0.In order to prove that the SDE (.) has a unique solution for all t ∈ (0,∞), it suffices toshow that for any T > 0

limδ↓0

Q(τ δ < T

)= 0. (.)

Indeed, by Chebyshev inequality, it holds that

limδ↓0

Q(τ δ < T

)≤ lim

δ↓0Q

(sup

t∈[0,T ]

∣∣Y ∗,εt

∣∣ > | log δ|

)

≤ limδ↓0

E[supt∈[0,T ]

∣∣Y ∗,εt

∣∣]| log δ|

= 0

which implies (.).

6

5.2 Estimates of moments and exit probability of (X∗,εt )

As a special case of (.), we have that

Etx[ sups∈[t,T ]

|X∗,εs |q] ≤ NeN(T−t)(1 + |x|q), (.)

for all q > 0, N = N(q, σ) and X∗,εt = x.

Given ρ > 0, define a stopping time

τρ := inf s ∈ [t, T ], such that|X∗,εs | ≥ ρ .

Conventionally, inf ∅ = ∞.By using the estimates of moments and Chebyshev inequality,

Qtx(τρ < T ) ≤ Qtx( sups∈[t,T ]

|X∗,εs | ≥ ρ) ≤ NeN(T−t)(1 + |x|)

ρ, (.)

where N = N(σ). This control on the exit probability enables us to use localization argu-ments in the sequel.

Remark 5.1. It is important to notice that the estimates in this section are independent ofε, due to our assumption ε ≤ 1.

6 Analysis of the error term

Define the error term of the suggested approximation

Zε = V ε − (V0 + εV1) .

LetL(σ) := ∂t +

12σ2x2∂2

x.

Next, we apply the operator L(α∗,ε) to the error term:

L(α∗,ε)Zε = −L(α∗,ε)(V0 + εV1)= − (L(α∗,ε)− L(σ))V0 − ε (L(α∗,ε)− L(σ))V1 − εL(σ)V1

=12

(σ2 − (σ + εg∗,ε)2

)x2∂2

xV0 + ε12

(σ2 − (σ + εg∗,ε)2

)x2∂2

xV1

+εgσx2∂2xV0

= −ε (g∗,ε − g) σx2∂2xV0 − ε2

(12

(g∗,ε)2 x2∂2xV0 + g∗,εσx2∂2

xV1

)−ε3

(12

(g∗,ε)2 x2∂2xV1

). (.)

Note that the terminal condition of Zε is

Zε(T ) = V ε(T )− V0(T )− εV1(T ) = 0.

From now on, we impose more regularity conditions on the terminal data ϕ, i.e. polyno-mial growth condition on the first four derivatives of ϕ(x):

ϕ′(x) ≤ K1,

ϕ′′(x) ≤ K2(1 + |x|m),ϕ′′′(x) ≤ K3(1 + |x|n),ϕ(4)(x) ≤ K4(1 + |x|l),

(.)

where Ki for i ∈ 1, 2, 3, 4, and m, n, and l are positive constants.

7

6.1 Feynman-Kac representation of the error term Zε

Given the equation (.) for Zε together with the existence and uniqueness of (X∗,εt ), we

have the following probabilistic representation of Zε:

Zε = −εEtx

[∫ T

t

(g∗,εs − gs) σ (X∗,ε

s )2 ∂2xV0 (s,X∗,ε

s )

ds

]

−ε2Etx

[∫ T

t

12

(g∗,εs )2 (X∗,εs )2 ∂2

xV0 (s,X∗,εs ) + g∗,εs σ (X∗,ε

s )2 ∂2xV1 (s,X∗,ε

s )

ds

]

−ε3Etx

[∫ T

t

12

(g∗,εs )2 (X∗,εs )2 ∂2

xV1 (s,X∗,εs )

ds

]= −εI1 − ε2I2 − ε3I3, (.)

where

I1 = Etx

[∫ T

t

(g∗,εs − gs) σ (X∗,ε

s )2 ∂2xV0 (s,X∗,ε

s )

ds

],

I2 = Etx

[∫ T

t

12

(g∗,εs )2 (X∗,εs )2 ∂2

xV0 (s,X∗,εs ) + g∗,εs σ (X∗,ε

s )2 ∂2xV1 (s,X∗,ε

s )

ds

],

I3 = Etx

[∫ T

t

12

(g∗,εs )2 (X∗,εs )2 ∂2

xV1 (s,X∗,εs )

ds

].

We shall derive the bounds of I2 and I3 in the next section. The term I1 will be dealtwith in Section 6.3.

6.2 Controls of the terms I2 and I3

As a special case of (.), the process (Xt) which solves

dXt = σXtdWt (.)

has the following estimates of moments

Etx

[sup

s∈[t,T ]

|Xs|q]≤ NeN(T−t)(1 + |x|q), (.)

for all q > 0, where N = N(q, σ).Because V0 is the solution of the Black-Scholes equation(.), the following lemma holds.

Lemma 6.1. Given ϕ(x) which satisfies the condition (.), there exist constants M1, M2,and M3 which only depend on σ, T , m, n, and l, such that

∣∣∂2xV0

∣∣ ≤ M1 (1 + |x|m),∣∣∂3

xV0

∣∣ ≤M2 (1 + |x|n), and

∣∣∂t∂2xV0

∣∣ ≤ M3 (1 + |x|p1), where p1 = max m,n + 1, l + 2.

Proof. We shall use the probabilistic representation of V0

V0(t, x) = Etx[ϕ(XT )], (.)

where the process (Xt) is the geometric Brownian motion (.). Then, starting at x at time

t, we have XT = xe−σ2(T−t)

2 +σ(WT−Wt). Therefore, using the notation τ = T − t, we obtain∣∣∂2xV0

∣∣ =∣∣∣∣∫

Re−σ2τ+2σ

√τyϕ′′(xe−

σ2τ2 +σ

√τy)

1√2π

e−y2

2 dy

∣∣∣∣≤ K2

∫R

e−σ2τ+2σ√

τy(1 + |x|me−

mσ2τ2 +mσ

√τy) 1√

2πe−

y2

2 dy

≤ M1 (1 + |x|m) ,

8

with M1 = M1(K2,m, T, σ).Similarly, we obtain controls of ∂3

xV0 and ∂4xV0:∣∣∂3

xV0

∣∣ =∣∣∣∣∫

Re−

32 σ2τ+3σ

√τyϕ′′′(xe−

σ2τ2 +σ

√τy)

1√2π

e−y2

2 dy

∣∣∣∣≤ K3

∫R

e−32 σ2τ+3σ

√τy(1 + |x|ne−

nσ2τ2 +nσ

√τy) 1√

2πe−

y2

2 dy

≤ M2 (1 + |x|n) ,

∣∣∂4xV0

∣∣ =∣∣∣∣∫

Re−2σ2τ+4σ

√τyϕ(4)(xe−

σ2τ2 +σ

√τy)

1√2π

e−y2

2 dy

∣∣∣∣≤ K4

∫R

e−2σ2τ+4σ√

τy(1 + |x|le− lσ2τ

2 +lσ√

τy) 1√

2πe−

y2

2 dy

≤ M4

(1 + |x|l

),

with M2 = M2(K3, n, T, σ) and M4 = M4(K4, l, T, σ).From the PDE of V0, it can be seen that ∂t∂

2xV0 = −∂2

x

(12σ2x2∂2

xV0

)= − 1

2σ2(2∂2

xV0 + 4x∂3xV0 + x2∂4

xV0

).

Therefore, the control of the mixed third partial derivative ∂t∂2xV0 is obtained through the

controls on the partial derivatives only with respect to x. Consequently, there exist twoconstants M3 = M3(M1,M2,M4,m, n, l) and p1 = maxm,n + 1, l + 2 such that

∣∣∂t∂2xV0

∣∣ ≤ σ2

22M1 (1 + |x|m) + 4|x|M2 (1 + |x|n) + x2M4

(1 + |x|l

)≤ M3(1 + |x|p1).

Given V0 which solves (.), we define

f(t, x) :=12σx2∂2

xV0(t, x).

The first order partial derivatives of f are

ft(t, x) =12σx2∂t∂

2xV0(t, x),

fx(t, x) =12σ[2x∂2

xV0(t, x) + x2∂3xV0(t, x)

].

Due to the lemma 6.1, it can be observed that

|f(t, x)| ≤ 12σx2M1(1 + |x|m) ≤ K5(1 + |x|m+2) (.)

for a constant K5 = K5(M1, σ) and that

|ft(t, x)|+ |fx(t, x)| ≤ σ

2(x2M4(1 + |x|p1) + 2|x|M1(1 + |x|m)

+x2M2(1 + |x|n))

≤ K6(1 + |x|p), (.)

where K6 = K6(M1,M2,M4, p1,m, n, σ) and p = maxp1 + 2,m + 1, n + 2.In order to analyze the partial derivatives of V1, we use (Xt,x

s ) to denote the solution ofthe SDE (.) with the initial condition Xt,x

0 = x. Similarly, (Xt+h,xs ) denotes the solution

of the same SDE with the initial condition Xt+h,x0 = x. We also define

∆Xs =Xt,x+h

s −Xt,xs

h. (.)

9

From D.10 in [9], we have that

Etx[|∆Xs|2] ≤ N, (.)

for any s ∈ [0, T − t], where N = N(σ).Due to (.), we can derive that∣∣∣∣f(s,Xt,x+h

s )− f(s,Xt,xs )

h

∣∣∣∣ ≤∫ 1

0

∣∣fx(s,Xλs )∣∣ |∆Xs| dλ

≤ K6

(1 +

∣∣Xt,xs

∣∣p +∣∣Xt,x+h

s

∣∣p) |∆Xs| , (.)

where Xλs = (1− λ)Xt,x

s + λXt,x+hs for λ ∈ (0, 1). Moreover, by the mean value theorem, it

holds that ∣∣∣∣f(s + h, Xt+h,xs )− f(s,Xt+h,x

s )h

∣∣∣∣ ≤ K6

(1 +

∣∣Xt+h,xs

∣∣p) . (.)

If we denote the source term in the equation of (.) as f+(t, x), then it is noted that

f+(t, x) := − supg∈[0,1]

12gσx2∂2

xV0

= −max

12σx2∂2

xV0, 0

= −max f, 0 .

Therefore, it is also clear that ∣∣f+(t, x)∣∣ ≤ |f(t, x)| ,∣∣f+(t, x)− f+(s, y)∣∣ ≤ |f(t, x)− f(s, y)| ,

(.)

for any (t, x), (s, y) ∈ [0, T ]× R.Note that the first order difference quotient of V1 with respect to the time variable t can

be represented as follows

V1(t + h, x)− V1(t, x)h

= Etx

∫ T−t−h

0

f+(s + h, Xt+h,xs )− f+(s,Xt+h,x

s )h

ds

+Etx

∫ T−t−h

0

f+(s,Xt+h,xs )− f+(s,Xt,x

s )h

ds

− 1h

Etx

∫ T−t

T−t−h

f+(s,Xt,xs )ds. (.)

Lemma 6.2. Given ϕ(x) which satisfies the condition (.), there exists a constant M5

which depends on K5, K6, T , p, m, and σ such that |∂tV1| ≤ M5(1 + |x|p).

Proof. Due to (.) and (.), the first term in (.) can be estimated as follows∣∣∣∣∣Etx

∫ T−t−h

0

f+(s + h, Xt+h,xs )− f+(s,Xt+h,x

s )h

ds

∣∣∣∣∣≤ Etx

∫ T−t−h

0

∣∣f(s + h, Xt+h,xs )− f(s,Xt+h,x

s )∣∣

hds

≤ K6Etx

∫ T−t−h

0

(1 +

∣∣Xt+h,xs

∣∣p) ds

≤ A1(1 + |x|p),

where A1 = A1(K6, T, p, σ). For the second term in (.), by noticing that the shiftedprocess (Xt,x

s ) is identical to (Xt+h,xs ) up to time T − t− h, we therefore can conclude that∣∣∣∣∣Etx

∫ T−t−h

0

f+(s,Xt+h,xs )− f+(s,Xt,x

s )h

ds

∣∣∣∣∣ = 0.

10

Due to (.) and (.), the third term in (.) can be controlled as follows,∣∣∣∣∣ 1hEtx

∫ T−t

T−t−h

f+(s,Xt,xs )ds

∣∣∣∣∣ ≤ 1h

Etx

∫ T−t

T−t−h

∣∣f+(s,Xt,xs )∣∣ ds

≤ 1h

K5Etx

∫ T−t

T−t−h

(1 +

∣∣Xt,xs

∣∣m+2)

ds

≤ A2(1 + |x|m+2),

where A2 = A2(K5, T, m, σ). Then, after summarizing the above three estimates the lemmafollows.

Proposition 6.1. Given ϕ(x) which satisfies the condition (.), there exist two constantsp′ = maxp, m+2 and M6 which depends on K5, M5, T , m, p, and σ, such that

∣∣x2∂2xV1

∣∣ ≤M6(1 + |x|p′).

Proof. By noticing that

∂tV1 +12σ2x2∂2

xV1 = f+,

we have ∣∣x2∂2xV1

∣∣ ≤ 2σ2

(|∂tV1|+

∣∣f+∣∣) .

The above fact together with (.), (.) and the lemma 6.2 results in the proposition.

Theorem 6.1. Given ϕ(x) which satisfies the condition (.), there exists a constant D1

which depends on M1, M6, T − t, m, p′, and σ, such that I2 and I3 in (.) satisfy

|I2|+ |I3| ≤ D1(1 + |x|p).

Proof. Due to the proposition 6.1 and (.), it follows that

|I3| = Etx

[∫ T

t

12

(g∗,εs )2 (X∗,εs )2

∣∣∂2xV1 (s,X∗,ε

s )∣∣ ds

]

≤ M6

2Etx

∫ T

t

(1 + |X∗,ε

s |p′)

ds

≤ M6

2

[∫ T

t

1 + N(σ, p′)eN(σ,p′)(T−t)(1 + |x|p′)ds

]

=M6

2

[T − t + N(σ, p′)eN(σ,p′)(T−t)(1 + |x|p

′)(T − t)

].

Similarly for I2, we have

|I2| ≤ Etx

[∫ T

t

12

(X∗,εs )2

∣∣∂2xV0 (s,X∗,ε

s )∣∣ ds

]

+Etx

[∫ T

t

σ (X∗,εs )2

∣∣∂2xV1 (s,X∗,ε

s )∣∣ ds

]

≤ M1

2Etx

∫ T

t

|X∗,εs |2 (1 + |X∗,ε

s |m) ds + M7σEtx

∫ T

t

(1 + |X∗,ε

s |p′)

ds

≤ M1

2(T − t)N(σ)eN(σ)(T−t)(1 + |x|2)

+M1

2(T − t)N(m + 2, σ)eN(m+2,σ)(T−t)(1 + |x|m+2)

+M6σ(T − t)N(p′, σ)eN(p′,σ)(T−t)(1 + |x|p′).

11

By summarizing the above controls on I2 and I3, there exists D1 = D1(M1,M6,T − t, m, p′, σ) such that |I2|+ |I3| < D1(1 + |x|p′).

6.3 Convergence of the term I1

Given the controls of I2 and I3 in the theorem 6.1, to prove

Zε(t, x) ∼ o(ε)

for a fixed point (t, x) ∈ [0, T ]× R, it suffices to prove that

limε↓0

I1 = 0.

Let Kρ := [−ρ, ρ] and gd,ε := g∗,ε − g which depends on ε through g∗,ε as follows

∣∣gd,ε(t, x)∣∣ = 1 ∂2

xV ε(t, x)∂2xV0(t, x) < 0,

0 ∂2xV ε(t, x)∂2

xV0(t, x) ≥ 0.

Intuitively, as ε approaches 0, V ε and its derivatives will get closer and closer to V0 andits corresponding derivatives. In order to lay out the analysis of this intuition, we decomposethe range of X∗,ε

s for each s ∈ [t, T ] into two parts: a compact set Kρ and its complement.Therefore, I1 can be written as the expectation of a sum of (i) the compact part (when X∗,ε

s

lies in the compact set) and (ii) the tail part (when X∗,εs lies outside of the compact set):

I1 = Etx

[∫ T

t

gd,εs σ (X∗,ε

s )2 ∂2xV0 (s,X∗,ε

s ) IKρ (X∗,εs ) ds

]

+Etx

[∫ T

t

gd,εs σ (X∗,ε

s )2 ∂2xV0 (s,X∗,ε

s ) IKcρ(X∗,ε

s ) ds

]= Etx[i] + Etx[ii]. (.)

In order to achieve limε↓0 I1 = 0, we shall use localization arguments to deal with Etx[ii]and the problem is reduced onto a compact set. On the compact set, it will be proved thatVε and its partial derivatives converge to V0 and its corresponding derivatives. Then, it isfollowed by the convergence of Etx[i] to 0.

6.3.1 Control of the tail part

We apply Holder’s inequality to the second term of (.)

Etx[|ii|] ≤ σM1

[Etx

∫ T

t

(X∗,εs )4 (1 + |X∗,ε

s |m)2 ds

]1/2

[Qtx (τρ < T )]1/2

where

Etx

∫ T

t

(X∗,εs )4 (1 + |X∗,ε

s |m)2 ds

≤∫ T

t

N(4, σ)eN(4,σ)(T−t)(1 + |x|4) + N(m + 4, σ)eN(m+4,σ)(T−t)(1 + |x|m+4)

+N(2m + 4, σ)eN(2m+4,σ)(T−t)(1 + |x|2m+4)ds

≤ M7(1 + |x|2m+4), (.)

for some constant M7 = M7(T − t, m, σ).

12

From (.) and the exit probability estimate of the process (X∗,εt ) in (.), it is concluded

that

Etx[|ii|] ≤ σM1

√M7(1 + |x|2m+4)

√N(σ)eN(σ)(T−t)(1 + |x|)(T − t)

ρ

≤ D2

(1 + |x|m+3/2

)√

ρ,

for some constant D2 = D2(M1, T − t, m, σ).As ρ increases, i.e. the compact set Kρ becomes larger and larger, the process will be

less and less likely to deviate outside of the set Kρ. Then, we would expect the tail part issmall enough for sufficiently large ρ. This result is summarized in the following proposition.

Proposition 6.2. Given ϕ(x) which satisfies the condition (.), Etx[|ii|] → 0 as ρ →∞.

6.3.2 Control of the compact part

In order to prove that the compact term is negligible when ε is sufficiently small, we needthe convergence of ∂2

xV ε to ∂2xV0 which would imply that gd,ε gradually vanishes as ε tends

to 0.Recall the result regarding the regularity of the solution of the BSB equation in [18] [6].

Theorem 6.2. If ϕ is locally Lipschitz continuous and ϕ, and ϕ′ have polynomial growth,then the solution V ε of (.) belongs to C1,2

p ([0, T )× R). Moreover, ∂2xV ε(t, x) is Holder

continuous in x with an exponent κ ∈ (0, 1] for any t ∈ [0, T ).

Remark 6.1. All the constants in the polynomial controls and Holder continuity only dependon the bounds of the volatility interval [σ, σ + ε]. Since we are only interested in the caseswhere ε′s are small, we can choose these constants including Holder exponent to be universal,i.e. independent of ε; see [18] [6] or [11].

We recapitulate the theorem as follows|V ε(t, x)| ≤ B0(1 + |x|b0),|∂xV ε(t, x)| ≤ B1(1 + |x|b1),∣∣∂2

xV ε(t, x)∣∣ ≤ B2(1 + |x|b2),∣∣∂2

xV ε(t, x)− ∂2xV ε(t, y)

∣∣ ≤ B3|x− y|κ,

(.)

for any t ∈ [0, T ) where all constants B0, B1, B2, B3 and b0, b1, b2, and κ are universal, i.e.independent of ε.

Lemma 6.3. Given ϕ(x) which satisfies the condition (.),∂2

xV ε(t, ·), as a family of

functions of x indexed by ε, uniformly converges to ∂2xV0(t, ·) on the compact set Kρ as ε

tends to 0 for any fixed t ∈ [0, T ).

Proof. The proof is similar to that of Theorem 5.2.5 in [10]. Indeed, because V ε has up tosecond order partial derivative in x, by following the arguments in the proof of the Theorem5.2.5 in [10] we conclude that ∂xV ε converges to ∂xV0 as ε tends to 0 , uniformly for allx ∈ Kρ.

However, V ε(t, ·) is only twice differentiable with respect to x. In order to obtain theuniform convergence of the sequence of second partial derivatives

∂2

xV ε, we first observe

from (.) that∂2

xV ε

is a family of uniformly bounded and Holder continuous functionsof x on Kρ. It implies that

∂2

xV ε

is equi-continuous. Then, there exists a sub-sequence∂2

xV ε′

which converges uniformly on Kρ. Together with the convergence of ∂xV ε to

∂xV0, we conclude that the

∂2xV ε′

uniformly converges to ∂2

xV0 for all x ∈ Kρ.

13

Since the limit ∂2xV0 is independent of the choice of sub-sequence

∂2

xV ε′

, we claim that∂2

xV ε

converges uniformly to ∂2xV0. Indeed, if there is a sub-sequence

∂2

xV ε′′

does not

converges to ∂2xV0, then according to the Arzela-Arscoli Theorem

∂2

xV ε′′

has a uniformly

convergent sub-sequence

∂2xV ε′′′

. Again, together with the convergence of ∂xV ε to ∂xV0,

∂2xV ε′′′

has to converge to ∂2

xV0, which is a contradiction with the assumption. Therefore,the lemma follows.

LetSε

t =

x∣∣∣∂2

xV ε′(t, x)∂2xV0(t, x) ≤ 0, ∃ε′ ≤ ε

∩Kρ.

Note that Sεt ε as a family of sets indexed by ε is non-increasing as ε decreases to 0.

DefineS0

t := limε↓0

Sεt .

Lemma 6.4. Given ϕ(x) which satisfies the condition (.), for any fixed t ∈ [0, T )

S0t =

x ∈ Kρ

∣∣∣ ∂2xV0(t, x) = 0

.

Proof. Notice that if x ∈ Kρ such that ∂2xV0(t, x) = 0, then x ∈ Sε

t for all ε > 0. It impliesthat

S0t ⊇

x ∈ Kρ

∣∣∣ ∂2xV0(t, x) = 0

.

On the other hand, if ∂2xV0(t, x) > 0 for any x ∈ Kρ, then due to the uniform conver-

gence of∂2

xV ε(t, ·)

there exists ε0 > 0 such that ∂2xV ε(t, x) > 0 for all ε < ε0. Hence,

∂2xV ε(t, x)∂2

xV0(t, x) > 0 for all ε < ε0, i.e., x /∈ Sεt ∀ε < ε0. It is followed by x /∈ S0

t .Similarly, we can prove that any x ∈ Kρ such that ∂2

xV0(t, x) < 0 does not lie in S0t either.

Therefore, we can claim that

S0t ⊆

x ∈ Kρ

∣∣∣ ∂2xV0(t, x) = 0

.

For any fixed t ∈ [0, T ), we take Sεt := the closure of Sε

t for each ε. For this sequence ofclosed, bounded and non-increasing sets, we define its limit

S0t := lim

ε↓0Sε

t .

Due to the continuity of ∂2xV0(t, x) in x, it is true that S0

t = S0t for any fixed t ∈ [0, T ). The

following lemma tells us that the same relationship holds between Sεt and Sε

t .

Lemma 6.5. Given ϕ(x) which satisfies the condition (.) and any fixed t ∈ [0, T ), it istrue that

Sεt = Sε

t .

Proof. To prove the lemma, it suffices to show that ∀x0 ∈ Sεt , either x0 ∈ Sε

t or x0 ∈ S0t .

Indeed, if it is true, then together with the fact that Sεt ⊇ S0

t (by the monotonicity of Sεt )

we can see that Sεt = Sε

t , i.e. Sεt is a closed set.

For any x0 ∈ Sεt , according to the definition of closure, there exists a sequence xn ⊂ Sε

t

such that limn→∞ xn = x0. For each xn, there is a εn ≤ ε such that

∂2xV εn(t, xn)∂2

xV0(t, xn) ≤ 0.

Since 0 < εn ≤ ε for all n > 0, there exist a ε0 ∈ [0, ε] and a sub-sequence εn′ such thatlimn→∞ εn′ = ε0.

14

We take the corresponding sub-sequence xn′ which converges to x0 by assumption.Then,

limn′→∞

∂2xV εn′ (t, xn′)∂2

xV0(t, xn′) = ∂2xV ε0(t, x0)∂2

xV0(t, x0) ≤ 0. (.)

Indeed, ∣∣∂2xV εn′ (t, xn′)∂2

xV0(t, xn′)− ∂2xV ε0(t, x0)∂2

xV0(t, x0)∣∣

≤∣∣∂2

xV εn′ (t, xn′)∂2xV0(t, xn′)− ∂2

xV εn′ (t, x0)∂2xV0(t, xn′)

∣∣+∣∣∂2

xV εn′ (t, x0)∂2xV0(t, xn′)− ∂2

xV εn′ (t, x0)∂2xV0(t, x0)

∣∣+∣∣∂2

xV εn′ (t, x0)∂2xV0(t, x0)− ∂2

xV ε0(t, x0)∂2xV0(t, x0)

∣∣≤ ∂2

xV0(t, xn′)B3 |xn′ − x0|κ + ∂2xV εn′ (t, x0)B3 |xn′ − x0|κ

+∂2xV0(t, x0)

∣∣∂2xV εn′ (t, x0)− ∂2

xV ε0(t, x0)∣∣ (.)

where the last inequality is due to Holder continuity of ∂2xV ε for all ε. Then, due to the

continuity of ∂2xV0(t, ·), ∂2

xV0(t, xn′) converges to ∂2xV0(t, x0) as n′ tends to ∞. Based on the

fact of the uniform boundedness of∂2

xV ε(t, ·)

on Kρ and convergence of∂2

xV εn′ (t, x0)

to ∂2xV ε0(t, x0), it is clear that all three terms in (.) converges to 0. Therefore, (.)

follows. We discuss two possible cases for the value of ε0.

1. Case ε0 > 0: We also know that ε0 ≤ ε, since εn′ ≤ ε. According to the definition ofSε

t , x0 ∈ Sεt .

2. Case ε0 = 0: From the fact (.),(∂2

xV0(t, x0))2 ≤ 0. Therefore, ∂2

xV0(t, x0)= 0, i.e. x ∈ S0

t due to the lemma 6.5.

Therefore, the lemma is proved.

Note that any x ∈ S0t is a zero point of ∂2

xV0 at time t. Let U(t, x) := ∂2xV0(t, x). We

shall consider zero set of U(t, x) for any fixed t ∈ [0, T ). With the assumption (.) on ϕ,V0 has up to fourth derivative with respect to x. Therefore, we can derive the equation forU(t, x) from (.) as follows

∂tU + σ2U + 2σx∂xU +12σ2x2∂2

xU = 0,

U(T ) = ϕ′′.

Let x := log y, U(t, y) := U(t, x). Then, U(t, x) solves the following PDE

∂tU +32σ2∂yU +

12σ2∂2

yU + σ2U = 0.

Notice that all coefficients in the above equation are constant. Therefore, Theorem B in[1] and the remark below it are applicable to U . They directly implies that the size of thezero set of U

Zt =

y∣∣∣U(t, y) = 0

,

is non-increasing as variable t goes from T to 0. Noting that the change of variable x = log yis a one-to-one mapping, we can conclude the following proposition.

Proposition 6.3. If ϕ satisfies (.), then ∂2xV0 has at most the same number of zero points

as ϕ′′ does for any fixed t.

At this point, we can conclude that if ϕ′′ has a finite number of zero points then thegd,ε will be vanishing as ε decreases to 0. However, to achieve the goal that expectation ofcompact part goes to 0, we still need to show that the law of variable X∗,ε

s does not give apositive probability to any single point for any fixed s ∈ [t, T ].

15

Recall the main Theorem 1.1 in [14] that for every t ∈ (0, T ), the marginal law of

Mt =∫ t

0

αsdWs

does not weight points, where (αt) is any progressively measurable process such that

0 < σ ≤ αt ≤ σ + ε.

The following lemma is a simple extension to the above result.

Lemma 6.6. Let (X∗,εt ) solves the SDE (.). For any t ∈ (0, T ], X∗,ε

t does not weightpoints.

Proof. Due to the transformation applied in the section 5.1, we only need to prove the claimfor the process (Y ∗,ε

t ) which solves

dY ∗,εt = −1

2(α∗,εt

)2dt + α∗,εt dWt,

on the probability space (Ω, F, Q).Let

ξt = exp

(−∫ t

0

(α∗,εs )2

8ds +

∫ t

0

α∗,εs

2dWs

), for t ≤ T.

Define a measure Q on FT bydQ = ξT dQ.

According to Girsanov’s theorem, under the measure Q

Wt = −∫ t

0

α∗,εs

2ds + Wt

is a Brownian motion and (Y ∗,εt ) has the following dynamic

dY ∗,εt = α∗,εt dWt.

Note that the worst case scenario volatility process for χ, (α∗,εt ) is a adapted and boundedprocess. According to the Novikov condition, (ξt) is a martingale, and therefore Q and Qare two equivalent measures.

From the Theorem 1.1 in [14], we learn that the law of Y ∗,εt does not weight points under

the measure Q, for any t ∈ [0, T ). Due to equivalence between Q and Q, we can claim thatY ∗,ε

t does not weight points under the measure Q. Therefore, the lemma follows.

For given s ∈ [t, T ], we can not directly use the continuity of a probability measure toclaim that limε↓0 Qtx

[X∗,ε

s ∈ Sεs

]= Qtx

[X∗,0

s ∈ S0s

], because both the process (X∗,ε

t ) andthe set Sε

t depend on ε. Therefore, we define a capacity from the laws of a family of randomvariables X∗,ε

s ε as follows

c(A) := supε∈[0,1]

Qtx (X∗,εs ∈ A) ,

for any A ∈ B(R).

Proposition 6.4. X∗,εs converges weakly to Xs for any s > 0.

Proof. It is a direct implication of the lemma 3.1.

If ϕ′′ has a finite number of zero points, then due to the lemma 6.6Qtx

(X∗,ε

s ∈ S0s

)= 0 for any ε > 0. This fact directly leads to c(S0

s ) = 0. Due to the weakconvergence of X∗,ε

s , the family of laws of X∗,εs is weakly compact.

16

Lemma 6.7. If ϕ satisfies the condition (.) and ϕ′′ has a finite number of zero points,then

limε↓0

Qtx

[X∗,ε

s ∈ Sεs

]= 0

for any s ∈ (t, T ).

Proof. First, we observe that

0 ≤ Qtx

(X∗,ε

s ∈ Sεs

)≤ c(Sε

s).

Notice thatSε

s

is a sequence of decreasing closed sets and converges to S0

s as ε goesto 0. Because of the weak compactness of the laws of X∗,ε

s and Lemma 8 in [7], it can beseen that

c(Sεs) ↓ c(S0

s ) = 0.

Then, the lemma is true.

Theorem 6.3. If ϕ satisfies (.) and ϕ′′ has a finite number of zero points, then thereexists ε0 > 0 such that Etx [|i|] < δ for any fixed ρ > 0, any fixed point (t, x) ∈ (0, T ]×R and∀ δ > 0 as long as ε < ε0.

Proof. Recall from (.) that

Etx[i] = Etx

[∫ T

t

gd,ε(t,X∗,εs )IKρ(X∗,ε

s )σ (X∗,εs )2 ∂2

xV0(s,X∗,εs )ds

].

Note that gd,ε(t, x) can only take three possible values: −1, 0, 1. Indeed,

∣∣gd,ε(t, x)∣∣ = 1 if ∂2

xV ε(t, x)∂2xV0(t, x) < 0,

0 if ∂2xV ε(t, x)∂2

xV0(t, x) ≥ 0.

Therefore, due to (.) it follows that

Etx[|i|] = Etx

[∫ T

t

∣∣gd,ε(s,X∗,εs )∣∣ IKρ(δ)(X

∗,εs )σ (X∗,ε

s )2∣∣∂2

xV0(s,X∗,εs )∣∣ ds

]

≤ σM1

[Etx

∫ T

t

(X∗,εs )4 (1 + |X∗,ε

s |m)2 ds

]1/2 [∫ T

t

Qtx(X∗,εs ∈ Sε

s)ds

]1/2

≤ σM1

√M8(1 + |x|2m+4)

[∫ T

t

Qtx(X∗,εs ∈ Sε

s)ds

]1/2

.

Due to the lemma 6.7, it follows that∫ T

t

Qtx(X∗,εs ∈ Sε

s)ds → 0,

as ε ↓ 0. Therefore, the lemma is concluded.

Now, we are ready to claim the main result in this section.

Theorem 6.4. If ϕ satisfies (.) and ϕ′′ has a finite number of zero points, then limε↓0 I1 =0 for any fixed (t, x) ∈ [0, T )× R.

Proof. Note that|I1| ≤ Etx[|i|] + Etx[|ii|].

For any δ > 0, due to the proposition 6.2 there exists ρ0(t, x, δ) > 0 such that Etx[|ii|] < δfor all ρ > ρ0(t, x, δ).

By the theorem 6.3, for the given ρ0(t, x, δ) and δ there exist ε0(t, x, ρ0(t, x,δ)) such that Etx[|i|] < δ for any ε < ε0(t, x, ρ0(t, x, δ)). Therefore, the theorem follows.

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From the proofs, we essentially derived that α = σ + εg is a good approximation of theworst case scenario volatility α∗,ε; see their definitions in the section 5. Together with theproperties of the law of the asset price process in the worst case scenario, we proved themain theorems that Zε(t, x) goes to 0 as ε ↓ 0, for any (t, x) ∈ [0, T ) × R. Therefore, wecan construct an approximation of V ε of order o(ε): V0 + εV1. The performance of thisapproximation procedure is studied numerically in the next section.

7 Numerical results

In this section, we will work on a non-trivial example: a symmetric European butterfly withthe following payoff function

ϕ(x) = (x− 90)+ − 2(100− x)+ + (x− 110)+ (.)

which is neither convex nor concave; see the Figure 1.

60 80 100 120 140

02

46

810

Asset Price

Pay

off:

ϕ

Figure 1: The payoff function of a symmetric European butterfly

Even though the payoff function (.) does not satisfy the conditions of the main theorem4.1, we can consider a regularization ϕ of ϕ, which satisfies the conditions of the Theorem4.1. Moreover, ϕ can be chosen sufficiently close to ϕ such that

‖V0(ϕ) + εV1(ϕ)− [V0(ϕ) + εV1(ϕ)]‖ ε.

That is, the approximation of ϕ: V0(ϕ) + εV1(ϕ) is a good proxy for that of ϕ.Indeed, we numerically compute the worst case scenario price V ε(ϕ), by the scheme

provided in [15]. It is proved by Barles [4] that the numerical solution from that schemeis locally uniformly convergent to V ε, the unique viscosity solution of (.), as the schemebecomes finer. We also compare the numerically computed worst case scenario price withits approximation: V0(ϕ) + εV1(ϕ), where V0(ϕ) is given by the Black-Schles formula andV1(ϕ) is numerically computed by a simple difference scheme according to the equation(.). Because the scheme for computing V ε(ϕ) uses Newton iteration in each time step todeal with the nonlinearity, our approximation is computed a lot more efficiently. For visualcomparison of the worst case scenario prices with corresponding approximations, we showcomplete numerical results for a very small ε = 0.006, a small ε = 0.01, and a ε = 0.05

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which is not so small. Throughout all the experiments, we set σ = 0.15 and T = 0.25. Whenε = 0.05, the upper bound of the volatility interval is σ + ε = 0.20, which is 1/3 larger thanthe base volatility level σ = 0.15. In other words, even if ε = 0.05 is small, 5% volatility issignificant. From Figure 2, we note that the worst case scenario prices are higher than theBlack-Scholes prices. That is, we need extra cash to super-replicate the option when facingthe model ambiguity. It also can be noted that the first order corrected prices capture themain feature of the worst case scenario prices V ε for different values of ε.

0 50 100 150 200

01

23

4

Asset Price

Price

0 50 100 150 200

0.00

0.02

0.04

0.06

0.08

Asset Price

Price

Case: ε =0.006

0 50 100 150 200

01

23

4

Asset Price

Price

0 50 100 150 200

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Asset Price

Price

Case: ε =0.01

0 50 100 150 200

01

23

45

Asset Price

Price

0 50 100 150 200

0.00.1

0.20.3

0.40.5

0.60.7

Asset Price

Price

Case: ε =0.05

Figure 2: The black curve represents the worst case scenario prices and the red curve rep-resents the Black-Scholes prices; the blue curve represents the difference between the worstcase scenario prices and its Black-Scholes prices and the green curve is ε ∗ V1; all curves areplotted against asset prices.

To see the trend of the error of our approximation as ε increases, we choose 8 equallyspaced values from 0 to 0.05 for ε. For each ε, we compute the error of the approximation,which is defined by

error(ε) = supx|V ε(0, x)− V0(0, x)− εV1(0, x)|. (.)

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0.01 0.02 0.03 0.04 0.05

0.02

0.04

0.06

0.08

ε = σ − σ

Err

or: V

ε −(V

0+εV

1)

Figure 3: Error for different values of ε;

As shown in the Figure 3 the error increases as ε becomes larger, so the second orderapproximation will be needed to improve accuracy for large values of ε. However, there isnot an abrupt change in the range of ε we choose.

8 Conclusion

In this paper, we have studied the asymptotic behavior of the worst case scenario option pricesas the degree of model ambiguity vanishes. This study not only helps us understand howa linear expectation turns into a sublinear expectation, but also gives us an approximationprocedure of worst case scenario option prices when ε is small. From the numerical results,we see that the approximation procedure works well even when the upper bound volatilityis 1/3 larger than the lower bound.

Note that the worst case scenario price is often computed to evaluate the risk in a portfolio.Our approximation procedure improves the efficiency of this evaluation, because it avoids theNewton iteration which is employed in the scheme for V ε. Moreover, the worst case scenarioprice V ε has to be re-computed for a new value of ε. However, the equations (.) and (.)for V0 and V1 are independent of ε, so the approximation only requires us to compute V0 andV1 once for all small values of ε.

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