Conclusion and final remarks

In this chapter, we have obtained a formula to calculate the vega index for options whose payoff functions may depend on the maximum or minimum of a one-dimensional SDE. The key technique is the Lamperti transform which enables us to calculate the directional derivatives with respect to the diffusion coefficient.

This formula gives a decomposition of the vega index into three sensitivities: extrema sensitivity, terminal sensitivity and drift sensitivity. Numerical tests illustrate that there are some important relationships between extrema sensitivity and terminal sensitivity in realistic options.

In [10], formulas of this type in the multi-dimensional case under some commutativity conditions on the diffusion coefficient are used to prove the smoothness of the density function concerning the supremum of a multi-dimensional diffusion. The authors obtain the formulas by means of Garsia-Rodemich-Rumsey’s lemma.

The numerical result on the comparison of the vega index in two different models tells us that the traditional Black-Scholes model is far away from the one-dimensional model dealt in this chapter as far as the vega index is concerned. Today, some practitioners are using so-called stochastic volatility models which deal with stochastic diffusion coefficients to express the dynamics of economy (see [8], for example).

There are many difficulties to deal with stochastic volatility models, however to compute the vega index for exotic options in stochastic volatility models is a challenging problem for the future. According to [4], there is a relationship between a one-dimensional model and a stochastic volatility model, when we consider some exotic options. Thus, the results obtained in [4] may be applied to this problem.

The kernel method has been used to compute the vega index for a specific option in this chapter.

The bandwidth selection problem in this chapter can be successfully solved, since it is much simpler than general bandwidth selection problems that appear in the kernel density estimation problems for multi-dimensional density functions. Our numerical result shows the optimal bandwidth works well for the problem considered in this chapter.

Appendix A

In Appendix A, we shall give the proof of Lemma 8, 7, 9 and 10 which we have used in Section 3.2.

We use some preparatory lemmas. We denote by Ki, i= 1,2 the appropriate Lipschitz constants that appear in the hypothesis(H1).

Appendix A.1

In this subsection, we shall prove some properties which are used in order to prove Lemma 8, 7, 9 and 10.

Lemma A1. Assume (H1)and(H2). Then, we have the following inequalities, 1

2K1

log (1 +z

2

) ≤ |F_ε(z)| ≤ 1

σ0|log(z)|,∀z >0 ∫ z

x

ˆ σ

σ^ε1σ^ε2(y)dy

≤ K₁ σ²₀

(1 z −1

x +

log (z

x ) )

,∀x, z >0.

Proof. In fact, the proof follows by using the lower bound forσ^εin the hypothesis(H2)and the upper bounds forσ^εand ˆσin the hypothesis(H1).

Lemma A2. Assume (H1),(H2) and(H3). Then,F_ε⁻¹ can be evaluated as follows, e^σ0z≤F_ε⁻¹(z)≤2e^2K1z, ifz≥0

r0e^σ1z≤F_ε⁻¹(z)≤e^σ0z, if z <0.

Proof. To prove this lemma, we consider two cases according to the sign of z. In the case that z is negative, we use hypothesis(H3)in addition to(H1)and(H2). In both cases, we use the monotonicity ofF_εandF_ε⁻¹. We leave the details to the reader.

Let us prove a lemma on the regularity ofF_ε⁻¹(z).

Lemma A3. Assume (H1)and(H2). Then,F_ε⁻¹(z)is differentiable with respect toε and

∂F_ε⁻¹

∂ε (z) =σ^ε(F_ε⁻¹(z))

∫ F_ε⁻¹(z) 1

ˆ σ

(σ^ε)²(y)dy.

Furthermore _∂ε^∂ (

F_ε⁻¹(Fε(x) +z))

exists and

∂

∂ε

(F_ε⁻¹(F_ε(x) +z))

= ∂F_ε⁻¹

∂z (F_ε(x) +z)∂F_ε

∂ε (x) +∂F_ε⁻¹

∂ε (F_ε(x) +z)

= σ^ε(F_ε⁻¹(Fε(x) +z))

∫ F_ε⁻¹(F_ε(x)+z) x

ˆ σ

(σ^ε)²(y)dy. (A.1) Proof. The proof follows from the implicit function theorem applied toz =F_ε(F_ε⁻¹(z)) and the chain rule for partial differentiation.

Now we shall prove Lemma 7, 8 and 9. For convenience, let us prove Lemma 8 first.

Appendix A.2

In this subsection, we prove Lemma 8.

The result in Theorem 5 implies that ifσand ˆσsatisfy(H1)and(H2), andε∈[0,1] andx >0 are fixed, then there exists a stochastic processS^ε such that it satisfies

S_t^ε=x+

∫ t 0

σ^ε(S^ε_u)◦dW˜_u,∀t∈[0, T], a.s. (A.2) However, the exceptional set where the above equality is not satisfied may depend on ε. In order to consider the differentiation of the solution to (3.3), we modify the solution so that the exceptional set does not depend onε. Let us prove that the solution to (3.3) is continuous with respect to εso that we can discuss its differentiability.

We refer [12] to obtain the continuity inε.

Lemma A4. Assume (H1)-(H2). LetSⁱ_t be the solution to { dS_tⁱ=σ^εi(S_tⁱ)◦dW˜t

(

=σ^εi(S_tⁱ)dW˜t+¹₂σ^εi(σ^εi)^′(S_tⁱ)dt )

S₀ⁱ =x, (A.3)

εi∈[0,1], i= 1,2. Then, forp >2 there exists a constantC >0 such that E^Q˜[|S_t²₂−S¹_t1|^p]≤C

{|ε2−ε1|^p+|t2−t1|^p2} .

Proof. We have

E^Q˜[|S_t²₂−S_t¹₁|^p] ≤ 2^p−1 {

E^Q˜

[|F_ε⁻₂¹(Fε₂(x) + ˜Wt₂)−F_ε⁻₂¹(Fε₁(x) + ˜Wt₁)|^p]

+E^Q˜

[|F_ε⁻₂¹(Fε₁(x) + ˜Wt₁)−F_ε⁻₁¹(Fε₁(x) + ˜Wt₁)|^p]}

=: 2^p−1 {

E^Q˜[|I1|^p] +E^Q˜[|I2|^p] }

.

The arguments to deal with the above expectations are standard and similar to the proof of continuity of flows associated with the solution of stochastic differential equations. We do not give all the arguments here and only stress that to obtain the result one uses the lemmas in Appendix Appendix A.1 together with hypotheses(H1)-(H2)and the fact that the Wiener process has finite exponential moments.

Lemma A4 implies that there exists a continuous modification of the solutionS^εwith respect to two variables (t, ε) so that (3.3) is satisfied for all (t, ε), a.s. Then we can consider the differentiation of the solution to (3.3).

Proof of Lemma 8. Combining Lemma A3 and Lemma A4, the proof is straightforward.

Appendix A.3

This subsection is devoted to the proof of Lemma 7.

Using the fact that Fε(z) and F_ε⁻¹(z) are continuous monotone increasing functions, the term max0≤t≤TS_t^ε can be expressed as max0≤t≤TS_t^ε =F_ε⁻¹(Fε(x) + max0≤t≤TW˜t) under ˜Q. We have the following continuity lemma.

Lemma A5. Assume (H1) and (H2). For i = 1,2, let S_tⁱ be the solutions to (A.3). Then, for any p >2, there exists a constant C >0 such that

E^Q˜[ max

0≤t≤TS_t²− max

0≤t≤TS_t^{1 p}]

≤C|ε2−ε1|^p.

Proof. The proof is similar to the proof of Lemma A4. We obtain the result by replacing exponential moments of ˜Wtby the respective ones of max0≤t≤TW˜t.

Proof of Lemma 7. Using the above lemma, the explicit expression for max0≤t≤TS_t^ε in Theorem 5 and (A.1), we obtain the result.

Appendix A.4

The goal of this subsection is to prove Lemma 9.

First, let us prove the following lemma aboutL^p-boundedness ofS_t^ε, logS_t^ε, ^∂S_∂ε^εt and ^∂_∂ε^2S₂^tε under the hypotheses(H1)-(H3).

Lemma A6. Assume (H1)-(H3). Let S^ε be the solution to (A.2). Then, for any p > 2 there exists C1>0 such that

E^Q˜[|S_t^ε|^p] +E^Q˜[|logS_t^ε|^p]≤C₁. Moreover, for any real number pthere exists C2>0 such that

E^Q˜[|S_t^ε|^p]≤C2.

Proof. The estimate forE^Q˜[|S_t^ε|^p] follows from Lemma A1 and Lemma A2. Let us consider the finiteness ofE^Q˜[|logS_t^ε|^p]. Using the fact thatS^ε_t ≥1⇔Fε(x) + ˜Wt≥0 andS_t^ε<1⇔Fε(x) + ˜Wt<0 together with Lemma A2, we have

E^Q˜[|logS_t^ε|^p]

= E^Q˜[(logS_t^ε)^p:S_t^ε≥1] +E^Q˜ [(

log 1 S^ε_t

)p

:S_t^ε<1 ]

≤ E^Q˜ [(

log (

2e^{2K1(Fε(x)+ ˜Wt)} ))p

:F_ε(x) + ˜W_t≥0 ] +E^Q˜

[(

log (1

r0

e^{−σ1(Fε(x)+ ˜Wt)} ))p

:Fε(x) + ˜Wt<0 ]

≤ 2^p−1

{|log 2|^p+ (2K₁)^pE^Q˜[F_ε(x) + ˜W_t^p]}

+ 2^p−1 {(

log (1

r0

))p

+σ₁^pE^Q˜[F_ε(x) + ˜W_t^p]}

.

Then by Lemma A1, we have E^Q˜[Fε(x) + ˜Wt^p]

≤2^p−1 ( 1

σ^p₀|log(x)|^p+E^Q˜[ max

0≤t≤T

W˜t

^p]) ,

and the result follows.

Now, we consider the L^p-boundedness of ^∂S_∂ε^tε and ^∂_∂ε^2S₂^εt. Note that as σ^ε(z) and ∫z x

1

(σ^ε)²(y)dy are continuously differentiable with respect toεandz, ^∂_∂ε^2S2^εt exists forε∈[0,1] andt∈[0, T], a.s. by Lemma 8.

Lemma A7. Assume (H1)-(H3). Let S^ε be the solution to (A.2). Then, for any p > 2 there exists C >0such that

E^Q˜[ ∂S_t^ε

∂ε ^p]

+E^Q˜[ ∂²S_t^ε

∂ε^{2 p}]

≤C.

Proof. E^Q˜[^∂Sεt

∂ε ^p]

part follows from the explicit form of ^∂S_∂ε^εt, Lemma A1, A6 and (H1). Then, the upper bound for E^Q˜[^∂2Stε

∂ε^{2 p}]

is calculated as follows. By the differentiability of ∫z x

ˆ σ

(σ^ε)²(y)dy with respect toεandz, we have

∂²S_t^ε

∂ε² = (∂S_t^ε

∂ε σ^ε′(S^ε_t) + ˆσ(S_t^ε) ) ∫ S^ε_t

x

ˆ σ

(σ^ε)²(y)dy−2σ^ε(S_t^ε)

∫ S^ε_t x

ˆ σ²

(σ^ε)³(y)dy+ σˆ

σ^ε(S_t^ε)∂S_t^ε

∂ε . Thus, the result follows from Lemma A6 and the inequality

∫ z x

ˆ σ²

(σ^ε)³(y)dy ≤ K₁²

σ³₀ (1

2 1

x² − 1 z²

+ 2 1

x−1 z

+ log

(z x

) ) .

For the proof of Lemma 9, we use the following theorem which is proved in Theorem 10.6 of [12]. We shall use this theorem without the proof.

Theorem A7. (Theorem 10.6 of [12]) Let (Ω,F, P) be a complete probability space equipped with a filtration σ-field {Ft, t ∈ [0, T]} satisfying the usual condition. Let ft(ε),(t, ε) ∈ [0, T]×[0,1]^d be a measurable random field satisfying the following properties.

(i) For each ε,f_t(ε) is predictable.

(ii) For any p >2, there is a positive constantC1 such that

∫ T 0

E^P[|f_t(ε)|^p]dt≤C₁, for any ε.

(iii) For anyp >2, there is a positive constant C2 such that

∫ T 0

E^P[|ft(ε1)−ft(ε2)|^p]dt≤C2|ε1−ε2|^αp, where 0< α≤1.

(iv) f_t(ε) is m-times continuously differentiable in ε for all t, a.s. and derivatives ^∂_∂ε^kf_k^t(ε),|k| ≤ m satisfy conditions (ii) and (iii).

Let Mt be a continuous local martingale such that⟨M⟩t− ⟨M⟩s≤t−s for any t > s, a.s. Then there is a modification of the integral which is continuous in (t, ε) and m-times continuously differentiable.

Furthermore, it holds that

∂^k

∂ε^k (∫ t

0

fs(ε)dMs

)

=

∫ t 0

∂^kfs

∂ε^k (ε)dMs

for any _∂ε^∂k_k such that|k| ≤m.

Proof of Lemma 9. Firstly, we prove the stochastic integral term of X_T^ε. For the proof, we use Theorem A7 on the differentiability of the stochastic integral withε-dependent integrand. Forft(ε) :=

−(^σ₂^ε′ −_σ^bε)(S^ε_t), it is straightforward to check thatft(ε) satisfies the sufficient conditions of Theorem A7 by (H1) and Lemma A6 and A7.

Secondly, let us consider the Lebesgue integral term of X_T^ε. It suffices to prove that there existsAt

such that

∂

∂ε ((σ^ε)^′

2 − b σ^ε

)2

(S_t^ε)

≤A_t,∀t∈[0, T], a.s., and∫T

0 A_tdt <∞, a.s., whereA_tdoes not depend onε. This follows from the fact that

∂

∂ε ((σ^ε)^′

2 − b σ^ε

)2

(S^ε_t)

=

((σ^ε)^′ 2 − b

σ^ε )

(S^ε_t) {

σ^′′(S_t^ε)∂S_t^ε

∂ε + ˆσ^′(S_t^ε) +εˆσ^′′(S_t^ε)∂S_t^ε

∂ε }

−2 ((σ^ε)^′

2 − b σ^ε

) (S_t^ε)

b^′(S^ε_t)^∂S_∂ε^εtσ^ε(S_t^ε)−b(S_t^ε) {

σ^′(S_t^ε)^∂S_∂ε^tε + ˆσ(S^ε_t) +εˆσ^′(S_t^ε)^∂S_∂ε^tε }

(σ^ε)²(S^ε_t) .

Then,S_t^ε, _S¹ε

t and ^∂S_∂ε^tε can be evaluated above due to Lemma A1 and A2.

Appendix A.5

In this subsection, we shall prove Lemma 10.

We have already checked that H(ε, h) := 1

h {

f( max

0≤t≤TS_t^ε+h, S_T^ε+h) exp(X_T^ε+h)−f( max

0≤t≤TS_t^ε, S_T^ε) exp(X_T^ε) }

→ ∂

∂ε (

f( max

0≤t≤TS_t^ε, S_T^ε) exp(X_T^ε) )

, ε∈[0,1], a.s.

ashtend to 0. Then it suffices to prove the uniform integrability ofH(ε, h) with respect toh.

To prove the uniform integrability, we need the following lemma.

Lemma A8. Assume (H1)-(H3). Then, forp >2, there existsC >0such that E^Q˜[|X_T^ε2−X_T^ε1|^p]≤C|ε₂−ε₁|^p.

Proof. It suffices to check that there existsC >0 such that E^Q˜[

∂X_T^ε

∂ε ^p]

≤C.

By using the proof of Lemma 9, we have the explicit form of^∂X_∂ε^Tε. Thus, the result follows by Burkholder-Davis-Gundy’s inequality and Lemma A6 and A7.

Now let us prove the uniform integrability.

Lemma A9. Assume (H1)-(H5). Letε∈[0,1)be fixed. Then, for anyp >2, we have sup

h∈(0,1−ε)

E^Q˜[|H(ε, h)|^p]<∞. Proof. A straighforward calculation yields

f( max

0≤t≤TS_t^ε+h, S^ε+h_T ) exp(X_T^ε+h)−f( max

0≤t≤TS_t^ε, S_T^ε) exp(X_T^ε) ^p

≤ 3^p−1 {

C₁ max

0≤t≤TS_t^ε+h− max

0≤t≤TS^ε_t^pe^pXε+hT +C2|S_T^ε+h−S_T^ε|^pe^pXε+hT +|f( max

0≤t≤TS_t^ε, S_T^ε)|^p|X_T^ε+h−X_T^ε|^p ∫ 1

0

e^{v((XTε+h−XTε))+XTε}dv ^p}

. The first and second term of the right-hand side can be evaluated as

E^Q˜[ max

0≤t≤TS_t^ε+h− max

0≤t≤TS_t^εpe^pXTε+h ]

+E^Q˜[|S_T^ε+h−S_T^ε|^pe^pXTε+h]≤C|h|^p. due to the boundedness ofσ^ε′, _σ^b_ε and Lemma A4, A5. For the third term, we have

E^Q˜ [

|f( max

0≤t≤TS_t^ε, S^ε_T)|^p|X_T^ε+h−X_T^ε|^p ∫ 1

0

e^{v((XTε+h−XTε))+XTε}dv ^p]

≤ E^Q˜[|X_T^ε+h−X_T^ε|^2p]¹²E^Q˜[|f( max

0≤t≤TS_t^ε+h, S_T^ε+h)|^4p]¹⁴ (∫ 1

0

E^Q˜ [

e^{4pvXTε+h+4p(1−v)XTε} ]

dv )¹₄

.

Moreover, by the boundedness ofσ^ε′, band _σ^bε we have E^Q˜

[

e^{4pvXTε+h+4p(1−v)XTε}

]≤C <∞.

Finally, by Lemma A8 we have E^Q˜

[

|f( max

0≤t≤TS_t^ε+h, S_T^ε+h)|^p|X_T^ε+h−X_T^ε|^p ∫ 1

0

e^{v((XTε+h−XTε))+XTε}dv ^p]

≤C|h|^p.

The proof is completed.

Proof of Lemma 10. The result in Lemma 10 follows from Lemma A9.

Appendix B

In this section, we shall prove equation (3.13).

In order to prove (3.13), we use the mollifier approximation off(z) = (z−K)+. Define j(z) =

{

Ce^z21−1 (|z|<1)

0 (|z| ≥1) (B.1)

whereC is a constant such that∫_∞

−∞j(z)dz= 1. We consider f_n(z) := (j_n∗f)(z) :=

∫ _∞

−∞

j_n(z−y)f(y)dy, (B.2)

where jn(z) = nj(nz). Then by the Lipschitz continuity of f, we have fn(z) ↗ f(z), uniformly as n→ ∞. Moreover,f_n^′ exists and we have

f_n^′(z) =

∫ n(z−K)

−∞

j(y)dy,

therefore, we have that limn→∞f_n^′(z)→I_(K,∞)(z), for almost everyz andf_n^′ is bounded.

Proofof (3.13). Forh∈(0,1] we define Π^h_n:=E^Q˜[fn(max0≤t≤TS_t^h) exp(X_T^h)] andϕn(h),ϕ(h) by ϕ_n(h) := Π^h_n−Π⁰_n

h , ϕ(h) := Π^h−Π⁰ h . Then, we have

nlim→∞ sup

0<h≤1|ϕn(h)−ϕ(h)|= 0. (B.3)

Let us prove (B.3). We defineg_n(z) :=f_n(z)−f(z), then we have ϕ_n(h)−ϕ(h) = 1

hE^Q˜[(g_n( max

0≤t≤TS^h_t)−g_n( max

0≤t≤TS_t))e^XTh]−1

hE^Q˜[g_n( max

0≤t≤TS_t)(e^XhT −e^XT)].

(B.4)

The second term on the right-hand side of (B.4) can be evaluated by Lemma A8 as 1

hE^Q˜[gn( max

0≤t≤TSt)(e^XhT −e^XT)]≤CE^Q˜[|gn( max

0≤t≤TSt)|²]¹², and this goes to 0 asn→ ∞by the uniform convergence of fn.

Now let us evaluate the first term on the right-hand side of (B.4). By the definition ofgn we have gn(z)−gn(z^′) = (z−z^′)

∫ 1 0

[f_n^′(v(z−z^′) +z^′)−I_(K,∞)(v(z−z^′) +z^′)]dv. (B.5) Using (B.5) we have

E^Q˜[|gn( max

0≤t≤TS^h_t)−gn( max

0≤t≤TSt)|²]≤E^Q˜[|gn( max

0≤t≤TS_t^h)−gn( max

0≤t≤TSt)|⁴]¹²

× E^Q˜ [

∫ 1 0

{

f_n^′(v max

0≤t≤TS_t^h+ (1−v) max

0≤t≤TSt)−I_(K,∞)(v max

0≤t≤TS_t^h+ (1−v) max

0≤t≤TSt) }

dv ⁴

]¹₂ .

The second expectation of right-hand side of the above inequality can be written as

∫ 1 0

∫ _∞

0

f_n^′(vF_h⁻¹(Fh(x) +z) + (1−v)F⁻¹(F(x) +z))

−I_(K,∞)(vF_h⁻¹(Fh(x) +z) + (1−v)F⁻¹(F(x) +z))

⁴pM(z)dzdv, (B.6) where p_M(z) is the density function of max_0≤t≤TW˜_t. We define g_v,h(z) := vF⁻¹(F_h(x) +z) + (1− v)F⁻¹(F(x) +z) and consider a change of variableg_v,h(z) =u, then, by Lemma A2, we have ^du_dz ≥C, whereC does not depend onv, handz. Thus, we can show that the term in (B.6) is bounded by

1 C

∫ 1 0

∫ _∞

x

|f_n^′(u)−I_(K,∞)(u)|⁴pM(g⁻_v,h¹(u))dudv,

which goes to 0 asn→ ∞by the dominated convergence theorem, therefore, we have (B.3).

The limit in (B.3) asserts that

∂Π^ε

∂ε

ε=0

= lim

n→∞

∂Π^ε_n

∂ε

ε=0

.

Then (3.13) follows from the dominated convergence theorem, the monotone convergence theorem and the existence of the density of max0≤t≤TSt.

Appendix C

This section is devoted to the proof of Lemma 11.

Define

ˆ σ_n(z) :=

∫ _∞

−∞

j_n(z−y)ˆσ(y)dy, (C.1)

where ˆσis defined by (3.10) andjn is defined by (B.2). Furthermore, we define Π^ε,n:=E^P[f( max

0≤t≤TS_t^ε,n, S_T^ε,n)],

where S^ε,n denotes the solution to (3.1) withb(z) = 0, σ(z) = ˜σz and ˆσis defined by (C.1). Then by (3.2), we have

∂Π^ε,n

∂ε

ε=0

= 1

˜ σE^P

[

∂₁f( max

0≤t≤TS_t, S_T) max

0≤t≤TS_t

∫ max

0≤t≤TSt x

ˆ σ_n(y)

y² dy ]

+1

˜ σE^P

[

∂₂f( max

0≤t≤TS_t, S_T)S_T

∫ St

x

ˆ σ_n(y)

y² dy ]

−σ˜ 2E^P

[

∂1f( max

0≤t≤TSt, ST)Sη

∫ η 0

ˆ

σ^′_n(St)dt+∂2f( max

0≤t≤TSt, ST)ST

∫ T 0

ˆ σ^′_n(St)dt

] . (C.2) From the fact that ˆσ∈C_b¹(R+;R+), we get ˆσ_n→σ, uniformly asˆ n→ ∞. Thus, one has

1

˜ σE^P

[

∂₁f( max

0≤t≤TS_t, S_T) max

0≤t≤TS_t

∫ max

0≤t≤TSt x

ˆ σn(y)

y² dy ]

→ 1

˜ σE^P

[

∂1f( max

0≤t≤TSt, ST) max

0≤t≤TSt

∫ max

0≤t≤TSt x

ˆ σ(y)

y² dy ]

,

1

˜ σE^P

[

∂2f( max

0≤t≤TSt, ST)ST

∫ S_t x

ˆ σ_n(y)

y² dy ]

→ 1

˜ σE^P

[

∂₂f( max

0≤t≤TS_t, S_T)S_T

∫ St

x

ˆ σ(y)

y² dy ]

,

asn→ ∞.

By the definition of ˆσ_n, one has ˆσ_n(z)→σ(z) for almost everyˆ z. Moreover, we have

E^P [

∂f( max

0≤t≤TS_t, S_T)S_η

∫ η 0

(ˆσ^′_n(S_t)−σˆ^′(S_t))dt ]

≤C (∫ T

0

E^P[

|σˆ^′_n(S_t)−σˆ^′(S_t)|²] dt

)¹₂ .

Due to the boundedness of ˆσ^′_n and ˆσ^′, and the existence of the density function ofSt, we get

˜ σ 2E^P

[

∂1f( max

0≤t≤TSt, ST)Sη

∫ η 0

ˆ

σ^′_n(St)dt+∂2f( max

0≤t≤TSt, ST)ST

∫ T 0

ˆ σ^′_n(St)dt

]

→ −σ˜ 2E^P

[

∂1f( max

0≤t≤TSt, ST)Sη

∫ η 0

ˆ

σ^′(St)dt+∂2f( max

0≤t≤TSt, ST)ST

∫ T 0

ˆ σ^′(St)dt

] ,

asn→ ∞.

Finally, let us prove that

nlim→∞lim

ε→0

Π^ε,n−Π^0,n

ε = lim

ε→0 lim

n→∞

Π^ε,n−Π^0,n

ε . (C.3)

We defineφn(ε) :=^Πε,n−_ε^Π0,n andφ(ε) := ^Πε−_ε^Π0, then φn(ε)−φ(ε) = 1

εE^P [

f( max

0≤t≤TS^ε,n_t , S^ε,n_T )−f( max

0≤t≤TS_t^ε, S_T^ε) ]

,

where S^ε denotes the solution to (3.1) with b(z) = 0, σ(z) = ˜σz and ˆσ is defined by (3.10). From Burkholder-Davis-Gundy’s inequality, we have

E^P[

|S_T^ε,n−S_T^ε|²]

≤ C1sup

y |σˆn(y)−σ(y)ˆ |²ε²+C2

∫ T 0

E^P[|S_t^ε,n−S_t^ε|²]dt, E^P

[

| max

0≤t≤TS_t^ε,n− max

0≤t≤TS_t^ε|² ]

≤ C₁sup

y |σˆ_n(y)−σ(y)ˆ |²ε²+C₂

∫ T 0

E^P [

max

0≤u≤t|S_u^ε,n−S_u^ε|² ]

dt.

This implies that limn→∞sup_0≤ε≤1|φn(ε)−φ(ε)|= 0 is true and, therefore, (C.3) holds. The proof is completed.

Appendix D

In this section, we shall prove Lemma 13.

The proof is divided in two steps: in the first step, we deal with the bias part of (3.19), then we consider the variance part of (3.19) in the second step.

step1. Let us consider the bias part of (3.19). By the change of variables we have E^P[ ˆE(U, K)] = σ˜2

˜ σ₁

∫ ^U−K_h

1

−∞ K1(y)(ψ₁p_M)(U−h₁y)dy. (D.1) From the explicit expression for (ψ₁p_M), we know thaty∈(K,∞)7→(ψ₁p_M)(y)∈R+ is smooth. We use the Taylor’s theorem and(K2), and obtain

˜ σ₂

˜ σ1

∫ ^U−K

h1

−∞ K1(y)ψ₁p_M(U−h₁y)dy

= σ˜2

˜ σ1

∫ ^U−K

h1

−∞ K1(y) [

ψ1pM(U)−h1y(ψ1pM)^′(U) +1

2h²₁y²(ψ1pM)^′′(U) ]

+o(h²₁).

Due to (3.16) and(K1)it is easy to see 1

h²₁

(∫ ^U−K_h

1

−∞ K1(y)dy−

∫ _∞

−∞

K1(y)dy )

→ 0, 1

h₁

∫ ^U−K_h

1

−∞

yK1(y)dy → 0,

∫ ^U−K_h

1

−∞ K1(y)y²dy → µ₂(K1),

ash1 tends to 0. This implies that

E^P[ ˆE(U, K)]−E(U, K) =1 2

˜ σ2

˜ σ1

(ψ1pM)^′′(U)µ2(K1)h²₁+o(h²₁).

By the same arguments, we have E^P[ ˆT(U, K)]−T(U, K) = 1

2

˜ σ2

˜ σ1

(ψ2pS)^′′(K)µ2(K2)h²₂+o(h²₂) E^P[ ˆD(U, K)]−D(U, K) = −σ˜1σ˜2

4

[(ϕ₁p_M)^′′(U)µ₂(K1)h²₁+ (ϕ₂p_S)^′′(K)µ₂(K2)h²₂]

+o(h²₁+h²₂).

step2. Now, let us consider the variance part of (3.19). The termV ar[ ˆV(U, K)] is calculated as V ar[ ˆV(U, K)]

= 1

NE^P [{ 1

h1

˜ σ₂

˜ σ1K1

(U−M_T h1

)

I_(K,∞)(S_T)M_Tlog (M_T

x )

+1 h2

˜ σ2

˜ σ1

K2

(K−ST

h2

)

I_(U,∞)(M_T)S_Tlog (ST

x )

−1 h1

˜ σ₁σ˜₂

2 K1

(U−M_T h1

)

I(K,∞)(ST)MTη− 1 h2

˜ σ₁σ˜₂

2 K2

(K−S_T h2

)

I(U,∞)(MT)STT }2]

−1 NE^P

[1 h₁

˜ σ2

˜ σ₁K1

(U−MT

h₁ )

I_(K,∞)(ST)MTlog (MT

x )

+1 h2

˜ σ2

˜ σ1K2

(K−ST

h2

)

I_(U,∞)(MT)STlog (ST

x )

−1 h1

˜ σ1σ˜2

2 K1

(U−MT

h1

)

I(K,∞)(ST)MTη− 1 h2

˜ σ1˜σ2

2 K2

(K−ST

h2

)

I(U,∞)(MT)STT ]2

=:I1−I2. (D.2)

Let us check thatI2 in (D.2) iso(_{N h}¹

1 +_{N h}¹

2). Using the same calculation as in (D.1), we have E^P

[ 1 h1

˜ σ₂

˜ σ1K1

(U −M_T h1

)

I(K,∞)(ST)MTlog (M_T

x )]

=σ˜₂

˜ σ1

∫ ^U−K

h1

−∞ K1(y)(ψ1pM)(U−h1y)dy.

Due to the fact that|ψ1(y)| ≤ ^|_|^y_x^|_|²,∀y∈(K,∞) and(K1), we obtain

lim

h₁→0

∫ ^U−K_h

1

−∞ K1(y)(ψ₁p_M)(U−h₁y)dy <∞.

We use the same calculations for the other terms of the second part of (D.2), then we can show that I2of (D.2) iso(_{N h}¹

1 +_{N h}¹

2).

Then, we focus on I1 part of (D.2). Clearly, we have

I1 = 1 NE^P

[ 1 h²₁K1

(U−MT

h1

)2

M_T²I_(K,∞)(ST) (˜σ2

˜ σ1

log (MT

x )

−σ˜1σ˜2

2 η )2]

+2 NE^P

[ 1 h₁h₂K1

(U−MT

h₁ )

M_TI_(K,∞)(S_T) (σ˜2

˜ σ₁log

(MT

x )

−σ˜1σ˜2

2 η )

×K2

(K−ST

h₂ )

STI_(U,∞)(MT) (σ˜2

˜ σ₁log

(ST

x )

−˜σ1σ˜2

2 T )]

+1 NE^P

[ 1 h²₂K2

(K−S_T h2

)2

S_T²I_(U,∞)(M_T) (˜σ₂

˜ σ1

log (S_T

x )

−σ˜₁σ˜₂ 2 T

)2]

=: I_1,1+I_1,2+I_1,3. (D.3)

Let us show thatI1,2=o(_{N h}¹

1+_{N h}¹

2). We define J1 := E^P

[ 1 h1h2

K1

(U −M_T h1

)

MTI(K,∞)(ST)K2

(K−S_T h2

)

×STI_(U,∞)(MT) (σ˜2

˜ σ₁

)2

log (MT

x )

log (ST

x )]

= (σ˜₂

˜ σ1

)2

1 h1h2

∫ _∞

U

K1

(U−y h1

)

(Φp_M)(y)dy, where Φ(y) := ylog(^y_x)E^P[I_(K,∞)(ST)K2(^K−_h^ST

2 )STlog(^S_x^T)|MT = y]. By the change of variables, we obtain

J1= (σ˜2

˜ σ₁

)2

1 h₂

∫ 0

−∞K1(y)(ΦpM)(U −h1y)dy.

Now let us focus onE^P[I(K,∞)(ST)K2(^K−_h^ST

2 )STlog(^S_x^T)|MT =y]. Fory∈(K,∞), again by the change of variables, we have

E^P [

I_(K,∞)(S_T)K2

(K−S_T h2

) S_Tlog

(S_T x

) M_T =y ]

=

∫ y K

K2

(K−z h₂

) zlog

(z x )

p_S|M(z|y)dz

= 1

p_M(y)

∫ y K

K2

(K−z h₂

) zlog

(z x )

pM,S(y, z)dz

= h₂

pM(y)

∫ 0

K−y h2

K2(z)(K−h2z) log

(K−h₂z x

)

pM,S(y, K−h2z)dz,

wherepS|M(y|z) denotes the conditional density function ofST given byMT andpM,S(y, z) denotes the

joint density function of (MT, ST). Therefore, we have J1 =

(˜σ₂

˜ σ1

)2∫ 0

−∞K1(y)(U−h1y) log

(U−h₁y x

)

×

∫ 0

K−y h2

K2(z)(K−h2z) log

(K−h2z x

)

pM,S(U−h1y, K−h2z)dzdy, andJ₁=o(_{N h}¹

1+_{N h}¹

2). The same calculation yields that the other terms ofI_1,2are alsoo(_{N h}¹

1+_{N h}¹

2).

Finally, we considerI_1,1andI_1,3of (D.3). By using Ψ₁, Ψ₂and Ψ₃, we have the following expression forI1,1,

I_1,1 = 1 N h₁

[(σ˜2

˜ σ₂

)2∫ ^U−K_h

1

−∞ K1(y)²(Ψ₁p_M)(U−h₁y)dy−σ˜₂²

∫ ^U−K_h

1

−∞ K1(y)²(Ψ₂p_M)(U−h₁y)dy +˜σ₁²σ˜₂²

4

∫ ^U−K_h

1

−∞ K1(y)²(Ψ₃p_M)(U−h₁y)dy ]

.

Thus, by applying the Taylor’s theorem for (Ψ1pM)(y), (Ψ2pM)(y) and (Ψ3pM)(y), due to(K2), we get I1,1=

[(σ˜2

˜ σ₁

)2

Ψ1(U)−˜σ₂²Ψ2(U) +σ˜²₁σ˜₂² 4 Ψ3(U)

]

pM(U)R2(K1) 1

N h₁+o( 1 N h₁).

ForI_1,3, the same calculation applies and we obtain I_1,3=

[(σ˜₂

˜ σ1

)2

Φ₁(K)−σ˜²₂Φ₂(K) +σ˜²₁σ˜₂² 4 Φ₃(K)

]

p_S(K)R₂(K2) 1 N h2

+o( 1 N h2

).

This completes the proof, due to(K4).

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