The critical price of the American put near maturity in the jump diffusion model

HAL Id: hal-00979936

https://hal-upec-upem.archives-ouvertes.fr/hal-00979936v2

Submitted on 7 Jul 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The critical price of the American put near maturity in the jump diffusion model

Aych Bouselmi, Damien Lamberton

To cite this version:

Aych Bouselmi, Damien Lamberton. The critical price of the American put near maturity in the

jump diffusion model. SIAM Journal on Financial Mathematics, Society for Industrial and Applied

Mathematics 2016, 7 (1), pp.236-272. �10.1137/140965910�. �hal-00979936v2�

The Critical Price of the American Put Near Maturity in the Jump Diffusion Model

^∗

Aych Bouselmi

^† and

Damien Lamberton

^†

Abstract. We study the behavior of the critical price of an American put option near maturity in the jump diffusion model when the underlying stock pays dividends at a continuous rate and the limit of the critical price is smaller than the stock price. In particular, we prove that, unlike the case where the limit is equal to the strike price, jumps can influence the convergence rate.

Key words. American put, L´ evy processes, critical price, free boundary, jump diffusion, convergence rate AMS subject classifications. 60H30, 60J75, 91G80

DOI. 10.1137/140965910

1. Introduction. The behavior of the critical price of the American put near maturity has been deeply investigated. Its limit was characterized in the Black–Scholes model (see [5, 13]) by

b(T ) := lim

t→T

b(t) = min r

δ K, K

,

where r and δ denote the interest rate and the dividend rate and b(t) is the critical price at time t.

This result was generalized to more general exponential L´ evy models in [7]. In fact, denoting ¯ d = r − δ − R

(e

^y

− 1)

⁺

ν (dy),

¹

with ν the L´ evy measure of the underlying L´ evy process, we have

b(T ) = K if d ¯ ≥ 0 and

b(T ) = ξ if d < ¯ 0, where ξ is the unique solution, in [0, K], of

(1) rK − δx −

Z

(xe

^y

− K)

⁺

ν (dy) = 0.

In the Black–Scholes model, the quantity ¯ d reduces to ¯ d = r − δ and we distinguish, according as ¯ d > 0, ¯ d = 0, and ¯ d < 0, different behaviors of the critical price near maturity.

∗Received by the editors April 21, 2014; accepted for publication (in revised form) February 2, 2016; published electronically May 5, 2016.

http://www.siam.org/journals/sifin/7/96591.html

†Universit´e Paris-Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050), UPEM, UPEC, CNRS, Projet Mathrisk INRIA, F-77454, Marne-la-vall´ee, France (aych.bouselmi@gmail.fr,damien.lamberton@u-pem.fr).

1

The quantity ¯

d

is denoted by

d⁺

in [7].

236

In fact, Barles et al in [1] (see also Lamberton [6]) established, in the case where ¯ d > 0 (which implies b(T ) = K), that

(2) K − b(t)

σK ∼

t→T

p (T − t)| ln(T − t)|, where the expression f ∼

_t→a

g (or f ∼

_a

g) is equivalent to lim

t→af(t)

g(t)

= 1. The cases ¯ d < 0 and ¯ d = 0 were investigated by Lamberton and by Villeneuve in [14] and they obtained as follows: If ¯ d = 0 (which also implies b(T ) = K)

K − b(t)

σK ∼

_t→T

p

2(T − t)| ln(T − t)|.

If ¯ d < 0 (b(T ) < K), there exists y

₀

∈ (0, 1), which is characterized thanks to an auxiliary optimal stopping problem, such that

b(T) − b(t)

σb(T ) ∼

_t→T

y

0

p (T − t).

Note that y

0

can also be characterized more explicitly as the solution of an equation (see [14]).

The critical price has also been studied in the jump diffusion model. In fact, Pham proved in [11] that the result (2), obtained in [1, 6], remains exactly the same in the jump diffusion model, in the case where ¯ d > 0 and δ = 0. This remains true if δ > 0 (see [10]).

The purpose of this paper is to study the convergence rate of the critical price of the American put, in the jump diffusion model, with ¯ d ≤ 0. Considering the results of Pham in [11], we expect to obtain the same results as the study performed by Lamberton and Villeneuve in the Black–Scholes model when ( ¯ d = r − δ ≤ 0 ), meaning that jumps do not have any influence on the convergence rate. Surprisingly, we obtain the expected result only for the case ¯ d = 0. Indeed, we obtain for ¯ d = 0 (see Theorem 5.1)

K − b(t)

σK ∼

_t→T

p

2(T − t)| ln(T − t)|, and for ¯ d < 0 (see Theorem 4.7)

b(T ) − b(t)

σb(T ) ∼

_t→T

y

_λ,β

p

(T − t),

where y

λ,β

is a real umber satisfying y

λ,β

≥ y

0

, and depending on ν({ln(K/b(T ))}) we can have y

_λ,β

> y

0

. This point will be discussed in more detail in section 4.3.

This study is composed of four sections. In section 1, we recall some useful results on the

American put which will be used throughout this study. In section 2, we give some results

on the regularity of the American put price and the early exercise premium. In section 3, we

investigate the case where the limit of the critical price is far from the singularity K (we refer

to this as the regular case). Indeed, we then have enough regularity to give an expansion of the

American put price near maturity from which the critical price behavior will be deduced. The

method is similar to the one used in [14] and is based on an expansion of the American put

price along parabolas. However, the possibility that the stock price jumps into a neighborhood of the exercise price produces a contribution of the local time in the expansion. Section 4 is devoted to the study of the case ¯ d = 0. In this case b(T) = K; hence we no longer have enough smoothness to obtain an expansion around the limit point (T, b(T )). Then we will study the behavior of the European critical price b

_e

(t) instead of b(t). Thereafter, we prove that b(t) and b

e

(t) have the same behavior.

2. Preliminaries. In the jump diffusion model, under a risk-neutral probability which is used as a pricing measure, the risky asset price is modeled by (S

_t

)

_t≥0

, given by

S

_t

= S

₀

e

^X˜t

with ˜ X

_t

= (r − δ)t + σB

_t

− σ

²

2 t + Z

_t

− t Z

(e

^y

− 1)ν(dy),

where r > 0 is the interest rate, δ ≥ 0 the dividend rate, (B

_t

)

_t≥0

a standard Brownian motion, and (Z

_t

)

_t≥0

a compound Poisson process and ν its L´ evy measure. We then have

dS

_t

= S

_t⁻

γ

₀

dt + σdB

_t

+ d Z ¯

_t

with ¯ Z

_t

= Σ

0<s≤t

(e

^∆Zs

− 1) and γ

₀

= r − δ − Z

(e

^y

− 1)ν(dy).

Denote by F the completed natural filtration of the process ˜ X

t

and suppose throughout this paper that the following assumptions are satisfied:

σ > 0, ν( R ) < ∞, Z

e

^y

ν(dy) < ∞, and d ¯ = r − δ − Z

y>0

(e

^y

− 1)ν(dy) ≤ 0.

The price of an American put with maturity T > 0 and strike price K > 0 is given, at t ∈ [0, T ], by P (t, S

_t

) with P defined for all (t, x) ∈ [0, T ] × R

⁺

by

P (t, x) = sup

τ∈T0,T−t

E

e

^−rτ

ψ(xe

^X˜τ

)

with ψ(y) = (K − y)

⁺

, where T

_0,T−t

is the set of all F -stopping times taking values in [0, T − t].

The value function P can also be characterized (see [7]) as the unique continuous and bounded solution of the variational inequality

max

ψ − P ; ∂P

∂t + AP − rP

= 0 (in the sense of distributions)

with the terminal condition P (T, .) = ψ. Here, A is the infinitesimal generator of the process S. The free boundary of this variational inequality is called the exercise boundary, and at each t ∈ [0, T ], the critical price is given by

b(t) = inf

x > 0 | P(t, x) > (K − x)

⁺

. It was proved in [7] that if ¯ d ≤ 0, then

(3) lim

t→T

b(t) = ξ := b(T ),

where ξ is the unique solution, in [0, K], of rK = δx + R

(xe

^y

− K)

⁺

ν(dy). Note that if ¯ d = 0, then b(T ) = ξ = K.

Finally, recall that the price of a European put with maturity T and strike price K is given, at time t, by

P

e

(t, x) = E

e

^−r(T−t)

(K − S

T−t

)

+

| S

0

= x

.

The quantity (P − P

e

) is called the early exercise premium; we then have P (t, x) = P

e

(t, x) + e(T − t, x). Setting θ = T − t, then the early exercise premium, e(θ, x), is characterized for the American put in the exponential Levy model as follows (see [10]):

e(θ, x) = E Z

θ

0

e

^−rs

×

rK − δS

_s^x

− Z

y>0

[P (t + s, S

_s^x

e

^y

) − (K − S

_s^x

e

^y

)] ν(dy)

1

_{S^x

s<b(t+s)}

ds

.

Here S

_s^x

= xe

^X˜s

. We also define, for all t ∈ (0, T ), the European critical price, b

_e

(t), as the unique solution of

F(t, x) = P

e

(t, x) − (K − x) = 0.

It easy to check that for all t ∈ (0, T ), b

e

(t) is well defined, b

e

(t) ∈ (0, K). It is also straight- forward that P

e

≤ P ; therefore b(t) ≤ b

e

(t) ≤ K.

3. Regularity estimate for the value function in the jump diffusion model. In this section, we study the spatial derivatives behavior of P, P

_e

, and e(θ, x) near (T, b(T)). We also give a lower bound for the second spatial derivative near (T, b(T )). These results will be proved in Appendix A.

Lemma 3.1. Under the model assumption, we have the following 1. For all x ∈ (0, b

e

(t) ∧ b(T )], we have, as θ(= T − t) goes to 0,

∂e

∂x (θ, x)

= 1 x o( √

θ)

with o( √

θ) uniform with respect to x.

2. For all x ∈ (0, b(T ) ∧ b

_e

(t)], we have

∂P

∂x (t, x) + 1 =

1 + 1 x

o(

√ θ)

with o( √

θ) uniform with respect to x.

Lemma 3.2. According to the hypotheses of the model, we have, for all b(t) ≤ x < b(T) ∧ b

e

(t) and for all θ = T − t small enough, the following inequality:

inf

b(t)<u<x

u

²

σ

²

2

∂

²

P

∂x

²

(t, u) ≥

δ ˆ − (θ)

(b(T ) − x) − λK E

σB

_θ

− ln b(T )

x

+

+ o(

√ θ)

with lim

θ↓0

(θ) = 0, ˆ δ = δ + R

{y>ln(K/b(T))}

e

^y

ν(dy), λ = ν{ln(

_b(T^K₎

)}.

4. Regular case. We begin this section by introducing an auxiliary optimal stopping problem which will be needed for deriving the expansion of the American put price near maturity along a parabolic branch. Once we have this expansion we will be able to derive the convergence rate of the critical price.

4.1. An auxiliary optimal stopping problem. Let β be a nonnegative number and (B

_s

)

s≥0

be a standard Brownian motion with local time at x denoted by ˜ L

^x

. We denote by T

_0,1

the set of all σ(B

_t

; t ≥ 0)-stopping times with values in [0, 1]. Consider also a Poisson process (N

_s

)

s≥0

, independent of B, with intensity λ; we denote by ˆ T

₁

its first jump time and by ˆ T

_0,1

the set of all σ ((N

t

, B

t

) ; t ≥ 0)-stopping times with values in [0, 1]. We define the functions υ

_λ,β

as follows:

v

λ,β

(y) = sup

τ∈Tˆ_0,1

E

e

^λτ

1

{N_τ=0}

Z

τ 0

f

λβ

(y + B

s

)ds + β

2 e

^λτ

1

{N_τ=1}

L

^−y_τ

(B) − L

^−y_ˆ

T1

(B)

,

where f

_a

(x) = x + ax

⁺

. Notice that v

_λ,β

is a nonnegative function. Moreover, we have the following.

Lemma 4.1. Define

y

_λ,β

= − inf{x ∈ R | v

_λ,β

(x) > 0}.

We have 0 < y

_λ,β

< 1 + λβ(2 + e

^λ

) and

∀y < −y

_λ,β

, v

_λ,β

(y) = 0.

We finish this subsection with an inequality, which will be used to derive a lower bound for the second derivative of P (see the proof of the upper bound in Theorem 4.7).

We define the function C on R by C(x) = x − λβ E(B

1

− x)

⁺

and we have the following lemma.

Lemma 4.2. For all x > y

_λ,β

, we have

C(x) > 0.

These results will be proved in Appendix B.

4.2. American put price expansion. Throughout this section, we assume ¯ d < 0, so that b(T ) < K . We then have enough regularity of the American put price to derive an expansion of P around b(T ) along a certain parabolic branch.

Theorem 4.3. Let a be a negative number (a < 0) and let b(T) denote the limit of b(t) when t goes to T , b(T ) = lim

t→T

b(t). If d < ¯ 0, we have

P (T − θ, b(T )e

^a

√

θ

) = (K − b(T )e

^a

√

θ

)

⁺

+ Cθ

³²

υ

_λ,β

a σ

+ o(θ

³²

), where C = σb(T )ˆ δ, with λ = ν{ln

_b(T^K₎

}, δ ˆ = δ + R

y>ln(K/b(T))

e

^y

ν(dy) and υ

λ,β

(y) as defined in the previous section with β =

^K

b(T)ˆδ

.

Remark 1. Notice that if ν does not charge ln(K/b(T )), meaning that λ = 0 and ˆ T

1

=

∞ a.s., then

υ

λ,β

(a) = υ

0

(a) = sup

τ∈T0,1

E Z

τ

0

(a + B

s

) ds

.

In this case, the American put price will have the same expansion as in the Black–Scholes model (see [9]).

The proof of Theorem 4.3 will require some estimates for the local time at K of the process (S

_t

)

t≥0

, which we denote by (L

^K_t

)

t≥0

. In fact, the main difference with the approach used in the Black–Scholes case lies in this analysis of the local time. The process L

^K

can be derived from the Ito–Tanaka formula (see [12]):

(K − S

_t

)

⁺

= (K − S

₀

)

⁺

+ Z

t

0

(−1

_{Ss≤K}

)S

_s

(γ

₀

ds + σdB

_s

) (4)

+ X

0<s≤t

(K − S

s

)

⁺

− (K − S

_s⁻

)

⁺

+ 1

2 L

^K_t

.

The following proposition provides an expansion of E L

^K_τ

for stopping times τ with values in [0, θ] with θ close to 0.

Proposition 4.4. We assume b(T ) < K . Let a be a fixed negative number, a < 0, and S

₀

= b(T )e

^a

√θ

.

• We have lim sup

θ↓0

E(L^K_θ )

θ^3/2

< ∞. Moreover, if ν{ln(

_b(T^K₎

} = 0, lim sup

θ↓0

E(L^K_θ) θ^3/2

= 0.

• If ν{ln(K/b(T))} 6= 0, then we have, for all F -stopping time τ with values in [0, θ], E L

^K_τ

= 2K E h

(−a √

θ − σB

τ

)

⁺

− (−a √

θ − σB

_Tˆ

1

)

⁺

1

_{Tˆ

1<τ}

i

+ o(θ

³²

), where T ˆ

1

= inf{s ≥ 0 ; ∆X

s

= ln(

_b(T^K₎

)} and o(θ

^3/2

) is uniform with respect to τ . For the proof of Proposition 4.4, we will need an elementary estimate for the expectation of the local time of Brownian motion.

Lemma 4.5. For all real number a and for all t > 0, we have 0 ≤ E (a − B

_t

)

⁺

− a

⁺

≤ √

t e

^−a

2

√

2t

2π .

Proof of Lemma 4.5. The first inequality follows from Jensen’s inequality. For the other inequality, we have

E (a − B

t

)

⁺

= Z

a/√

t

−∞

(a − √

ty)e

^−y2/2

dy

√ 2π

= a Z

a/√

t

−∞

e

^−y2/2

dy

√

2π + √ t e

^−a

2

√

2t

2π . Then, if a ≤ 0,

E(a − B

t

)

⁺

≤ √ t e

^−a

2

√

2t

2π . If a ≥ 0, we can write

E (a − B

t

)

⁺

− a = −a Z

+∞

a/√ t

e

^−y2/2

dy

√

2π + √ t e

^−a

2

√

2t

2π ≤ √ t e

^−a

2

√

2t

2π .

Before proving Proposition 4.4, we state and prove an estimate for E (L

^K_θ

| S

₀

= x).

Lemma 4.6. There exists a positive constant C such that for all x > 0 and for all θ > 0, we have

E L

^K_θ

| S

0

= x

≤ Cx √

θ exp

− (K − x)

²

2x

²

σ

²

θ

+ θ

.

Proof of Lemma 4.6. We will use the notation E

x

for E (. | S

0

= x). Taking expectations in (4) and using the compensation formula (see, for instance, [3]), we have

1

2 E

x

L

^K_θ

= E

x

(K − S

_θ

)

⁺

− (K − S

₀

)

⁺

+ E

x

Z

θ 0

γ

₀

S

_s

1

_{Ss≤K}

− Z

Ψ(S

_s

, y)ν(dy)

ds

,

where Ψ(x, y) = (K − xe

^y

)

⁺

− (K − x)

⁺

. We deduce easily from this equality that

1

2 E

x

L

^K_θ

= E

x

(K − S

θ

)

⁺

− (K − x)

⁺

+ xO(θ) with O(θ) independent of x. We have, with the notation ˜ Z

θ

= Z

θ

− θ R

(e

^y

− 1)ν(dy), E

x

(K − S

θ

)

⁺

− (K − x)

⁺

= E

x

(K − xe

^(r−δ−σ

2

2 )θ+σBθ+ ˜Zθ

)

⁺

− (K − x)

⁺

. We also have

E

e

^(r−δ−σ

2

2 )θ+σBθ+ ˜Zθ

− e

^σBθ

= e

^σ2θ/2

E

e

^(r−δ−σ

2

2 )θ+ ˜Zθ

− 1

= O(θ).

Therefore

E

x

(K − S

_θ

)

⁺

− (K − x)

⁺

= E (K − xe

^σBθ

)

⁺

− (K − x)

⁺

+ xO(θ)

= E (K − x(1 + σB

_θ

))

⁺

− (K − x)

⁺

+ xO(θ)

= xσ E

K − x xσ − B

θ

+

−

K − x xσ

+

!

+ xO(θ).

Hence, using Lemma 4.5 above,

E

x

(K − S

_θ

)

⁺

− (K − x)

⁺

≤ xσ p

θ/(2π) exp

− (K − x)

²

2x

²

σ

²

θ

+ xO(θ).

Proof of Proposition 4.4. Let T

1

be the first jump time of the process Z . We will use the following decomposition:

L

^K_θ

= L

^K_θ∧T1

+ L

^K_θ

− L

^K_θ∧T1

= L

^K_θ∧T1

+ 1

{T₁<θ}

L

^K_θ

− L

^K_T1

.

Estimating E L

^K_θ∧T

1

. In the stochastic interval [0, T

1

[, the process (S

t

) matches the process ( ˇ S

_t

) defined by

S ˇ

_t

= S

₀

e

^(γ0−σ

2

2 )t+σBt

.

We deduce (when observing that the process L

^K

is continuous) that L

^K_θ∧T1

= ˇ L

^K_θ∧T1

≤ L ˇ

^K_θ

,

where ˇ L

^K

is the local time at K of the process ˇ S. Note that 1

2

L ˇ

^K_θ

= (K − S ˇ

_θ

)

₊

− (K − S

₀

)

₊

− Z

θ

0

(−1

_{Sˇ

s≤K}

) ˇ S

_s

(γ

₀

ds + σdB

_s

).

As the process ( ˇ L

^K_θ

) increases only on { S ˇ

_t

= K}, we have L ˇ

^K_θ

= ˇ L

^K_θ

1

_{ˇτK<θ}

,

where ˇ τ

K

= inf {t ≥ 0; ˇ S

t

> K}. Note that ˇ τ

K

= inf {t ≥ 0; (γ

0

−

^σ₂²

)t + σB

t

> ln(K/S

0

)}, so that, with our assumptions on S

₀

, we have P (ˇ τ

_K

≤ θ) = o(θ

ⁿ

) for all n > 0. By H¨ older,

E L ˇ

^K_θ

≤ ( P (ˇ τ

_K

< θ))

^1−1p

|| L ˇ

^K_θ

||

_p

, p > 1.

We easily deduce that E L ˇ

^K_θ

= o(θ

ⁿ

) for all n > 0, so that EL

^K_θ∧T1

= o(θ

ⁿ

) for all n > 0.

Estimating E

1

{T₁<θ}

L

^K_θ

− L

^K_T

1

. By the strong Markov property, we have E

1

_{T1<θ}

L

^K_θ

− L

^K_T1

≤ E

1

_{T1<θ}

L

^K_T1+θ

− L

^K_T1

= E

1

_{T1<θ}

E

ST1

(L

^K_θ

) , (5)

where E

x

is the expectation associated to P

x

and P

x

defines the law of S

_t

when S

₀

= x.

Using Lemma 4.6, we deduce E

1

_{T1<θ}

L

^K_T1+θ

− L

^K_T1

≤ C E 1

_{T1<θ}

S

_T1

√

θ exp − (K − S

_T1

)

²

2S

_T²

1

σ

²

θ

! + θ

!!

. (6)

At this stage, we notice that P (T

1

≤ θ) = O(θ) and that, conditionally on {T

₁

≤ θ}, T

1

is uniformly distributed on [0, θ].

As Z

_T1

is independent of both T

₁

and B , we see that, conditionally to {T

₁

< θ}, S

_T1

has the same law as the random variable ζ

θ

defined by

ζ

θ

= K exp

(V − ln(K/b(T ))) +

√ θ

a +

γ

0

− σ

²

2

√

θU + σg

√ U

,

where U , g, and V are independent random variables, U is uniform on [0, 1], g is standard Gaussian, and V has the same law as Z

T1

. Therefore, the estimate (6) can be rewritten as follows:

E

1

_{T1<θ}

L

^K_T1+θ

− L

^K_T1

≤ C √

θ P (T

₁

< θ) E

ζ

_θ

exp

− (K − ζ

_θ

)

²

2ζ

_θ²

σ

²

θ

+ √

θ (7)

≤ C

√

θ P (T

₁

< θ) E

ζ

_θ

1 +

√ θ

.

(8)

Clearly, with probability one,

θ→0

lim ζ

_θ

= K exp

V − ln K

b(T )

,

and we easily deduce from (8) that E

1

{T₁<θ}

L

^K_T1+θ

− L

^K_T1

= O(θ

^3/2

). We can now conclude that E (L

^K_θ

) = O(θ

^3/2

).

Moreover, if we assume ν{ln(

_b(T^K₎

)} = 0, we have V − ln(

_b(T^K₎

) 6= 0 a.s., so that lim

_θ→0

ζ

_θ

6=

K a.s. and, by dominated convergence, lim

θ↓0

E

ζ

_θ

exp

− (K − ζ

_θ

)

²

2ζ

_θ²

σ

²

θ

= 0.

Therefore, using (7), E

1

_{T1<θ}

L

^K_T

1+θ

− L

^K_T

1

= o(θ

^3/2

), hence E

L

^K_θ

| S

0

= b(T )e

^a

√ θ

= o(θ

^3/2

).

Expansion of E L

^K_τ

, in the case where ν{ln(

_b(T^K₎

)} > 0. For the proof of the second part of the proposition, we assume ν{ln(

_b(T^K₎

)} > 0, and we introduce the processes ˆ X and ˆ Z such that

Z ˆ

t

= X

s<t

∆ ˜ X

s

1

_{{∆ ˜X}

s=ln_b(T)^K }

and X ˆ = ˜ X − Z, ˆ and ˆ T

₁

= inf {s ≥ 0, Z ˆ

_t

6= 0}.

For any stopping time with values in [0, θ], we have E L

^K_τ

= E L

^K_τ∧T1

+ E L

^K_τ

− L

^K_τ∧T1

= E L

^K_τ

− L

^K_τ∧T1

+ o(θ

^3/2

), where we have used the inequality E L

^K_τ∧T1

≤ E L

^K_θ∧T1

and the fact (observed in the first step of the proof) that E L

^K_θ∧T

1

= o(θ

^3/2

).

We now observe that since T

₁

≤ T ˆ

₁

and τ ≤ θ, 0 ≤ E

L

^K_τ∧Tˆ

1

− L

^K_τ∧T1

≤ E

L

^K_θ∧Tˆ

1

.

On the stochastic interval [0, T ˆ

₁

), the process ˜ X matches the process ˆ X whose L´ evy measure does not charge the point ln(K/b(T )). So, we have E (L

^K_θ∧ˆ

T1

) = E ( ˆ L

^K_θ∧ˆ

T1

) ≤ E ( ˆ L

^K_θ

), where L ˆ

^K

is the local time at K of the process S obtained by replacing ˜ X with ˆ X in the definition of S. Since the L´ evy measure of ˆ X does not charge ln(K/b(T )), we deduce from the above discussion that E ( ˆ L

^K_θ

) = o(θ

^3/2

). Hence

E L

^K_τ

= E

L

^K_τ

− L

^K_τ∧Tˆ

1

+ o(θ

^3/2

).

Going back to (4) and using again the compensation formula, we have 1

2 E

L

^K_τ

− L

^K_τ∧Tˆ

1

= E

(K − S

τ

)

⁺

− (K − S

_τ∧Tˆ1

)

⁺

+E

Z

τ τ∧Tˆ1

γ

0

S

s

1

{S_s≤K}

− Z

Ψ(S

s

, y)ν(dy)

ds

with Ψ(x, y) = (K − xe

^y

)

⁺

− (K − x)

⁺

. Note that

E

Z

τ τ∧Tˆ1

γ

0

S

s

1

{S_s≤K}

− Z

Ψ(S

s

, y)ν(dy)

ds

≤ E

1

_{Tˆ1<θ}

Z

θ 0

j(S

s

)ds

with j(z) = |γ

₀

|z + R

|Ψ(z, y)|ν (dy). Since the function Ψ is bounded, we easily derive E (1

_{Tˆ

1<θ}

R

θ

0

j(S

_s

)ds) = O(θ

²

), so that 1

2 E

L

^K_τ

− L

^K_τ∧ˆ

T1

= E

(K − S

_τ

)

⁺

− (K − S

_τ∧Tˆ

1

)

⁺

+ O(θ

²

)

= E h

1

_{Tˆ

1<τ}

(K − S

_τ

)

⁺

− (K − S

_Tˆ

1

)

⁺

i

+ O(θ

²

).

We now argue that up to O(θ

²

), we have at most one jump before θ. More precisely, let (N

t

)

t≥0

be the counting process of the jumps of Z, so that

N

_θ

= X

0<s≤θ

1

_{{∆ ˜X}

s6=0}

= X

s≤θ

1

_{∆Zs6=0}

.

We have P (N

_θ

≥ 2) = O(θ

²

), so that E

h 1

_{Tˆ

1<τ}

(K − S

_τ

)

⁺

− (K − S

_Tˆ

1

)

⁺

i

= E h

1

_{Tˆ1<τ,Nθ≤1}

(K − S

τ

)

⁺

− (K − S

Tˆ1

)

⁺

i

+ O(θ

²

).

On { T ˆ

₁

< τ, N

_θ

≤ 1}, we have, for ˆ T

₁

≤ s ≤ θ,

S

_s

= S

₀

e

^{Xˆs+ ˆZTˆ1}

= S

₀

e

^Xˆs

K/b(T ) = Ke

^a

√

θ+ ˆXs

= Ke

^a

√

θ+µs+σBs

,

where µ = γ

0

−

^σ₂²

. Therefore 1

2 E

L

^K_τ

− L

^K_τ∧Tˆ

1

= K E h

1

_{Tˆ

1<τ}

(1 − e

^a

√

θ+µτ+σBτ

)

⁺

− (1 − e

^a

√

θ+µTˆ1+σBTˆ1

)

⁺

i

+ O(θ

²

)

= K E h

(−a √

θ − µτ − σB

τ

)

⁺

−(−a √

θ − µ T ˆ

1

− σB

_Tˆ

1

)

⁺

1

_{Tˆ

1<τ}

i

+O(θ

²

)

= K E

h (−a √

θ − σB

_τ

)

⁺

− (−a √

θ − σB

_Tˆ

1

)

⁺

1

_{Tˆ

1<τ}

i

+ O(θ

²

).

The last two equalities follow from P ( ˆ T

₁

< τ ) = O(θ),

(1 − e

^x

+ x)1

_{x≤0}

≤

^x₂²

, and the fact

that E (B

_τ

)

²

≤ θ.

Proof of Theorem 4.3. First of all, we recall our notation ˇ X

_t

= ˜ X

_t

− Z

_t

, ˇ S

_t

= ˜ S

_t

/e

^Zt

, and T

1

the first jump time T

1

= inf{t > 0|Z

_t

6= 0}, and from now on we consider S

0

as a function of θ. More precisely, we denote by S

₀^θ

= b(T )e

^a

√θ

= e

^x0+a

√θ

with a < 0 and x

₀

= ln(b(T )).

We deduce from (4) and the compensation formula that for all stopping times τ ∈ T

_0,θ

, E

e

^−rτ

(K − S

_τ

)

⁺

− (K − S

₀

)

⁺

= E Z

τ

0

e

^−rs

1

{S_s≤K}

−rK + δS

s

+ S

s

Z

(e

^y

− 1)ν(dy)

+ e

^−rs

Z

(K − S

_s

e

^y

)

⁺

− (K − S

_s

)

⁺

ν(dy)

ds

+ 1

2 E Z

τ

0

e

^−rs

dL

^K_s

= I

^a

(τ ) + J

^a

(τ ), (9)

where, with the notation Ψ(x, y) = (K − xe

^y

)

⁺

− (K − x)

⁺

, I

^a

(τ ) = E

Z

τ 0

e

^−rs

1

_{Ss≤K}

−rK + δS

_s

+ S

_s

Z

(e

^y

− 1)ν(dy)

+ e

^−rs

Z

Ψ(S

_s

, y)ν(dy)

ds and

J

^a

(τ ) = 1 2 E

Z

τ 0

e

^−rs

dL

^K_s

.

Note that since τ ≤ θ, E R

τ

0

(1 − e

^−rs

)dL

^K_s

≤ rθE(L

^K_θ

), so that using Proposition 4.4, J

^a

(τ ) = 1

2 E L

^K_τ

+ O(θ

¹⁺³²

)

= KE h

(−a √

θ − σB

τ

)

⁺

− (−a √

θ − σB

Tˆ1

)

⁺

1

_{Tˆ1<τ}

i

+ o(θ

³²

).

(10)

Estimating I

^a

. First of all, note that we have E

Z

τ 0

e

^−rs

1

_{Ss>K}

Z

(K − S

_s

e

^y

)

⁺

− (K − S

_s

)

⁺

ν(dy)

ds

≤ Kν(R) Z

θ

0

P {S

_s

> K}ds

≤ Kν( R ) Z

θ

0

P {S

_s

> K, T

₁

> θ} + P {S

_s

> K, T

₁

≤ θ}ds

≤ Kν( R ) Z

θ

0

P { S ˇ

s

> K}ds + θ P {T

₁

≤ θ}

= O(θ

²

).

Here, we have used the fact that with the notation ˇ τ

K

= inf{t ≥ 0; ˇ S

t

> K }, P( ˇ S

s

> K ) = P (ˇ τ

_K

≤ s) ≤ P (ˇ τ

_K

≤ θ) = o(θ

ⁿ

) for all n > 0, as observed in the proof of Proposition 4.4.

We can now write I

^a

(τ ) = E

Z

τ 0

e

^−rs

1

_{Ss≤K}

−rK + δS

s

+ Z

(S

s

(e

^y

− 1) + Ψ(S

s

, y)) ν (dy)

ds

+ O(θ

²

)

= E Z

τ

0

e

^−rs

1

_{Ss≤K}

−rK + δS

_s

+ Z

(S

_s

e

^y

− K)

⁺

ν (dy)

ds

+ O(θ

²

),

where the last equality follows from 1

{x≤K}

x(e

^y

− 1) +

(K − xe

^y

)

⁺

− (K − x)

⁺

= (xe

^y

− K)

⁺

1

{x≤K}

.

We can also omit e

^−rs

in the expression as an error of the order of O(θ

²

). Then we obtain, for all stopping times τ with values in [0, θ],

I

^a

(τ ) = E Z

τ

0

1

_{Ss≤K}

−rK + δS

_s

+ Z

(S

_s

e

^y

− K)

⁺

ν (dy)

ds

+ O(θ

²

).

We denote

h(x) = −rK + δe

^x

+ Z

(e

^x

e

^y

− K)

⁺

ν(dy) and recall that S

t

= S

₀^θ

e

^X˜t

= b(T )e

^a

√

θ+ ˜Xt

= b(T)e

^X˜a

√ θ

t

= e

^{x0+ ˜Xa}

√ θ

t

, where ˜ X

_t^y

= y + ˜ X

t

. We thus have

I

^a

(τ ) = E Z

τ

0

1

_{a^√_{θ+ ˜X}

s≤ln_b(T)^K }

h(x

0

+ a

√

θ + ˜ X

s

)ds

| {z }

(I)

+o(θ

³²

).

(11)

Now, we will try to express the quantity (I ) under a more appropriate form. The first step is to neglect the contribution of the finite variation part of the process ˜ X. Notice that

1

_{x≤ln(K)}

h(x)

≤ K(r ∨ | d|) ¯ and |h(x) − h(y)| ≤ |e

^x

− e

^y

|

δ + Z

e

^y

ν(dy)

.

Moreover, for all (x, y) ∈ R

²

, we have

1

_{x≤ln(K)}

h(x) − 1

_{y≤ln(K)}

h(y)

=

(h(x) − h(y)) 1

{x∨y≤ln(K)}

+ h(x)1

{x≤ln(K)<y}

− h(y)1

{y≤ln(K)<x}

≤ A

₀

|e

^x

− e

^y

| 1

{x∨y≤ln(K)}

+ A

₁

1

{ln(K)<y}

+ 1

{ln(K)<x}

, where A

₁

= K(r ∨ | d|) and ¯ A

₀

= δ + R

e

^y

ν(du). Let k

_b

= ln(

_b(T^K₎

) > 0 and recall that X ˜

t

− σB

t

= (γ

0

−

^σ₂²

)t + Z

t

; then

1

_{x

0+a√

θ+ ˜Xs≤lnK}

h(x

₀

+ a

√

θ + ˜ X

_s

) − 1

_{x

0+a√

θ+σBs≤lnK}

h(x

₀

+ a

√

θ + σB

_s

)

≤ A

0

e

^x0+a

√

θ+ ˜Xs

− e

^x0+a

√ θ+σBs

1

_{X˜s∨σBs≤kb−a√

θ}

+ A

1

(1

_{k

b−a√

θ<σBs}

+ 1

_{k

b−a√ θ<X˜s}

)

≤ A

0

b(T )e

^σBs

e

^(γ0−σ

2

2 )s+Zs

− 1

+ A

1

(1

_{kb<σBs}

+ 1

_{k

b<X˜s}

), where the last inequality is due to a < 0 and e

^x0

= b(T ).

Note that for all s ∈ [0, θ],

P (k

_b

< σB

_s

) ≤ P k

_b

σ √ θ < B

₁

≤ C

√ θe

⁻

k2 b 2σ2θ

.

Moreover, for θ small enough, we have

^k₂^b

< k

b

− (γ

0

−

^σ₂²

)s, so that

P (k

_b

< X ˜

_s

) ≤ P

k

_b

− (γ

₀

−

^σ₂²

)s σ √

θ < B

₁

!

+ P (T

₁

≤ θ) ≤ C

√ θe

⁻

k2 b 8σ2θ

+ Aθ and

E

e

^σBs

e

^(γ0−σ

2

2 )s+Zs

− 1

≤ e

^σ

2 2 s

e

^(r−δ−σ

2 2 )s

− 1

+ e

^σ

2 2 s

E e

^Zs−s

R(e^y−1)ν(dy)

− 1 ≤ Dθ.

Hence,

Z

θ 0

E

1

_{x

0+a√

θ+ ˜Xs≤lnK}

h(x

0

+ a

√

θ + ˜ X

s

)

− 1

_{x

0+a√

θ+σBs≤lnK}

h(x

0

+ a

√

θ + σB

s

)

ds = O(θ

²

).

Thanks to this estimation, (11) becomes I

^a

(τ ) = E

Z

τ 0

1

_{ξ^a,θ

s ≤ln ^K

b(T)}

h(x

0

+ ξ

_s^a,θ

)ds

+ o(θ

³²

), (12)

where

ξ

_s^a,θ

= a √

θ + σB

_s

.

We will now use an expansion of h around x

0

. When h is differentiable at x

0

, we have h(x

0

+ y) = h(x

0

) + yh

⁰

(x

0

) + o(y). As we will see, when ν{ln(K/b(T ))} >, there is a jump in the derivative that modifies the expansion.

The function h is convex; therefore it has right-hand and left-hand derivatives everywhere.

Particularly, we have, for x < ln(K ), h

⁰_g

(x) = e

^x

δ +

Z

e

^y

1

{y>ln(K)−x}

ν(dy)

and

h

⁰_d

(x) = e

^x

δ + Z

e

^y

1

{y≥ln(K)−x}

ν (dy)

.

Hence, we can write

h

⁰_d

(x

₀

)(x − x

₀

)

⁺

− h

⁰_g

(x

₀

)(x − x

₀

)

⁻

≤ h(x) − h(x

₀

) ≤ h

⁰_g

(x)(x − x

₀

)

⁺

− h

⁰_d

(x)(x − x

₀

)

⁻

, and hence

0 ≤ h(x) − h(x

₀

) + h

⁰_d

(x

₀

)(x − x

₀

)

⁺

− h

⁰_g

(x

₀

)(x − x

₀

)

⁻

≤ h

⁰_g

(x) − h

⁰_d

(x

₀

)

(x − x

₀

)

⁺

+ h

⁰_g

(x

₀

) − h

⁰_d

(x)

(x − x

₀

)

⁻

= h

⁰_g

(x ∨ x

₀

) − h

⁰_d

(x ∧ x

₀

)

|x − x

₀

|.

Thanks to the equation characterizing b(T ) when ¯ d < 0, we have h(x

₀

) = h(ln(b(T )) = 0. We thus obtain, by setting ∆h

⁰

(x

₀

) = h

⁰_d

(x

₀

) − h

⁰_g

(x

₀

),

h(x

0

+ x) = ∆h

⁰

(x

0

)x

⁺

+ h

⁰_g

(x

0

)x + |x| R(x), ˜ where ˜ R(x) −→

_x→0

0, and

0 ≤ R(x) ˜ ≤ h

⁰_g

(x

0

+ x

⁺

) − h

⁰_d

(x

0

− x

⁻

)

≤ L (1 + e

^x

) , with L a positive constant. We can then write

1

_{ξ^a,θ

s ≤ln_b(T)^K }

h(x

0

+ ξ

_s^a,θ

)

=

∆h

⁰

(x

0

)(ξ

_s^a,θ

)

⁺

+ h

⁰_g

(x

0

)ξ

_s^a,θ

1 − 1

_{ξa,θ

s >ln_b(T)^K }

+

ξ

_s^a,θ

R(ξ ˜

^a,θ_s

)1

_{ξa,θ

s ≤ln_b(T)^K }

. (13)

We claim that

E Z

θ

0

ξ

_s^a,θ

R(ξ ˜

_s^a,θ

)1

_{ξa,θ s ≤ln ^K

b(T)}

ds = o(θ

³²

) (14)

and

E Z

_θ

0

∆h

⁰

(x

₀

)(ξ

_s^a,θ

)

⁺

+ h

⁰_g

(x

₀

)ξ

_s^a,θ

1

_{ξa,θ

s >ln_b(T)^K }

ds = o(θ

³²

).

(15)

Indeed, for (14), we have, by setting s = uθ and using the fact that ξ

_θu^a,θ

has the same distribution as √

θξ

^a,1_u

, E

Z

θ 0

ξ

_s^a,θ

R(ξ ˜

^a,θ_s

)1

_{ξa,θ s ≤ln ^K

b(T)}

ds

= θ

³²

Z

₁

0

E

ξ

_u^a,1

R( ˜

√

θξ

_u^a,1

)1

_{^√_θξ^a,1

u ≤ln ^K

b(T)}

du.

As | R(x)| ≤ ˜ L(e

^x

+ 1) and | R(x)| −→ ˜

x→0

0, we have by dominated convergence lim

θ↓0

Z

1 0

E h

ξ

^a,1_u

R( ˜

√ θξ

^a,1_u

)

i du

= 0.

This proves (14). For (15), we have, for some positive constant C, E

Z

_θ

0

∆h

⁰

(x

₀

)(ξ

_s^a,θ

)

⁺

+ h

⁰_g

(x

₀

)ξ

_s^a,θ

1

_{ξa,θ

s >ln_b(T)^K }

ds

≤ C

√ θ

Z

θ 0

E

|a| + σ r s

θ |B

₁

|

1

_{a+σ

√

s/θB1>^√1

θln_b(T)^K }

ds

≤ Cθ

³²

q

E (|a| + |B

₁

|)

²

s

P

a + σB

₁

> 1

√ θ ln K b(T )

= O(θ

ⁿ

).

Going back to (13), we deduce from (14) and (15) I

^a

(τ ) = h

⁰_g

(x

0

) E

Z

τ 0

ξ

_s^a,θ

ds + ∆h

⁰

(x

0

) E Z

τ

0

ξ

_s^a,θ

+

ds + o(θ

³²

),

= b(T )ˆ δ E Z

τ

0

a √

θ + σB

_s

+ λβ a √

θ + σB

_s

+

ds

+ o(θ

³²

) (16)

with ˆ δ = δ + R

y>ln_b(T)^K

e

^y

ν(dy), β =

^K

b(T)ˆδ

, λ = ν{ln

_b(T^K₎

} and we recall that h

⁰_g

(x

0

) = b(T)ˆ δ and ∆h

⁰

(x

₀

) = Kν {ln

_b(T^K₎

}, so that λβ =

^∆h_h0^0(x0)

g(x0)

.

Going back to (9) and using (10) and (16), we obtain E e

^−rτ

(K − S

_τ

)

⁺

= (K − S

₀

)

⁺

+ E

b(T )ˆ δ Z

τ

0

a √

θ + σB

_s

+ λβ(a √

θ + σB

_s

)

⁺

ds

+K1

_{Tˆ1<τ}

(a

√

θ + σB

τ

)

⁺

− (a

√

θ + σB

Tˆ1

)

⁺

+ o(θ

^3/2

) with o(θ

^3/2

) uniform with respect to τ . Hence

P (T − θ, b(T)e

^a

√

θ

) = (K − b(T )e

^a

√

θ

)

⁺

+ σb(T )ˆ δ v ¯

_λ,β,θ

(a/σ) + o(θ

^3/2

), where ¯ v

_λ,β,θ

is defined by

¯

v

λ,β,θ

(y) = sup

τ∈T_0,θ

E Z

τ

0

f

λβ

(y

√

θ + B

s

)ds + β1

_{Tˆ1<τ}

(y

√

θ + B

τ

)

⁺

− (y

√

θ + B

Tˆ1

)

⁺

with f

a

(x) = x + ax

⁺

. Recall that β = K/(b(T )ˆ δ). To simplify the expression of ¯ v

_λ,β,θ

, we notice first that if we set B

_t^θ

= B

_θt

/ √

θ, we can write

¯ v

_λ,β,θ

=

√ θ sup

τ∈T0,θ

E Z

τ

0

f

_λβ

(y + B

^θ_s/θ

)ds + β1

_{Tˆ

1<τ}

(y + B

_{τ /θ}^θ

)

⁺

− (y + B

_T^θ_ˆ

1/θ

)

⁺

= √ θ sup

τ∈T_0,θ

E θ Z

τ /θ

0

f

_λβ

(y + B

_s^θ

)ds + β1

_{Tˆ

1<τ}

(y + B

_{τ /θ}^θ

)

⁺

− (y + B

_T^θ_ˆ

1/θ

)

⁺

! .

We also notice that τ ∈ T

_0,θ

if and only if τ /θ ∈ T

_0,1^θ

, where T

_0,1^θ

is the set of the stopping times of the filtration (F

_θt

)

t≥0

, with values in [0, 1], so that

¯

v

_λ,β,θ

= √ θ sup

τ∈T_0,1^θ

E

θ Z

τ

0

f

_λβ

(y + B

_s^θ

)ds + β1

_{Tˆ

1<θτ}

(y + B

^θ_τ

)

⁺

− (y + B

^θ_ˆ

T1/θ

)

⁺

.

Note that ¯ v

_λ,β,θ

(y) does not change if we replace T

_0,1^θ

by ˆ T

_0,1

the set of the stopping times of the natural filtration of the couple (B

^θ_t

, N ˆ

_θt

), where ˆ N is defined by

N ˆ

_t

= X

0<s≤t

1

_{{∆Zs=ln(K/b(T))}}

.

The process ( ˆ N

θt

)

t≥0

is a Poisson process with intensity θλ, where λ = ν{ln(K/b(T ))}. Under the probability ˆ P , defined by

d P ˆ

dP = θ

^Nˆ1

e

^{−λ(θ−1)}

,

the process (B

t

, N ˆ

t

)

0≤t≤1

has the same law as (B

_t^θ

, N ˆ

θt

)

0≤t≤1

. Observe that (θ

^Nˆt

e

^{−λt(θ−1)}

)

t≥0

is a martingale. Hence,

¯

v

_λ,β,θ

(y) = √ θ sup

τ∈T0,1

E

θ

^Nˆ1

e

^{−λ(θ−1)}

θ Z

τ

0

f

_λβ

(y + B

_s

)ds + β1

_{Tˆ

1<τ}

× ·

(y + B

_τ

)

⁺

− (y + B

_Tˆ

1

)

⁺

i

=

√ θ sup

τ∈T0,1

E

θ

^Nˆτ

e

^{−λτ(θ−1)}

θ Z

τ

0

f

λβ

(y + B

s

)ds + β

2 1

_{Tˆ1<τ}

L

^−y_τ

(B) − L

^−y_ˆ

T1

(B)

,

where L

^−y

(B) denotes the local time of B at −y. We have for τ ∈ T

_0,1

, E

θ

^Nˆτ

e

^{−λτ(θ−1)}

θ Z

_τ

0

f

_λβ

(y + B

_s

)ds

= θ E

1

_{Nˆ

τ=0}

e

^{−λτ(θ−1)}

Z

τ

0

f

_λβ

(y + B

_s

)ds

+θR

_τ

, and if θ ≤ 1

|R

_τ

| ≤ θ E

1

_{Nˆ

τ≥1}

e

^{−λτ(θ−1)}

Z

1

0

|f

_λβ

(y + B

_s

)|ds

= O(θ).

Hence, E

θ

^Nˆτ

e

^{−λτ(θ−1)}

θ Z

τ

0

f

_λβ

(y + B

_s

)ds

= θ E

1

_{Nˆ

τ=0}

e

^λτ

Z

τ

0

f

_λβ

(y + B

_s

)ds

+ O(θ

²

).

Besides, E

h

θ

^Nˆ1

e

^{−λ(θ−1)}

1

_{Tˆ

1<τ}

L

^−y_τ

(B) − L

^−y_ˆ

T1

(B) i

= E h

θ

^Nˆτ

e

^{−λτ(θ−1)}

1

_{Tˆ

1<τ}

L

^−y_τ

(B) − L

^−y_ˆ

T1

(B ) i

= θ E h

e

^λτ

1

_{Nˆ

τ=1}

L

^−y_τ

(B ) − L

^−y_ˆ

T1

(B) i

+ O(θ

²

).

We then have

¯

v

_λ,β,θ

(y) = θ

^3/2

v

_λ,β

(y) + o(θ

^3/2

) with

v

_λ,β

(y) = sup

τ∈T0,1

E

e

^λτ

1

_{Nˆ

τ=0}

Z

τ 0

f

_λβ

(y + B

_s

)ds + β 2 e

^λτ

1

_{Nˆ

τ=1}

L

^−y_τ

(B) − L

^−y_ˆ

T1

(B) .

Finally, we obtain P (T − θ, b(T )e

^a

√

θ

) − (K − b(T)e

^a

√

θ

) = θ

³²

(σb(T)ˆ δ)υ

_λ,β

a σ

+ o(θ

³²

).

4.3. Convergence rate of the critical price. Thanks to the expansion given in Theo- rem 4.3, we are now able to state the first main result of this paper.

Theorem 4.7. Under the hypothesis of the model and d < ¯ 0, we have as follows.

If ν{ln

_b(T^K₎

} = 0, then we have

t→T

lim

b(T ) − b(t) σb(T ) p

(T − t) = y

0

with y

0

= − sup{x ∈ R ; v

0

(x) = sup

_τ∈T0,1

E ( R

_τ

0

(x + B

s

)ds) = 0}.

If ν{ln

_b(T^K₎

} > 0, we then have

t→T

lim

b(T ) − b(t) σb(T) p

(T − t) = y

_λ,β

with y

λ,β

as defined in Lemma 4.1, with

λ = ν

ln K b(T )

, β = K

b(T )ˆ δ and δ ˆ = δ + Z

y>ln(K/b(T))

e

^y

ν (dy).

Proof of Theorem 4.7.

According to Theorem 4.3, we have for all a < 0, P (T − θ, b(T )e

^a

√

θ

) = (K − b(T )e

^a

√

θ

)

⁺

+ Cθ

³²

υ

λ,β

a σ

+ o(θ

³²

).

Lower bound for b(T ) − b(t). Specifically, we have for all a > −σy

_λ,β

, where y

_λ,β

is defined by Lemma 4.1,

υ

_λ,β

a σ

> 0;

we thus obtain for θ close to 0,

P(t, b(T )e

^a

√θ

) > (K − b(T )e

^a

√θ

), and then

ln(b(T)) + a

√

θ > ln(b(t)), hence

b(T ) − b(t) b(t) √

θ > −a.

Noting that since r > 0 we have b(T ) > 0, and by making t tend to T, then a to −σy

_λ,β

, we obtain

lim inf

t→T

b(T ) − b(t) b(T) √

T − t ≥ σy

_λ,β

.