Simulation of Lévy processes and option pricing

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Simulation of Lévy processes and option pricing

El Hadj Aly Dia

To cite this version:

El Hadj Aly Dia. Simulation of Lévy processes and option pricing. The Journal of Computational

Finance, Incisive Media, 2013, 17 (é), pp.41-69. �hal-00551972v4�

EL HADJ ALY DIA^∗

Abstract. We approximate an infinite activity L´evy process by either truncating its small jumps or replacing them by a Brownian motion with the same variance. Then we derive the errors resulting from these approximations for some exotic options (Asian, barrier, lookback and American). We also propose a simple method to evaluate these options using the approximated L´evy process.

Key words. Exponential L´evy model, Exotic options, Monte-Carlo method AMS subject classifications. 60G51, 65C05, 65N15, 60J75, 91G20 JEL classification. C02, C63, G13

1. Introduction. General exponential L´ evy models (see [2, 5, 11]) are widely used, nowadays, in option valuation. Many numerical methods based on Fourier anal- ysis have been subsequently developed to evaluate exotic options (see [4, 12, 13, 16]).

However, in many situations, Monte-Carlo methods have to be used, especially when the underlying L´ evy process has an infinite L´ evy measure. Because the simulation of such a L´ evy process is not straightforward, except in some special cases (for example gamma or inverse Gaussian processes), in practice, the small jumps of the L´ evy pro- cess are either truncated or replaced by a Brownian motion with the same variance.

The latter approach was introduced by Asmussen and Rosinsky, who showed that, un- der suitable conditions, the normalized cumulated small jumps asymptotically behave like a Brownian motion (see [1]).

Many others authors were also interested in the issue of small jump approxi- mations. We can mention Cont and Tankov (2004), Cont and Voltchkova (2005), Kohatsu-Higa and Tankov (2010), Signahl (2003) and Rydberg (1997). To our best knowledge, the behaviour of the errors resulting from these approximations for path- dependent options has not been studied yet.

The purpose of this article is to derive bounds for the errors generated by these two methods of approximation in the valuation of exotic options (Asian, barrier, lookback, and American options) in exponential L´ evy models, and to propose a new method to evaluate these options.

The paper is organized as follows. In the next section, we recall some basic facts about real L´ evy processes and give some useful notations. In section 3 we will study the errors resulting from the small jump approximations for Asian, barrier, lookback, and American options. The results of this section are the applications of [10]. In Section 4 we will propose a new method to evaluate exotic options when the underlying L´ evy process has an infinite L´ evy measure. The main idea is to reduce the valuation of these options to the simulation of the big jumps. The last section is devoted to numerical applications. We will give an example of pricing using our method and we will study the optimality of the bounds derived in Section 3.

2. Preliminaries. Consider a filtered probability space Ω, F, (F

t

)

_t∈[0,T]

, P .

∗Universit´e Paris-Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR CNRS 8050, 5 bd. Descartes, Champs-sur-Marne,77454Marne-la-Vall´ee, France (dia.eha@gmail.com).

1

The distribution of a L´ evy process X is characterized by its generating triplet (γ, σ

²

, ν) where (γ, σ) ∈ R × R

⁺

, and ν is a Radon measure on R \{0} satisfying

Z

R

1 ∧ x

²

ν (dx) < ∞.

By the L´ evy-Itˆ o decomposition, X can be written in the form X

_t

= γt + σB

_t

+

Z

|x|>1,s∈[0,t]

xJ

_X

(dx × ds) + lim

δ↓0

Z

δ≤|x|≤1,s∈[0,t]

x J e

_X

(dx × ds) (2.1) Here J

X

is a Poisson measure on R × [0, ∞) with intensity ν(dx)dt, J e

X

(dx × ds) = J

X

(dx × ds) − ν(dx)ds and B is a standard Brownian motion. The process X has finite activity if ν( R ) < ∞. If ν( R ) = +∞, the process X is called an infinite activity L´ evy process. Given > 0, we define the process R

by

R

_t

= Z

0≤|x|≤,s∈[0,t]

x J e

X

(dx × ds), t ≥ 0. (2.2) Note that we have

E R

_t

= 0

Var (R

_t

) = σ

^X

()

²

t, where

σ

^X

() = s Z

|x|≤

x

²

ν(dx).

The process X

is then defined by

X

_t

= X

_t

− R

_t

, t ≥ 0. (2.3)

We also define the processes X ˆ

by

X ˆ

_t

= X

_t

+ σ() ˆ W

t

, t ≥ 0, (2.4) where W ˆ is a standard Brownian motion independent of X. Set, for any t ≥ 0,

M

_t^X

= sup

0≤s≤t

X

s

, M

_t^,X

= sup

0≤s≤t

X

_s

, M ˆ

_t^,X

= sup

0≤s≤t

X ˆ

_s

.

The following notations will be used for the results in the next section. We define σ

^X₀

() = max σ

^X

(),

. β

^X

() =

R

|x|<

x

⁴

ν(dx) (σ

0

())

⁴

β

₁^X,t

() = β

^X

()

¹⁶

s

log t

β ()

¹³

+ 3

+ 1

!

β

_p,θ^X

() = β

^X

()

p+4θ^pθ

log 1 (β

^X

())

pθ p+4θ

!!

^p

β ˜

_ρ,θ^X

() = β

^X1 ρ−1,θ

().

When there is no ambiguity we can remove the super index X. Recall that Asmussen and Rosinski proved ([1], Theorem 2.1) that, if X is a L´ evy process, then the process σ()

⁻¹

R

converges in distribution to a standard Brownian motion, when → 0, if and only if for any k > 0

→0

lim

σ (kσ() ∧ )

σ () = 1. (2.5)

This result is implied by the condition

→0

lim σ()

= +∞. (2.6)

The conditions (2.5) and (2.6) are equivalent, if ν does not have atoms in some neigh- borhood of zero ([1], Proposition 2.1). On the other hand we approximate X by X ˆ

when the condition (2.6) is satisfied. In this case the functions β , β

1

, β

p,θ

and β ˜

ρ,θ

converge to 0 when goes to 0.

Remark 2.1. Note that, we always have lim

→0

σ() = 0, because all L´ evy pro- cesses satisfy the condition R

R

1 ∧ |x|

²

ν(dx) < ∞.

3. Error bounds for exotic options. Let (S

_t

)

_t∈[0,T]

be the price of a security.

The σ-algebra F

t

will represent here the historical information on the price until time t. Under the exponential L´ evy model, the process S is an exponential of a L´ evy process

S

t

= S

0

e

^Xt

, ∀t ≥ 0.

The considered probability will be a risk-neutral probability, under which the process e

^−(r−δ)t

S

t

t∈[0,T]

is a martingale. The parameter r is the risk-free interest rate, and δ is the dividend rate. The options we will consider in the sequel will have as underlying the asset with price S. The option price will be denoted by V .

To compute V by Monte Carlo methods, we need to simulate many paths of X.

This is quite difficult, in general, for infinite activity L´ evy processes. In practice we approximate X by a finite activity L´ evy process, by truncating its small jumps (and get X

) or replacing them by a Brownian motion with the same variance (and get X ˆ

). The simulations of X

and X ˆ

can be quite different in terms of computation time if the initial L´ evy process has no Brownian component. This is why one needs to know what is the best approximation. If the condition (2.6) is satisfied, it is better to approximate X by X ˆ

, otherwise the errors generated by the two approximations will be similar.

Set V

(resp. V ˆ

) as the price of the option obtained by replacing X by X

(resp. X ˆ

). We will call V

(resp. V ˆ

) the approximated price by truncation (resp.

the approximated price by Brownian approximation). The resulting errors will be expressed in terms of σ

0

() = max (σ(), ).

The behaviour of σ() when goes to 0 depends on the L´ evy measure. If there exists a real α such that the L´ evy measure behaves like

_|x|¹1+α

when x goes to 0, then σ() behaves like

^1−α2

when goes to 0. This is the case for CGMY, normal inverse gaussian and varince gamma processes.

Remark 3.1. Notice that for all infinite activity L´ evy processes we have

_σ()

is

bounded.

Call fixed strike floating strike Arithmetic mean

1 T

R

T

0

S

0

e

^Xs

ds − K

⁺

S

_T

−

_T¹

R

T

0

S

₀

e

^Xs

ds

+

Geometric mean S

0

exp

1 T

R

T 0

X

s

ds

− K

+

S

T

− S

0

exp

1 T

R

T

0

X

s

ds

+ Table 3.1

Payoffs of Asian call options.

Put fixed strike floating strike

Arithmetic mean

K −

_T¹

R

T

0

S

0

e

^Xs

ds

⁺

1 T

R

T

0

S

0

e

^Xs

ds − S

T

+

Geometric mean

K − S

0

exp

1

T

R

T 0

X

s

ds

+

S

0

exp

1 T

R

T 0

X

s

ds

− S

T

⁺

Table 3.2

Payoffs of Asian put options.

3.1. Asian options. The payoffs of arithmetic and geometric Asian options are given in Table 3.1 and Table 3.2. We will only consider the arithmetic case. The geometric case can be deduced using the same arguments.

Proposition 3.2. Let X be an infinite activity L´ evy process with generating triplet (γ, σ

²

, ν). We assume there exists p > 1 such that E e

^pMT

< ∞. Then, the price of a continuous arithmetic Asian option with fixed strike and its approximation by truncation satisfy

V = V

+ O (σ

₀

()) .

Proof. To simplifiy the proof we assume that r = 0 and S

₀

= 1. We have

|V − V

| ≤ E 1 T

Z

T 0

e

^Xs

− e

^Xs

ds

= E 1 T

Z

T 0

|X

_s

− X

_s

| e

^X˜s

ds,

where X ¯

_s

is between X

s

and X

_s

. Define p

^∗

such that

¹_p

+

_p¹∗

= 1. We have

|V − V

| ≤ E sup

0≤s≤T

|R

_s

| max

e

^MT

, e

^MT

≤ E

sup

0≤s≤T

|R

_s

|

^p∗

_p¹∗

E max

e

^pMT

, e

^pMT

¹_p

≤ p

^∗

p

^∗

− 1 E

|R

_T

|

^p∗

_p¹∗

E e

^pMT

+ E e

^pMT

¹_p

= O (σ

₀

()) , by Proposition 1 and Lemma 1 of [10].

Proposition 3.3. Let X be an infinite activity L´ evy process with generating

triplet (γ, σ

²

, ν). We assume that there exists p > 1 such that E e

^pMT

< ∞. Then

the price of a continuous arithmetic Asian option with fixed strike and its Brownian approximation satisfy, for any θ ∈ (0, 1)

V = ˆ V

+ O

σ

0

() β

^p

p−1,θ

()

1−¹_p

.

Note that, under the condition (2.6), we have lim

→0

β

^p

p−1,θ

() = 0.

Proof. To simplifiy the proof we assume that r = 0 and S

0

= 1. Define by f the payoff function of the Asian option, and set

V

_n

= E f 1 n

n

X

k=1

e

^XkTn

!

V

_n

= E f 1 n

n

X

k=1

e

^X

kTn

+σ() ˆW_kT

n

! .

The sequence V

n

(resp. V

_n

) converges to V (resp. V

). On the other hand, for k ∈ {1, . . . , n}, we have

R

kt n

= 1

√ n

k

X

j=1

V

_jⁿ

,

where

V

_jⁿ

= √ n

R

jT n

− R

_(j−1)T n

. The r.v. V

_jⁿ

j∈{1,...,n}

are i.i.d. and have the same distribution as √ nR

t

n

. But E V

₁

= 0, and var (V

₁

) = σ()

²

t, by Theorem 1 of [20] (see pp. 163) there ex- ists positive i.i.d. r.v., (τ

_j

)

_j∈1,...,n

, and a standard Brownian motion, B, such that ˆ P

k

j=1

V

_jⁿ

, k ∈ {1, . . . , n}

and

B ˆ

τ₁+···+τk

, k ∈ {1, . . . , n}

have the same joint dis- tribution. Thus

R

_kT

n

, k ∈ {1, . . . , n}

and

B ˆ

τ1 +···+τk n

, k ∈ {1, . . . , n}

have the same joint distribution. We also have

σ() ˆ W

kT

n

, k ∈ {1, . . . , n}

=

^d

B ˆ

σ()2kT

n

, k ∈ {1, . . . , n}

Set

T

k

= τ

1

+ · · · + τ

k

n T

_k

= σ()

²

kT

n . Thus

n

X

k=1

e

^XkTn

=

^d

n

X

k=1

e

^X

kT

n

+ ˆB_Tk

n

X

k=1

e

^X

kT

n

+σ() ˆWkT n

=

^d

n

X

k=1

e

^X

kT

n

+ ˆB_T

k

.

So

|V

n

− V

_n

| ≤ E 1 n

n

X

k=1

e

^X

kTn

+ ˆB_Tk

− e

^X

kTn

+ ˆB_T

k

= E 1 n

n

X

k=1

B ˆ

T_k

− B ˆ

T_k

e

^X¯k

, where X ¯

_k

is between X

kT

n

+ ˆ B

T_k

and X

kT n

+ ˆ B

T_k

. Define p

^∗

such that

¹_p

+

_p¹∗

= 1.

We have

|V

n

− V

_n

| ≤ E 1 n

n

X

k=1

B ˆ

_Tk

− B ˆ

_T

k

p^∗

!

_p¹∗

E 1 n

n

X

k=1

e

^pX¯k

!

_p¹

≤

E sup

1≤k≤n

B ˆ

T_k

− B ˆ

T_k

p^∗

_p¹∗

E 1 n

n

X

k=1

e

^pX¯k

!

¹_p

.

But

E 1 n

n

X

k=1

e

^pX¯k

≤ E 1 n

n

X

k=1

e

^p

X_kT

n

+ ˆB_Tk

+ E

1 n

n

X

k=1

e

^p

X_kT

n

+ ˆB_T

k

= E 1 n

n

X

k=1

e

^p

X_kT

n

+R_kT

n

+ E 1

n

n

X

k=1

e

^p

X_kT

n

+σ() ˆWkT n

≤ E e

^pMT

+ E e

^pMˆT

≤ E

e

^pMt

+ e

^{pσ() sup0≤s≤TWˆs}

e

^pMT

≤ E e

^pMT

+ 2e

^p

2 2σ()²T

E e

^pMT

≤ 2e

^p

2 2σ()²T

E

e

^pMT

+ e

^pMT

.

The term in the right hand side of the last inequality is bounded by a constant, say C, thanks to Lemma 1 of [10]. So

|V

n

− V

_n

| ≤ C

E sup

1≤k≤n

B ˆ

T_k

− B ˆ

T_k

p^∗

_p¹∗

.

Hence by Theorem 3 of [10]

|V − V

| = O

σ

0

() β

^p

p−1,θ

()

1−¹_p

∀θ ∈ (0, 1).

The above results on fixed strike Asian options are, obviously applicable for float- ing strike Asian options. If the mean is geometric, we will also get the same results.

Corollary 3.4. The results of Proposition 3.2 and Proposition 3.3 are true for either arithmetic or geometric Asian options whether the strike is fixed or floating.

The proof is similar to the proofs of the above propositions.

Barrier call put

Up Out (S

_T

− K)

⁺

1 _{

S₀eMT<H

} (K − S

_T

)

⁺

1 _{

S₀eMT<H

} Up In (S

T

− K)

⁺

1 _{

S0e^MT≥H

} (K − S

T

)

⁺

1 _{

S0e^MT≥H

} Down Out (S

T

− K)

⁺

1

^{S0e^mT>H}

(K − S

T

)

⁺

1

^{S0e^mT>H}

Down In (S

T

− K)

⁺

1

^{S0e^mT≤H}

(K − S

T

)

⁺

1

^{S0e^mT≤H}

Table 3.3 Payoffs of barrier options.

3.2. Barrier options. The approximation error is bigger in the barrier case compared to the other types of option. This is due to the fact that the payoff is less smooth.

Proposition 3.5. Let X be an integrable L´ evy process with generating triplet (γ, σ

²

, ν). We assume that M

T

has a locally bounded probability density function.

Then the price of a continuous barrier option and its approximated value by truncation satisfy, for any q ∈ (0, 1)

V = V

+ O σ

₀

()

^1−q

.

The real q is used to show that the error in Proposition 3.5 is arbitrarily close to O (σ

0

()).

Proposition 3.6. Let X be an integrable L´ evy process with generating triplet (γ, σ

²

, ν). We assume that M

_T

has a locally bounded probability density function, then for any ρ, θ ∈ (0, 1), the price of a continuous barrier option and its approximated value by Brownian approximation satisfy

V = ˆ V

+ O

σ

₀

()

^1−ρ

β ˜

_ρ,θ

()

^ρ

.

Here, the presence of ρ and θ is due to the fact that we use Holder inequality in the proof of the intermediate result. The existence of a probability density function for M

_T

and its regularity are studied in [6, 9].

If the arbitrary parameter (q) in Proposition 3.5 is fixed, we can get a better rate in Proposition 3.6 by chosing ρ = q. The reverse is not true.

Lemma 3.7. If E e

^Xt

< ∞, then E

e

^Xt

− e

^Xt

≤ Cσ

0

(), where C is a constant independent of .

Proof. We have E

e

^Xt

− e

^Xt

= E

e

^Xt+Rt

− e

^Xt

= E e

^Xt

E

e

^Rt

− 1

≤ E e

^Xt

E |R

_t

| e

^(Rt)+

≤ E e

^Xt

E |R

_t

|

e

^Rt

+ 1

.

We can prove that for any β ∈ R , e

^βRδt

0≤δ≤1

is uniformly integrable, and if e

^Xs

is integrable for some s > 0, then

e

^Xtδ

0≤δ≤1

is also uniformly integrable. Then, we use H¨ older inequality and Proposition 1 of [10].

Proof of Proposition 3.5. We denote by C any constant independent of . We will only prove the Up and Out put case. The Down case works in the same way. For the call, the problem can be reduced to a put case by a measure change. And finally the In case can be deduced from the relation between In and Out options. We have

|V − V

| ≤

E e

^−rT

K − S

₀

e

^XT

⁺

1

S₀e^MT<H

− E e

^−rT

K − S

₀

e

^XT

+

1

_S0e^M T<H

≤

E e

^−rT

K − S

0

e

^XT

⁺

1

S₀eMT<H

− 1

_S0e^M T<H

+

E e

^−rT

K − S

0

e

^XT

⁺

−

K − S

0

e

^XT

+

1

_S0e^M T<H

.

We define by I

1

(resp. I

2

) the first (resp. the second) term in the right hand side of the last inequality. Set h = log

H S0

, we have

I

₁

=

E e

^−rT

K − S

₀

e

^XT

⁺

1

^MT<h

− 1

^M_T^<h

=

E e

^−rT

K − S

0

e

^XT

⁺

1

^MT<h,M_T≥h

− 1

^MT≥h,M_T<h

≤ K P [M

T

< h, M

_T

≥ h] + K P [M

T

≥ h, M

_T

< h] . So, using the proof of Proposition 5 of [10], we get for any q ∈ (0, 1)

I

1

= O σ

0

()

^1−q

. On the other hand

I

₂

≤ S

₀

E

e

^XT

− e

^XT

= O (σ

0

()) , by Lemma 3.7.

Hence, for any q ∈ (0, 1)

|V − V

| = O σ

0

()

^1−q

.

Proof of Proposition 3.6. We consider only the Up and Out put option for the reasons mentioned in the beginning of the proof of Proposition 3.5. We set

M

_Tⁿ

= sup

0≤k≤n

X

_kT

n

, M ˆ

_T^,n

= sup

0≤k≤n

X ˆ

_kT n

. By the proof of Proposition 3.3, we know that

(X

T

, M

_Tⁿ

) =

^d

(Y

T

, U

_Tⁿ

) and

X ˆ

_T

, M ˆ

_T^,n

=

^d

Y ˆ

_T

, U ˆ

_T^,n

,

where

U

_Tⁿ

= sup

0≤k≤n

X

_kT

n

+ ˆ B

T_k

, Y

T

= X

_T

+ ˆ B

T_n

U ˆ

_T^,n

= sup

0≤k≤n

X

_kT

n

+ ˆ B

T_k

, Y ˆ

_T

= X

_T

+ ˆ B

T_n

and (T

_k

)

_0≤k≤n

, (T

_k

)

_0≤k≤n

and the Brownian motion B ˆ are defined in the proof of Proposition 3.3. So the discrete versions of V and V ˆ are given by

V

ⁿ

= e

^−rT

E K − S

₀

e

^YT

⁺

1

_S0e^{U n}T<H

V ˆ

^,n

= e

^−rT

E

K − S

₀

e

^YˆT

+

1 ⁿ

S₀e^Uˆ

,n T <H

o.

We will first study the quantity

V

ⁿ

− V ˆ

^,n

as for continuous prices in the proof of Proposition 3.5, and then we use the proof of the the third result of Proposition 10 of [10] and make the limit when n goes to the infinity.

3.3. Lookback and hindsight options. We will focus only on the lookback case. For the hindsight options we will use the relations between lookback and hind- sight options. The quantity S

+

(resp. S

₋

) in Table 3.4 is the predetermined maximum

Option call put

Lookback S

_T

− min (S

₋

, S

₀

e

^mT

) max S

₊

, S

₀

e

^MT

− S

_T

Hindsight max S

+

, S

0

e

^MT

− K

+

(K − min (S

−

, S

0

e

^mT

))

⁺

Table 3.4

The payoffs of lookback and hindsight options.

(resp. minimum) of the option. We denote by V

_l^c

(S

−

) and V

_l^p

(S

+

) the call and the put prices of lookback options. We also denote by V

_h^c

(S

₊

, K) and V

_h^p

(S

₋

, K) the call and the put prices of hindsight options. So, we have the following relations between lookback and hindsight options.

V

_h^c

(S

+

, K) = V

_l^p

(max (S

+

, K)) + S

0

− e

^−rT

K V

_h^p

(S

₋

, K) = V

_l^c

(min (S

₋

, K)) − S

₀

+ e

^−rT

K.

Proposition 3.8. Let X be an infinite activity L´ evy process with generating triplet (γ, σ

²

, ν). We assume that there exists p > 1 such that R

|x|>1

e

^px

ν(dx) < ∞.

Then the price of a continuous lookback call option and its approximation satisfy

V (S

₋

) = V

(S

₋

) + O (σ

₀

()) V (S

₋

) = ˆ V

(S

₋

) + O σ

₀

()β

^T₁

()

.

Proposition 3.9. Let X be an infinite activity L´ evy process with generating triplet (γ, σ

²

, ν). We assume that there exists p > 1 such that R

|x|>1

e

^p|x|

ν(dx) < ∞.

Then the price of a continuous lookback put option and its approximation satisfy V (S

₊

) = V

(S

₊

) + O (σ

₀

())

V (S

+

) = ˆ V

(S

+

) + O

σ

0

() β

^p

p−1,θ

()

1−¹_p

∀θ ∈ (0, 1).

The above results show that replacing small jumps by a Brownian motion leads to a better approximation, if the condition (2.6) is satisfied.

Proof of Proposition 3.8. To simplify the proof we assume, with no loss of generality, that r = 0 and S

0

= 1. Recall that m

T

= − M ˜

T

. We have

|V (S

₋

) − V

(S

₋

)| ≤ E

e

^−M˜T

− e

^−M˜T

+ E

e

^XT

− e

^XT

= O (σ

0

()) , by Proposition 3 of [10] and Lemma 3.7.

On the other hand, we have

V (S

₋

) − V ˆ

(S

₋

) ≤

E

e

^−M˜T

− S

₋

+

− E

e

^−Mˆ

X,˜

T

− S

₋

⁺

+ E

e

^XT

− e

^XˆT

= O σ

0

()β

₁^T

()

, by Theorem 2 and Proposition 6 of [10].

Proof of Proposition 3.9. As in the proof of Proposition 3.8, we assume that r = 0 and S

0

= 1. We have

|V (S

+

) − V

(S

+

)| ≤ E

e

^MT

− e

^MT

+ E

e

^XT

− e

^XT

= O (σ

0

()) , by Proposition 4 of [10] and Lemma 3.7.

On the other hand, we have

V (S

₊

) − V ˆ

(S

₊

) ≤

E e

^MT

− S

₊

⁺

− E

e

^MˆT

− S

₊

+

+ E

e

^XT

− e

^XˆT

= O

σ

0

() β

^p

p−1,θ

()

1−¹_p

∀θ ∈ (0, 1),

thanks to Propositions 6 and 9 of [10].

Corollary 3.10. Propositions 3.8 and 3.9 are true for hindsight option provided that we replace, call by put, and put by call.

3.4. American options. Recall that the price of an American option with strike K and maturity T is given by

sup

τ∈T_[0,T]

E e

^−rτ

S

₀

e

^Xτ

− K

⁺

, for the call sup

τ∈T_[0,T]

E e

^−rτ

K − S

₀

e

^Xτ

⁺

, for the put,

where T

_[0,T]

is the of stopping times with values in [0, T ].

Proposition 3.11. Let X be an infinite activity L´ evy process with generating triplet (γ, σ

²

, ν). Then the price of a continuous American option and its approxima- tion by truncation satisfy

V = V

+ O (σ()) .

Proposition 3.12. Let X an infinite activity L´ evy process with generating triplet (γ, σ

²

, ν). Then of a continuous American option and its Brownian approximation satisfy

V = ˆ V

+ O (σ()) .

Proof of Proposition 3.11 and Proposition 3.12. For the put, we use the proof of Proposition 4.6 of [9] (pp. 67-68). Furthermore, using the proof of Proposition 3.5, the call price is given by

sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

⁺

,

where E ¯ is the expectation under P ¯ , an equivalent measure of P which comes from the Esscher transform with Radon-Nikodym derivative

d P ¯ d P

_F

t

= e

^Xt

E e

^Xt

.

We will denote by (¯ γ, ¯ σ

²

, ν) ¯ the generating triplet of X ˜ under P. In particular we ¯ have

¯

ν(dx) = e

^−x

ν(−dx). (3.1)

See Section 9.4 and Section 9.5 of [7] for more details. Hence

V − V ˆ

=

sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

+

− sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

+

=

sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

+

− sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

+

+

sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

⁺

− sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

⁺

.

Note by I

₁

(resp. I

₂

) the first (resp. the second) term in the right hand side of the last equality. Set

I

₁¹

= sup

τ∈T_[0,T]

e

^−rτ

E e

^Xτ

− E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

+

I

₁²

= sup

τ∈T[0,T]

e

^−rτ

E e

^Xτ

− E e

^Xτ

E ¯

S

0

− Ke

^X˜τ

⁺

.

We can show that

|I

1

| ≤

max I

₁¹

, I

₁²

. Using the proof of Proposition 4.3 of [9], we get

I

₁

= O σ()

²

. Furthermore by Remark 2 of [10], we have

I

2

= O (σ()) .

This proves the second result of Proposition 3.11. For the second result of Proposi- tion 3.12, we use the same arguments, but we use this time the proof of Proposition 4.3 of [9] by replacing X

by X ˆ

and Remark 2 of [10].

4. Simulation of infinite activity L´ evy processes. In this section we will show a new method to simulate an infinite activity L´ evy process. The first step is to approximate the L´ evy process by a finite activity L´ evy process as seen above. Then we will simulate the big jumps. Not that others methods of simulation of L´ evy processes are proposed in [7, 21, 22].

We have (see Section 2) X

_t

= X

t

− R

_t

= γt + Z

|x|>1,s∈[0,t]

xJ

X

(dx × ds) + Z

≤|x|≤1,s∈[0,t]

x J ˜

X

(dx × ds)

= γ − Z

≤|x|≤1

xν(dx)

! t +

Z

|x|>,s∈[0,t]

xJ

_X

(dx × ds)

= γ − Z

≤|x|≤1

xν(dx)

! t +

Z

x>,s∈[0,t]

xJ

_X

(dx × ds) +

Z

x<−,s∈[0,t]

xJ

_X

(dx × ds)

= γ

₀

t +

N_t⁺

X

i=1

Y

_i⁺

−

N_t⁻

X

i=1

Y

_i⁻

, where

γ

₀

= γ − Z

≤|x|≤1

xν(dx). (4.1)

The r.v. Y

_i⁺

i≥1

are i.i.d. with common law

_ν(,+∞)^ν+(dx)

, the r.v. Y

_i⁻

i≥1

are i.i.d.

with common law

^ν_ν(−∞,)^−(−dx)

. The measures ν

⁺

and ν

⁻

correspond respectively to ν restricted to (0, +∞) and (−∞, 0). The process X

is a compound Poisson process.

4.1. Valuation of barrier and lookback options. Since the supremum of

a compound Poisson process is reached at its jump times or its extremities, we can

simulate the supremum of X

without any discretization. So we can compute the

barrier and lookback options without discretizing the time axis. When we replace

X by X ˆ

, to evaluate barrier option we can use Example 6.1 of [7] to avoid time

discretization. For lookback options, we can use the jump times as discretization points of the Brownian motion. So, the supremum and the infimum of X ˆ

should be replaced by

M ¯

_T

= max

0≤j≤N_t+1

X ˆ

Tˆ_j

¯

m

_T

= min

0≤j≤N_t+1

X ˆ

Tˆ_j

, where

T ˆ

₀

= 0 T ˆ

_j

= T

_j

∧ T, and T

_j

j≥1

are the jump times of X

. We will denote by V ¯

the corresponding price of the lookback option.

Proposition 4.1. Let X be an infinite activity L´ evy process with generating triplet (γ, 0, ν). We assume that there exists p > 1 such that R

|x|>1

e

^px

ν(dx) < ∞.

Then

V ˆ

= ¯ V

+ o (σ()) ,

where V ˆ

denotes the price of the lookback option where X is replaced by X ˆ

. The condition R

|x|>1

e

^px

ν(dx) < ∞ is necessary only for the put.

Proof. We will only prove the put case (the call case is quite easy). We have

V ˆ

− V ¯

≤ S

0

E

e

^MˆT

− e

^M¯T

≤ S

0

E e

^max

(

^MˆT,M¯_T

)

M ˆ

_T

− M ¯

_T

. So using Cauchy-Schwarz inequality, we get

V ˆ

− V ¯

≤ S

0

E e

^pmax

(

^MˆT,M¯_T

)

¹_p

E

M ˆ

T

− M ¯

T

p p−1

^p−1_p

. Note that, we have

M ¯

_T

≤ M ˆ

_T

. On the other hand

M ˆ

_T

= sup

0≤s≤T

σ() ˆ W

s

+ X

_s

≤ σ() sup

0≤s≤T

W ˆ

s

+ M

_T

. So

E e

^pmax

(

^MˆT,M¯T

) ≤ 2e

^p

2σ()2T

2

E e

^pMT

.

Hence, using Lemma 1 of [10], there exists a constant C > 0 (independent of ) such that

V ˆ

− V ¯

≤ C

E

M ˆ

_T

− M ¯

_T

p p−1

^p−1_p

.

The following lemma concludes the proof.

Lemma 4.2. For any p ≥ 1 E

M ˆ

_T

− M ¯

_T

p

= o ((σ())

^p

) . The proof of Lemma 4.2 can be found in the appendix.

4.2. Valuation of Asian options. We will focus on the fixed strike Asian put option. The call case can be easily deduced. Floating Asian options, can be evaluated using fixed strike options and symmetry. Consider the following payoffs

K − 1 T

Z

T 0

S

0

e

^Xs

ds

!

⁺

, arithmetic Asian put

K − S

0

e

^T1

R

T

0 X_sds

⁺

, geometric Asian put.

We set

V

a

= e

^−rT

E K − 1 T

Z

T 0

S

0

e

^Xs

ds

!

+

V

g

= e

^−rT

E

K − S

0

e

^T1

R

T

0 X_sds

⁺

.

The L´ evy process X considered has the generating triplet (γ, 0, ν). In fact we will estimate the quantities V

_a

, V ˆ

_a

, V

_g

and V ˆ

_g

obtained by replacing X by X

or X ˆ

.

We have

V

_a

= e

^−rT

E K − 1 T

Z

T 0

S

₀

e

^Xs

ds

!

⁺

= e

^−rT

E



K − S

₀

T

N_T+1

X

j=1

Z

T^ˆ_j Tˆ_j−1

e

^Xs

ds





+

= e

^−rT

E



K − S

0

T

N_T+1

X

j=1

Z

T^ˆ_j Tˆ_j−1

e

^γ0s+

P

j−1 i=1Y_i

ds





+

, see (4.1).

So

V

_a

= e

^−rT

E



K − S

₀

T

N_T+1

X

j=1

e P

j−1 i=1Y_i

Z

T^ˆ_j

Tˆ_j−1

e

^γ0s

ds





+

= e

^−rT

E



K − S

0

T

N_T+1

X

j=1

e P

j−1

i=1Y_i

e

^γ0Tˆj

− e

^γ0Tˆj−1

γ

₀





+

= e

^−rT

E



K − S

₀

T

N_T+1

X

j=1

e

^γ0Tˆj+

P

j−1 i=1Y_i

− e

^{γ0Tˆj−1 +}

P

j−1 i=1Y_i

γ

₀





+

.