Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
Подождите немного. Документ загружается.
580 10 Mixed Processes
and N an inhomogeneous Poisson process with intensity λ. We denote by i
the indicator function
i(t)=1
{φ(t)=0}
We assume that B and N are independent and that lim
t→∞
λ(t)=∞ and
λ(t) < ∞, ∀t. The process (X
t
,t≥ 0) defined by
dX
t
= i(t)dB
t
+
φ(t)
α(t)
(dN
t
− λ(t)dt) ,X
0
= 0 (10.6.3)
satisfies the structure equation
d[X, X]
t
= dt +
φ(t)
α(t)
dX
t
.
From X, we construct a martingale Z with predictable quadratic variation
dZ, Z
t
= α
2
(t)dt, by setting
dZ
t
= α(t)dX
t
,Z
0
=0,
that is,
dZ
t
= i(t)α(t)dB
t
+ φ(t)(dN
t
− λ(t)dt) ,Z
0
=0.
Proposition 10.6.2.1 The martingale Z satisfies
d[Z, Z]
t
= α
2
(t)dt + φ(t)dZ
t
, (10.6.4)
and dZ, Z
t
= α
2
(t)dt.
Proof: Using the relations d[B,N]
t
=0andi(t)φ(t)=0,wehave
d[Z, Z]
t
= i(t)α
2
(t)dt + φ
2
(t)dN
t
= i(t)α
2
(t)dt + φ(t)
dZ
t
− i(t)α(t)dB
t
+(1− i(t))
α
2
(t)
φ(t)
dt
= α
2
(t)dt + φ(t)dZ
t
.
From the general results on structure equations which we have already
invoked, the martingale Z enjoys the predictable representation property.
Let S denote the solution of the equation
dS
t
= S
t
−
(μ(t)dt + σ(t)dZ
t
),
with initial condition S
0
where the coefficients μ and σ are assumed to be
deterministic. Then,
S
t
= S
0
exp
t
0
σ(s)α(s)i(s)dB
s
−
1
2
t
0
i(s)σ
2
(s)α
2
(s)ds
×exp
t
0
(μ(s) − φ(s)λ(s)σ(s))ds
N
t
k=1
(1 + σ(T
k
)φ(T
k
)) ,
where (T
k
)
k≥1
denotes the sequence of jump times of N.
10.6 Complete Markets with Jumps 581
Proposition 10.6.2.2 Let us assume that φ(t)
r(t)−μ(t)
σ(t)α
2
(t)
> −1,andlet
ψ(t):=φ(t)
r(t) − μ(t)
σ(t)α
2
(t)
.
Then, the unique e.m.m. is the probability Q such that Q|
F
t
= L
t
P|
F
t
,where
dL
t
= L
t
−
ψ(t)dZ
t
, L
0
=1.
Proof: It is easy to check that
d(S
t
L
t
R(t)) = S
t
−
L
t
−
R(t)σ(t)dZ
t
,
where R(t)=e
rt
, hence SRL is a P-local martingale.
We now compute the price of a European call written on the underlying
asset S. Note that the two processes (
B
t
= B
t
−
t
0
ψ(s)i(s)α(s)ds, t ≥ 0) and
(N
t
−
t
0
λ(s)(1 + φ(s))ds, t ≥ 0) are Q-martingales. Furthermore,
S
t
= S
0
exp
t
0
σ(s)α(s)i(s)d
B
s
−
1
2
t
0
i(s)σ
2
(s)α
2
(s)ds
× exp
t
0
(r(s) − φ(s)λ(s)σ(s)(1 + ψ(s))) ds
N
t
k=1
(1 + σ(T
k
)φ(T
k
)) .
In order to price a European option, we compute E
Q
[R(T )(S
T
− K)
+
].
Let
BS (x, T; r, σ
2
; K)=E[e
−rT
(xe
rT−σ
2
T/2+σW
T
− K)
+
]
denote the classical Black-Scholes function, where W
T
is a Gaussian centered
random variable with variance T . In the case of deterministic volatility
(σ(s),s ≥ 0) and interest rate (r(s),s ≥ 0), the price of a call in the Black-
Scholes model is
BS (x, T; R, Σ(T); K), with R =
1
T
T
0
r(s)ds and Σ(T )=
1
T
T
0
σ
2
(s)ds.
Let Γ
σ
(t)=
t
0
i(s)α
2
(s)σ
2
(s)ds denote the variance of
t
0
i(s)α(s)σ(s)d
B
s
,
and Γ (t)=
t
0
γ(s)ds,whereγ(t)=λ(t)(1+φ(t)ψ(t)) denote the compensator
of N under Q.
Proposition 10.6.2.3 In this model, the price of a European option is
E
Q
exp
−
T
0
r(s)ds
(S
T
− K)
+
=exp(−Γ
T
)
∞
k=0
1
k!
T
0
···
T
0
dt
1
···dt
k
γ
t
1
···γ
t
k
BS
S
0
exp
−
T
0
(φγσ)(s)ds
k
i=1
(1 + σ(t
i
)φ(t
i
)) ,T; R,
Γ
σ
T
T
; K
.
582 10 Mixed Processes
Proof: We have
E
Q
R(T )(S
T
− K)
+
=
∞
k=0
E
Q
R(T )(S
T
− K)
+
| N
T
= k
Q(N
T
= k),
with Q(N
T
= k)=exp(−Γ
T
)(Γ
T
)
k
/k!. Conditionally on {N
T
= k},thejump
times (T
1
,...,T
n
)havethelaw
1
Γ
k
T
1
{0<t
1
<···<t
k
<T }
γ
t
1
···γ
t
k
dt
1
···dt
k
,
since the process (N
Γ
−1
t
,t ≥ 0) is a standard Poisson process. Hence,
conditionally on {N(Γ
−1
(Γ
T
)) = k} = {N
T
= k}, its jump times
(Γ
T
1
,...,Γ
T
k
) have a uniform law on [0,Γ
T
]
k
(see Subsection 8.2.1). We then
use the fact that
˜
B and N are also independent under Q, since (μ(t),t ≥ 0)
is deterministic, and the identity in law
S
T
law
= S
0
X
T
e
−
R
T
0
φ(s)λ(s)σ(s)ds
N
T
k=1
(1 + σ(T
k
)φ(T
k
)) ,
where
X
T
=exp
T
0
r(s)ds − Γ
σ
T
/2+
Γ
σ
T
T
1/2
W
T
,
with W
T
is independent of N.
Comment 10.6.2.4 Within the framework of Privault, the valuation and
hedging of European options is presented in Jeanblanc and Privault [485], the
valuation of exotic options is studied in El-Khatib [312].
The reader can refer to Hobson and Rogers [442] for a different model of
complete market with jumps.
10.7 Valuation of Options
Several articles have focused on the valuation of European options when the
underlying value follows a jump-diffusion process. Merton [643]wasthefirst
one to obtain a closed form solution assuming that the market price of jump
is null (see Subsection 10.5.1). Kou and Wang [542]havestudiedthecaseof
double exponential jumps. Jump-diffusion models with stochastic volatility
and interest rate have also been developed (see for example Scott [776]). The
problem of American option valuation is more complex. It was investigated
by several authors. Bates [59] derived the early exercise premium by relying
on an extension of the MacMillan [608] and Barone-Adesi and Whaley [56]
approaches in a jump-diffusion setting.
10.7 Valuation of Options 583
As shown by Chesney and Jeanblanc [174], this extension generates good
results only if the underlying process is continuous at the exercise boundary
with probability one (e.g., in the case of perpetual currency calls, when jumps
are negative). Otherwise, if the overshoot is strictly positive at the exercise
boundary (positive jumps for the currency in the call case) the pricing problem
is more difficult to tackle and one should be very cautious when applying a
MacMillan approximation. Therefore, a new approach is developed in the
paper [174].
Pham [709] considers the American put option valuation in a jump-
diffusion model (Merton’s assumptions), and relates this problem (which is
indeed an optimal stopping problem) to a parabolic integro-differential free
boundary problem. By extending the Riesz decomposition obtained by Carr et
al. [154] for a diffusion model, Pham derives a decomposition of the American
put price as the sum of a European price and an early exercise premium. The
latter term requires the identification of the exercise boundary. In the same
context, Zhang [873, 875] relies on variational inequalities and shows how to
use numerical methods, (finite difference methods), to price the American
put. Zhang [874] describes this problem as a free boundary problem, and by
using the MacMillan approximation obtains a price for the perpetual put, and
an approximation of the finite maturity put price. These results are obtained
only when jumps are positive. Mastroeni and Matzeu [628, 627] obtain an
extension of Zhang results in a multidimensional state space.
Boyarchenko and Levendorskii [119], Mordecki [659, 658]andGerber
and Shiu [388] also consider the American option pricing problem. They
obtain solutions which are explicit only if the distribution of the jump size is
exponential or if the jump size is negative for a call (resp. positive for a put).
Mordecki [659] establishes the value of the boundary for a perpetual option
in terms of the law of the extrema of the underlying L´evy process.
The structure of this section is as follows. We first present the valuation of
European calls. We give the explicit solution when the jumps are log-normally
distributed. In the case of constant jump size, the PDE for option pricing is
then obtained. In Subsection 10.7.2, the perpetual American currency call
option and its exercise boundary are considered. Exact analytical solutions
are derived when the jump size is negative, i.e., when the overshoot of the
exercise boundary is equal to 0. They are based on the computation of the
Laplace transform of the first passage time of the process at the exercise
boundary, which is obtained by use of the optimal sampling theorem. As in
Zhang [874]orBates[59], we then show, in Subsection 10.7.2, how an accurate
approximation for the valuation of finite maturity options can be obtained by
relying on the perpetual maturity case. See also Chapter 11 for a study if
the jumps can take positive values.
584 10 Mixed Processes
10.7.1 The Valuation of European Options
In this subsection, we suppose that the price of the risky asset has dynamics
dS
t
= S
t
−
(μdt + σdW
t
+ dM
t
)
where W is a Brownian motion and M is the compensated martingale
associated with a compound Poisson process, i.e., M
t
=
N
t
k=1
Y
k
− tλE(Y
1
).
The coefficients μ, σ, λ are constant. We assume that the law of the jumps Y
k
has support in ] −1, ∞[, so that S remains strictly positive. In a closed form,
S
t
= S
0
e
μt
exp
σW
t
−
1
2
σ
2
t
exp(−tλE(Y
1
))
N
t
k=1
(1 + Y
k
) .
When S is an exchange rate, Merton [643], Lamberton and Lapeyre [559]
and Nahum [664] choose to evaluate the derivatives under the e.m.m. Q
under which the underlying spot foreign exchange rate (S
t
,t ≥ 0) follows
the dynamics
dS
t
= S
t
−
((r − δ)dt + σd
W
t
+ dM
t
)
where
W is a Q-Brownian motion and M
t
=
N
t
k=1
Y
k
− tλE(Y
1
)isaQ-
martingale. In an explicit form,
S
t
= S
0
exp
r − m −
1
2
σ
2
t + σ
W
t
N
t
k=1
(1 + Y
k
) , (10.7.1)
where m = δ + λE(Y
1
). In particular, the Poisson process N has the same
intensity under P and Q,thelawofY
1
is the same under both probabilities
(note that, in particular, E
Q
(Y
1
)=E
P
(Y
1
)) and
N
t
k=1
Y
k
is a Q-compound
Poisson process. As in Merton [643], there is no risk premium for jumps.
The price of a European call can be evaluated as follows: from (10.7.1),
we deduce
e
−rT
E
Q
(S
T
− K)
+
)
=
∞
n=0
e
−(r+λ)T
(λT )
n
n!
E
Q
S
0
e
(r−m)T
E(σ
W )
T
n
k=1
(1 + Y
k
) − K
+
(10.7.2)
=
∞
n=0
e
−λT
(λT )
n
n!
∞
−1
∞
−1
F (dy
1
) ···F (dy
n
)
× e
−rT
E
Q
S
0
e
(r−m)T
E(σ
W )
T
n
k=1
(1 + y
k
) − K
+
=
∞
n=0
e
−λT
(λT )
n
n!
∞
−1
∞
−1
F (dy
1
) ···F (dy
n
)BS
S
0
e
−mT
n
k=1
(1 + y
k
),r,σ,T
10.7 Valuation of Options 585
where BS is the value of a plain vanilla call given by
BS (S
0
,θ,Σ,T)=S
0
N(d
1
) − Ke
−θT
N(d
2
) , (10.7.3)
d
1
=
ln(S
0
/K)+(θ + Σ
2
/2)T
Σ
√
T
,d
2
= d
1
− Σ
√
T.
Inthecasewhere1+Y
1
law
= e
Z
,whereZ is a Gaussian r.v., we obtain, as
in Merton, a more pleasant form, using the stability of independent Gaussian
random variables under addition in the equality (10.7.2).
Proposition 10.7.1.1 When ln(1 + Y )
law
= N(μ, α
2
), the price of a European
call with maturity T is given by
C(S
0
,T)=
∞
n=0
e
−λT
(λT )
n
n!
e
(θ
n
−r)T
BS (S
0
,θ
n
,Σ
n
,T) (10.7.4)
where BS is the value of a plain vanilla call defined in (10.7.3), with
⎧
⎨
⎩
θ
n
= r − δ −λE(Y
1
)+
n ln(1+E(Y
1
))
T
Σ
2
n
= σ
2
+ nα
2
/T ,
. (10.7.5)
Proof: Assume that the 1 + Y
j
are independent log-normally distributed
r.v’s, i.e., ln(1 + Y
1
)
law
= N(μ, α
2
). The expectation
E
Q
e
(r−m)T
E(σ
W )
t
n
j=1
(1 + Y
j
) − K
+
in equation (10.7.2)isequalto
E
Q
⎡
⎣
exp
r − m −
σ
2
2
T +
n
k=1
ln(1 + Y
k
)+σ
W
T
− K
+
⎤
⎦
.
The r.v.
n
k=1
ln(1 + Y
k
)+σ
W
T
is normally distributed with expectation and
variance respectively equal to nμ and nα
2
+ σ
2
T = TΣ
2
n
. Therefore,
r − m −
σ
2
2
T +
n
k=1
ln(1 + Y
k
)+σ
W
T
=
r − δ − λE(Y
1
) −
σ
2
2
T +
n
k=1
ln(1 + Y
k
)+σ
W
T
law
= θ
n
T + Σ
n
√
TG−
Σ
2
n
T
2
586 10 Mixed Processes
where Σ
n
and θ
n
are given by equations (10.7.5)andG is a standard Gaussian
variable.
It is straightforward to obtain the result given by formula (10.7.4). Indeed,
the latter expectation corresponds to the payoff of a standard European call,
with underlying adjusted drift and volatility respectively equal to θ
n
and Σ
n
.
The term e
(θ
n
−r)T
BS (S
0
,θ
n
,Σ
n
,T) in formula (10.7.4) is, therefore, the
value of the option conditional on knowing that exactly n Poisson jumps will
occur during the life of the option. The call price is a weighted average of
these prices. The weights are the probabilities that a Poisson random variable
takes the value n,forn ∈ N.
Comments 10.7.1.2 (a) See Lipton [597] for a general discussion.
(b) The valuation of lookback options is studied in Nahum [664].
(c) Sepp [780] studies the barrier options by means of the Laplace
transform in time of the price.
(d) Asian options are presented in Bayraktar and Xing [62]andin
Boughamoura et al. [113]inaL´evy setting.
PDE for Option Prices
Suppose now that the dynamics of the risky asset (the currency) are
dS
t
= S
t
−
((r − δ)dt + σdW
t
+ φdM
t
) (10.7.6)
under the chosen risk-neutral probability. Here, W is a BM and M is the
compensated martingale of a Poisson process with constant intensity λ and
the coefficients r, δ, σ and φ>−1 are supposed to be constant. Our aim is to
compute C(S
t
,T − t)=E(e
−r(T −t)
h(S
T
)|F
t
)whereh is a smooth function.
Applying the Feynman-Kac formula we obtain that, if C satisfies
σ
2
2
x
2
∂
2
C
∂x
2
(x, τ )+(r − δ − φλ)x
∂C
∂x
(x, τ ) − rC(x, τ) −
∂C
∂τ
(x, τ )
+ λ[C((1 + φ)x, τ ) − C(x, τ)] = 0 (10.7.7)
C(x, 0) = h(x)
where τ = T −t,thenC(S
t
,T −t) is the value of the European call. However
this kind of PDE is difficult to solve.
10.7.2 American Option
We apply the previous computation to the case of an American option. We
suppose that the dynamics of the underlying asset are given by (10.7.6)and
that the jump size is constant and equal to φ. The discounted American call
price is a martingale in the continuation region. This means that if the spot
10.7 Valuation of Options 587
price x is lower than the exercise boundary value at time t, the American call
value C
A
satisfies (10.7.7).
If the jump is positive, and if x belongs to the interval [
¯x
1+φ
, ¯x], where ¯x
is the exercise boundary level at time t, the American call value satisfies
σ
2
2
x
2
∂
2
C
A
∂x
2
(x, τ )+(r − δ − φλ)x
∂C
A
∂x
(x, τ ) − rC
A
(x, τ )
−
∂C
A
∂τ
(x, τ )+λ((1 + φ)x − K − C
A
(x, τ )) = 0 ,
because in this case, after the jump, the American call value is equal to its
intrinsic value.
The Valuation of the Perpetual American Option: Negative Jumps
We now assume that the jump is negative −1 <φ≤ 0. Zhang’s results
[874] concerning the perpetual option value and the exercise boundary will be
rederived in this context.
The exercise boundary is defined as follows: for a given time t ∈ [0,T]
b
c
(T − t)=inf{x ≥ 0:x − K = C
A
(x, T − t)} . (10.7.8)
It is worth mentioning that, with a strictly positive foreign interest rate,
American and European call options have different prices. When the maturity
tends to infinity, the exercise boundary admits a limit b
∗
c
(see Mordecki [659]).
As shown below, this limit b
∗
c
is finite (for δ>0). The option value is given
by:
C
A
(S
0
, +∞)=sup
τ
E((S
τ
− K)e
−rτ
):=C
A
(S
0
) (10.7.9)
where τ runs over stopping times. Here, the +∞ indicates that the maturity
is infinite. It is proved in Mordecki [659] that one can restrict attention to the
case of first passage time stopping times. Since the jump size is negative, the
process is continuous on the boundary, therefore,
C
A
(S
0
)= sup
L≥S
0
[(L − K)E(e
−rT
L
)] = (b
∗
c
− K)E(e
−rT
b
∗
c
) (10.7.10)
where T
L
is the first passage time of the process S out of the continuation
region :
T
L
=inf{t ≥ 0:S
t
≥ L}.
From (10.4.6), with =ln(L/S
0
), the Laplace transform of the hitting time
is
E
Q
(e
−rT
L
)=
S
0
L
ρ
∧ 1
where ρ is equal to g
−1
(r) (defined in 10.4.6) and is strictly greater than 1.
Indeed, in the case of the call, we use the positive root k of g(k)=r because we
588 10 Mixed Processes
want the call value to be an increasing function of the underlying asset value.
(See (10.4.5) for the definition of the Laplace exponent g(k).) We can thus
derive the value of the exercise boundary as the value where the supremum
in equation (10.7.10) is attained:
b
∗
c
=
ρK
ρ − 1
. (10.7.11)
In the continuation region, (i.e., if S
0
<
ρK
ρ−1
), the option value is:
C
A
(S
0
)=
S
0
b
∗
c
ρ
(b
∗
c
− K) . (10.7.12)
Without jumps (λ = 0), we obtain the known formula (3.11.13). Indeed,
in this case g(k)=(r − δ)k +
1
2
σ
2
k(k − 1), hence ρ =
−ν+
√
ν
2
+2r
σ
with
ν =
r − δ − σ
2
/2
/σ .
Put prices can be obtained by the symmetry formula (see Subsection
10.4.2), used in the case φ>0:
P
A
(S
0
,K,r,δ; σ, φ, λ)=KS
0
C
A
(1/S
0
, 1/K,δ,r; σ, −
φ
1+φ
,λ(1 + φ)) .
Do not forget, when applying this formula, that the price of the put on the
left-hand side is evaluated under the risk-neutral probability in the domestic
economy, whereas on the right-hand side, the call should be evaluated under
the foreign risk-neutral probability. By relying on Subsection 10.4.2,the
exercise boundary of the put can be obtained:
b
p
(K, T − t, r, δ; φ, λ) b
c
(K, T − t, δ; r,
−φ
1+φ
,λ(1 + φ)) = K
2
, (10.7.13)
where now φ>0.
An Approximation of the American Option Value
Let us now rely on the Barone-Adesi and Whaley approach [56]andonBates’s
article [59]. If the American and European option values satisfy the same linear
PDE (10.7.7) in the continuation region, their difference DC (the American
premium) must also satisfy this PDE in the same region. Write:
DC(S
0
,T)=yf(S
0
,y)
where y =1− e
−rT
, and where f is a function of two arguments that has
to be determined. In the continuation region, f satisfies the following PDE
which is obtained by a change of variables:
10.7 Valuation of Options 589
σ
2
2
x
2
∂
2
f
∂x
2
+(r−δ)x
∂f
∂x
−
rf
y
−(1−y)r
∂f
∂y
−λ[φx
∂f
∂x
−f((1+φ)x, y)+f(x, y)] = 0 .
Let us now assume that the derivative of f with respect to y may be neglected.
Whether or not this is a good approximation is an empirical issue that we do
not discuss here.
The equation now becomes an ODE
σ
2
2
x
2
∂
2
f
∂x
2
+(r − δ)x
∂f
∂x
−
rf
y
− λ[φx
∂f
∂x
− f ((1 + φ)x, y)+f (x, y)] = 0 .
The value of the perpetual option satisfies almost the same ODE. The
only difference is that y is equal to 1 in the perpetual case, and therefore we
have r/y instead of r in the third term of the left-hand side. The form of the
solution is known (see (10.7.12)):
f(S
0
,y)=zS
ρ
0
.
Here, z is still unknown, and ρ is the positive solution of equation g(k)=r/y.
When S
0
tends to the exercise boundary b
c
(T ), by the continuity of the
option value, the following equation is satisfied from the definition of the
American premium DC(x, T ):
b
c
(T ) − K = C
E
(b
c
(T ),T)+yz b
c
(T )
ρ
, (10.7.14)
and by use of the smooth-fit condition
1
, the following equation is obtained:
1=
∂C
E
∂x
(b
c
(T )),T)+yzρ(b
c
(T ))
ρ−1
. (10.7.15)
In a jump-diffusion model this condition was derived by Zhang [873]inthe
context of variational inequalities and by Pham [709] with a free boundary
formulation. We thus have a system of two equations (10.7.14)and(10.7.15)
and two unknowns z and b
c
(T ). This system can be solved: b
c
(T ) is the implicit
solution of
b
c
(T )=K −C
E
(b
c
(T ),T)+
1 −
∂C
E
∂x
(b
c
(T ),T)
b
c
(T )
ρ
.
If S
0
>b
c
(T ), C
A
(S
0
,T)=S
0
−K. Otherwise, if S
0
<b
c
(T ), the approximate
formula is
C
A
(S
0
,T)=C
E
(S
0
,T)+A(S
0
/b
c
(T ))
ρ
(10.7.16)
with
A =
1 −
∂C
E
∂x
(b
c
(T ),T)
b
c
(T )
ρ
.
1
The smooth-fit condition ensures that the solution of the PDE is C
1
at the
boundary. See Villeneuve [830].