Partial Differential Equations - Modelling and ... - ResearchGate

260 Y. Achdou
European options are similar contracts, except that they can be exercised only
at maturity t.
Consider an American option with payoff P ◦ and maturity t. Under the assumptions
that the market is complete and rules arbitrage out, Black–Scholes’
theory predicts that the price of this option at time τ is
( )
P τ = sup E ∗ e −r(s−τ) P ◦ (S s ) ∣ F τ , (2)
s∈T τ,t
where T τ,t denotes the set of stopping times in [τ,t] (see [LL97] for the proof
of this formula). It can also be proved, see, e.g., [BL84, JLL90] that P τ =
P (τ,S τ ), where the two variables function P is found by solving a parabolic
linear complementarity problem
∂P
∂τ + σ2 S 2 ∂ 2 P ∂P
+ rS
2 ∂S2 ∂S − rP ≤ 0, P(τ,S) ≥ P ◦(S), τ ∈ [0,t), S > 0,
( ∂P
∂τ + σ2 S 2 ∂ 2 )
P ∂P
+ rS
2 ∂S2 ∂S − rP (P − P ◦ (S)) = 0, τ ∈ [0,t), S > 0,
P (τ = t, S) =P ◦ (S).
(3)
The critical parameter in the Black–Scholes model is the volatility σ. Unfortunately,
taking σ to be constant and using (2) or (3) often leads to poor
predictions of the prices of the options which are available on the markets.
One possible fix is to assume that the process driving S t is a more general
Lévy process: Lévy processes are processes with stationary and independent
increments which are continuous in probability, see, for example, the book by
Cont and Tankov [CT04] and the references therein.
For a Lévy process X τ on a filtered probability space with probability P ∗ ,
the Lévy–Khintchine formula says that there exists a function χ : R → C such
that
E ∗ (e iuXτ )=e −τχ(u) , (4)
χ(u) = σ2 u 2 ∫
∫
− iβu + (e iuz − 1 − iuz)ν(dz)+ (e iuz − 1)ν(dz),
2
|z|1
(5)
for σ ≥ 0, β ∈ R and a positive measure ν on R \{0} such that
∫
min(1,z 2 )ν(dz) < +∞.
R
The measure ν is called the Lévy measure of X.
We assume that under P ∗ , the discounted price of the risky asset is a
martingale, and that it is represented as the exponential of a Lévy process:
e −rτ S τ = S 0 e Xτ .