Processus de Lévy en Finance - Laboratoire de Probabilités et ...

1.2. EXPONENTIAL LEVY MODELS 41
(b) C n → C, where C ∗ = A ∗ + ∫ ∞
−∞ h2 (x)ν ∗ (dx).
(c) ∫ ∞
−∞ f(x)ν n(dx) → ∫ ∞
−∞
f(x)ν(dx) for every continuous bounded function f such that
f(x) ≡ 0 on a neighborhood of zero, or, equivalently, for every continuous bounded
function f satisfying f(x) = o(|x| 2 ) when x → 0.
1.2 Exponential Lévy models: definition and main properties
Exponential Lévy models are obtained by replacing the Brownian motion with drift in the
classical Black-Scholes-Samuelson model of asset price, by a Lévy process:
S t = S 0 e rt+Xt , (1.10)
where X is a Lévy process on (Ω, F, P ), and the interest rate term rt is introduced to simplify
the notation below. When P is the probability that describes the evolution of stock prices
in the real world, also called the historical probability, the model (1.10) is called a historical
exponential Lévy model.
By the first fundamental theorem of asset pricing (see [34]), a financial market does not
allow for arbitrage opportunity (more precisely, satisfies the No Free Lunch with Vanishing Risk
condition) if there exists a probability measure Q, equivalent to P , such that the discounted
prices e −rt V t of all assets are Q-local martingales. Q is called a risk-neutral probability. The
absence of arbitrage in the model (1.10) is therefore equivalent to the existence of a probability
Q ∼ P , such that e X is a Q-local martingale. The following result shows that if X is a Lévy
process under P , one can almost always find a probability Q ∼ P , under which X is still a Lévy
process and e X is a martingale.
Proposition 1.8 (Absence of arbitrage in exp-Lévy models). Let {X t } t≥0 be a Lévy
process on (Ω, F, P ) with characteristic triplet (A, ν, γ). If the trajectories of X are neither
almost surely increasing nor almost surely decreasing, then there exists a probability measure Q
equivalent to P such that under Q, {X t } t≥0 is a Lévy process and {e Xt } t≥0 is a martingale.
Q may be chosen in such way that (X, Q) will have the characteristic triplet (A Q , ν Q , γ Q ),