Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
then call options still verify a forward PDE where the diffusion coefficient is
given by the local (or effective) volatility function σ(t,S) given by
σ(t,S) = E[δ2 t|St = S].
This method is linked to the “Markovian projection” problem: the construction
of a Markov process which mimicks the marginal distributions of
a martingale [15, 51, 74]. Such “mimicking processes” provide a method to
extend the Dupire equation to non-Markovian settings.
We show in Chapter 3 that the forward equation for call prices holds
in a more general setting, where the dynamics of the underlying asset is
described by a – possibly discontinuous – semimartingale. Namely, we consider
a (strictlypositive) semimartingale S whose dynamics under thepricing
measure P is given by
T
ST = S0 +
0
r(t)St −dt+
T
0
St −δtdWt +
T +∞
0
−∞
St −(ey −1) ˜ M(dt dy),
(1.17)
where r(t) > 0 represents a (deterministic) bounded discount rate, δt the
(random) volatility process and M is an integer-valued random measure with
compensator µ(dtdy;ω)= m(t,dy,ω)dt, representing jumps in the log-price,
and ˜ M = M−µ is the compensated random measure associated to M. Both
the volatility δt and m(t,dy), which represents the intensity of jumps of size
y at time t, are allowed to be stochastic. In particular, we do not assume
the jumps to be driven by a Lévy process or a process with independent
increments. The specification (1.17) thus includes most stochastic volatility
models with jumps. Also, our derivation does not require ellipticity or nondegeneracy
ofthediffusion coefficient andunder someintegrability condition,
we show that the call option price (T,K) ↦→ Ct0(T,K), as a function of
maturity and strike, is a solution (in the sense of distributions) of the partial
integro-differential equation
∂Ct0
∂T
0
∂ 2 Ct0
(T,K) = −r(T)K∂Ct0
∂K (T,K)+ K2σ(T,K) 2
2
+∞
+ y ∂2
Ct0 K
(T,dy)χT,y ln
∂K2 y
(T,K)
∂K2
, (1.18)
on [t0,∞[×]0,∞[ with the initial condition: ∀K > 0 Ct0(t0,K) = (St0 −
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