Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
where
• δ i is an adapted process taking values in R representing the volatility of
the asset i, W is a d-dimensional Wiener process : for all 1 ≤ (i,j) ≤ d,
〈W i ,W j 〉t = ρijt,
• N isaPoissonrandommeasureon[0,T]×R d withcompensatorν(dy)dt,
•
Ñ denotes its compensated random measure.
We consider an index, defined as a weighted sum of the asset prices:
It =
d
i=1
wiS i t , d ≥ 2.
The pricing of index options involves the computation of quantities of the
form E[f(It)|Ft0] and Chapter 4 shows that the short time asymptotics for
these quantitie that we have characterized explicitly in terms of the characteristic
triplet of the discontinuous semimartingale It in Chapter 3.
Short time asymptotics of index call option prices have been computed
by Avellaneda & al [5] in the case where S is a continuous process. Results
of Chapter 4 show that this asymptotic behavior is quite diferent for at
the money or out of the money options. At the money options exhibit a
bahavior in O( √ t) which involves the diffusion component of It whereas out
of the money options exhibit a linear behavior in t which only involves the
jumps of It.
In this Chapter, we propose an analytical approximation for short maturity
index options, generalizing the approach by Avellaneda & al. [5] to the
multivariate jump-diffusion case. We implement this method in the case of
the Merton model in dimension d = 2 and d = 30 and study its numerical
precision.
The main difficulty is that, even when the joint dynamics of the index
components (S 1 ,...,S d ) is Markovian, the index It is not a Markov process
but only a semimartingale. The idea is to consider the Markovian projection
of the index process, an auxiliary Markov process which has the same
marginals as It, and use it to derive the asymptotics of index options, using
the results of Chapter 4. This approximation is shown to depend only on
the coefficients of this Markovian projection, so the problem boils down to
computing effectively these coefficients: the local volatility function and the
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