Mathematics – Probability
Scientific paper
2011-08-03
Mathematics
Probability
31 pages, 1 figures
Scientific paper
Consider W a multidimensional continuous Gaussian process with independent components such that a geometric rough path exists over it and X the solution (in rough paths sense) of a stochastic differential equation driven by W on [0,T] with bounded coefficients (T > 0). In this article, we prove the existence and compute the sensitivity of E[F(X_T)] to any variation of the initial condition and then to any variation of the volatility function as well. On one hand, the theory of rough differential equations allows us to conclude when F is differentiable. On the other hand, using Malliavin calculus, the condition "F is differentiable" can be dropped under assumptions on the Cameron-Martin's space of W when F belongs to L^2. Finally, we provide two applications in finance in order to illustrate the link with the "usual" computation of Greeks.
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