PR ] 2 6 O ct 2 00 6 Stochastic flows approach to Dupire ’ s formula

The probabilistic equivalent formulation of Dupire’s PDE is the Put-Call duality equality. In local volatility models including exponential Lévy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire’s PDE recently obtained by Pironneau [7] by the adjoint equation technique in the case of complex options. Introduction The second order derivative of the price of a Call option with respect to the strike variable is equal to the risk-neutral density of the underlying stock at maturity multiplied by the actualization factor. In a stock model with a local volatility function and a proportional dividend rate ((0.1) with μ = m = 0), Dupire [4] takes advantage of this specificity to obtain a PDE (see (2.3) for m = 0) satisfied by the Call pricing function in the maturity and strike variables. His proof consists in integrating twice in space the Fokker-Planck equation governing the time evolution of the density of the stock price. Alternatively, one may use the Green function of the problem or the adjoint equation technique [7]. For calibration purposes, Dupire’s PDE permits to express the local volatility function in terms of the function giving the Call prices for all strikes and maturities. Dupire’s PDE can be interpretated as the pricing PDE for a Put option. This leads to the PutCall duality (equality (2.2) for μ̃ = μ = m = 0) which is in fact an equivalent formulation : the Call price is transformed into the Put price by simultaneous exchange of the interest and dividend rates and of the spot and strike prices in addition to time-reversal of the local volatility function. To our knowledge, no direct probabilistic proof is available for the equality of the expectations giving the Call and Put prices. In [2], in models including exponential Lévy jumps, Carr and Andreasen derive a PIDE generalizing Dupire’s PDE by computing the evolution of the Call payoff with respect to maturity thanks to the Itô-Tanaka formula and taking expectations. The present paper deals with such models (see (0.1)). In the second section, we give a probabilistic proof of the Put-Call duality (2.2) equality equivalent to this PIDE. We check the equality of the expectations by an argument based on stochastic flows of diffeomorphisms. The flow properties of the SDE (0.1) involved in this argument are introduced in the first section and proved in the appendix. ∗CERMICS, project-team Mathfi, École des Ponts, ParisTech, 6-8 av Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France e-mail : jourdain@cermics.enpc. fr 1 In a recent paper, Pironneau [7] obtains various generalizations of Dupire’s PDE to complex options by the adjoint equation technique. More precisely, for a given complex option, he shows that it is possible to compute the pricing function for all strikes and maturities by solving a single PDE. In calibration procedures, solving this PDE instead of one pricing PDE for the maturity and strike of each quoted option permits important computation time reduction. Most of these generalized Dupire’s PDEs have equivalent probabilistic interpretations similar to the Put-Call duality. In the third and fourth sections of the paper, we use stochastic flows to check the equivalent interpretations corresponding to binary and options written on two assets. The fifth section deals with barrier options in local volatility models without jumps. In section 1.1 [7], Pironneau addresses two-barriers options but we have only been able to give a probabilistic equivalent interpretation (see (5.1)) in the one-barrier case. Moreover, besides particular cases, it seems challenging to give a probabilistic proof of this equivalent formulation. The case of American options is not addressed in [7]. In [1], we deal with the case of perpetual options when the local volatility function does not depend on time. For the perpetual American Call price to be equal to the perpetual American Put price, in addition to the exchanges of the interest and dividend rates and of the spot and strike prices, the volatility function has to be modified. Our approach consists in deriving and studying an ODE satisfied by the exercise boundary as a function of the strike variable. Again, a direct probabilistic proof of the duality results appears challenging. The stochastic flow approach presented in the present paper does not seem suited to deal with options involving stopping times like barrier and American options. Notations : For T > 0 and m a measure on R such that ∫ R (1 + e) ∧ l2m(dl) < +∞, let (Wt)t∈[0,T ] be a standard Brownian motion and μ denote an independent Poisson random point measure on (0, T ] ×R with intensity m(dl)dt. We consider the following risk-neutral evolution for the underlying stock price dX t = σ(t,X x t )X x t dWt + (r− δ)X x t dt+X x t ∫ R (e − 1)(μ(dt, dl)−m(dl)dt), X 0 = x > 0 (0.1) where r denotes the interest rate and δ the dividend rate. The local volatility function σ(t, x) is assumed to belong to the space V = { f : [0, T ]× (0,+∞) → R : sup [0,T ]×R 3 ∑ k=0 |x∂ 2f(t, x)| < +∞ } where ∂ 2f denotes the k-th order derivative of the function f with respect to its second variable. The process (Ŵt = WT−t −WT )t∈[0,T ] obtained by time-reversal of W is a Brownian motion independent from the image μ̂ of μ by the mapping (t, l) ∈ (0, T ] × R → (T − t,−l) which is a Poisson random point measure on [0, T )×R with intensity m̂(dl)dt where m̂ denotes the image of m by l ∈ R → −l. Let us also introduce the Lévy processes Lt = ∫ (0,t]×R l(μ(ds, dl) − 1{|l|≤1}m(dl)ds)− t ∫ R (e − 1− l1{|l|≤1})m(dl) and L̂t = LT−t − LT = ∫ [0,t)×R l(μ̂(ds, dl)− 1{|l|≤1}m̂(dl)ds) + t ∫ R (e − 1− l1{|l|≤1})m(dl). (0.2)