Emmanuel Gobet

Time Dependent Heston Model

The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [S. Heston, Rev. Financ. Stud., 6 (1993), pp. 327-343] or piecewise constant [S. Mikhailov and U. Nogel, Wilmott Magazine, July (2003), pp. 74-79]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier-based methods is its rapidity (gain by a factor 100 or more) while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalent Heston models (extending some work of Piterbarg on stochastic volatility models [V. Piterbarg, Risk Magazine, 18 (2005), pp. 71-75]) and second, to the calibration procedure in terms of ill-posed problems.

LAN property for ergodic diffusions with discrete observations

Abstract We consider a multidimensional elliptic diffusion Xα,β, whose drift b(α,x) and diffusion coefficients S(β,x) depend on multidimensional parameters α and β. We assume some various hypotheses on b and S, which ensure that Xα,β is ergodic, and we address the problem of the validity of the Local Asymptotic Normality (LAN in short) property for the likelihoods, when the sample is (XkΔn)0⩽k⩽n, under the conditions Δn→0 and nΔn→+∞. We prove that the LAN property is satisfied, at rate nΔ n for α and n for β: our approach is based on a Malliavin calculus transformation of the likelihoods.

Nonparametric estimation of scalar diffusions based on low frequency data

We study the problem of estimating the coefficients of a diffusion (X t , t ≥ 0); the estimation is based on discrete data X n Δ, n = 0, 1,..., N. The sampling frequency Δ -1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X nΔ , n = 0, 1,..., N) in a suitable Sobolev norm, together with an estimation of its invariant density.

Discrete time hedging errors for options with irregular payoffs

Abstract. In a complete market with a constant interest rate and a risky asset, which is a linear diffusion process, we are interested in the discrete time hedging of a European vanilla option with payoff function f. As regards the perfect continuous hedging, this discrete time strategy induces, for the trader, a risk which we analyze w.r.t. n, the number of discrete times of rebalancing. We prove that the rate of convergence of this risk (when

Expansion Formulas For European Options In A Local Volatility Model

n \rightarrow + \infty

DISCRETIZATION AND SIMULATION OF THE ZAKAI EQUATION

) strongly depends on the regularity properties of f: the results cover the cases of standard options.

Sequential Control Variates for Functionals of Markov Processes

Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the re\-pre\-sentation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order

Advanced Monte Carlo Methods for Barrier and Related Exotic Options

\sqrt{\delta}

Analytical Formulas for Local Volatility Model with Stochastic Rates

(