Showing papers by "Benjamin Jourdain published in 2009"

Robust adaptive importance sampling for normal random vectors.

Abstract: Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.

Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process

Abstract: We prove existence and uniqueness for some nonlinear stochastic differential equation used in molecular dynamics, whose nonlinearity comes from a conditional expectation term. We also introduce an interacting particle system in order to approximate this conditional expectation, providing a discretization scheme for this equation.

Coupling Index and Stocks

Abstract: In this paper, we are interested in continuous time models in which the index level induces some feedback on the dynamics of its composing stocks. More precisely, we propose a model in which the log-returns of each stock may be decomposed into a systemic part proportional to the log-returns of the index plus an idiosyncratic part. We show that, when the number of stocks in the index is large, this model may be approximated by a local volatility model for the index and a stochastic volatility model for each stock with volatility driven by the index. This result is useful in a calibration perspective : it suggests that one should first calibrate the local volatility of the index and then calibrate the dynamics of each stock. We explain how to do so in the limiting simplified model and in the original model.

Adaptive variance reduction techniques in finance

Abstract: This paper gives an overview of adaptive variance reduction techniques recently devel- oped for financial applications. More precisely, we explain how informa tion available in the random drawings made to compute the expectation of interest may be used at the same time to optimize control variates, importance sampling or stratified sampling.

Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

Abstract: We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is non-increasing in a left-hand neighbourhood of $t_d$, and tends to $0$ as time tends to $t_d^-$ with a speed that we can characterize. When the dividend function is linear in a neighbourhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighbourhood of $t_d$. When it is globally linear, then right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.

Exact volatility calibration based on a Dupire-type Call–Put duality for perpetual American options

Abstract: This paper investigates the calibration of a model with a time-homogeneous local volatility function to the market prices of the perpetual American Call and Put options. The main step is the derivation of a Call–Put duality equality for perpetual American options similar to the equality which is equivalent to Dupire’s formula (Dupire in Risk 7(1):18–20, 1994) in the European case. It turns out that in addition to the simultaneous exchanges between the spot price and the strike and between the interest and dividend rates which already appear in the European case, one has to modify the local volatility function in the American case. To show this duality equality, we exhibit non-autonomous nonlinear ODEs satisfied by the perpetual Call and Put exercise boundaries as functions of the strike variable. We obtain uniqueness for these ODEs and deduce that the mapping associating the exercise boundary with the local volatility function is one-to-one onto. Thanks to this Dupire-type duality result, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price x0. The knowledge of the Put (resp. Call) prices for all strikes enables to recover the local volatility function on the interval (0, x0) (resp. (x0, +∞)). We last prove that equality of the dual volatility functions only holds in the standard Black-Scholes model with constant volatility.

High order discretization schemes for stochastic volatility models

Abstract: In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using It\^o's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a, 2008b].