# Abstracts

**Analytical formulas for multidimensional diffusion process**

*Emmanuel Gobet (Ecole Polytechnique, France)*

Thursday June 5, 14:00-14:30 | session 8.1 | Computational Finance | room AB

We derive an analytical weak approximation of a multidimensional diffusion process as coefficients or time are small. Our methodology combines the use of Gaussian proxys to approximate the law of the diffusion and a Finite Element interpolation of the terminal function applied to the diffusion. We call this method Stochastic Approximation Finite Element (SAFE for short) method. We provide error bounds of our global approximation depending on the diffusion process coefficients, the time horizon and the regularity of the terminal function. Then we give estimates of the computational cost of our algorithm. This shows an improved efficiency compared to Monte-Carlo methods in small and medium dimensions (up to 10), which is confirmed by numerical experiments. Applications to fast generic option pricing are presented.

**Efficient simulations for estimating Value-at-Risk in incremental risk charge**

*Cheng-Der Fuh (National Central University, Taiwan)*

Thursday June 5, 14:30-15:00 | session 8.1 | Computational Finance | room AB

In this paper, we describe, analyze and evaluate an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. This investigation is motivated by the incremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision, in which one needs to calculate $99.9\%$ confidence Value-at Risk (VaR) over a 1-year horizon in the presence of constraints on credit default and migration risks. The IRC assumes that a portfolio is managed dynamically to a target level of risk, with constraints on discrete rebalancing intervals (e.g., monthly or quarterly) as a rough measure of potential illiquidity in underlying assets. By taking these factors into consideration, we propose an efficient Monte Carlo simulation algorithm to estimate such loss probabilities which is essential to calculating VaR, a quantile of the loss distribution. The method employs importance sampling for Markov chains and a spherical Monte Carlo method for further variance reduction.

To have efficient importance sampling for Markov chains, we develop a general account for finding the optimal tilting probability measure. Our approach consists of constructing an exponential tilting family of probability distributions around the original probability and then minimizing the variance of the importance sampling estimator within this family.

By using the device of conjugate probability measure and a solution of a Poisson equation, we obtain a simple and explicit expression of the optimal alternative distribution. Simulation results indicate that an appropriate combination of importance sampling for Markov chains and spherical Monte Carlo can result in large variance reductions when estimating the probability of portfolio losses.