Jean-Pierre Fouque

Multiscale stochastic volatility for equity, interest rate, and credit derivatives

Introduction 1. The Black-Scholes theory of derivative pricing 2. Introduction to stochastic volatility models 3. Volatility time scales 4. First order perturbation theory 5. Implied volatility formulas and calibration 6. Application to exotic derivatives 7. Application to American derivatives 8. Hedging strategies 9. Extensions 10. Around the Heston model 11. Other applications 12. Interest rate models 13. Credit risk I: structural models with stochastic volatility 14. Credit risk II: multiscale intensity-based models 15. Epilogue Bibliography Index.

Singular Perturbations in Option Pricing

After the celebrated Black--Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101--142] that, in the presence of a separation of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff...

Multiscale Stochastic Volatility Asymptotics

In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black--Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index, say, and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena make it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work (see, for instance, [J. P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000]), we considered the situation when the volatility is fast mean reverting. Using a...

Stochastic Volatility Effects on Defaultable Bonds

This paper studies the effect of introducing stochastic volatility in the first‐passage structural approach to default risk. The impact of volatility time scales on the yield spread curve is analyzed. In particular it is shown that the presence of a short time scale in the volatility raises the yield spreads at short maturities. It is argued that combining first passage default modelling with multiscale stochastic volatility produces more realistic yield spreads. Moreover, this framework enables the use of perturbation techniques to derive explicit approximations which facilitate the complicated issue of calibration of parameters.

Mean Field Games and systemic risk

We propose a simple model of inter-bank borrowing and lending where the evolution of the log-monetary reserves of

Short-Maturity Asymptotics for a Fast Mean-Reverting Heston Stochastic Volatility Model

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Interacting particle systems for the computation of rare credit portfolio losses

banks is described by a system of diffusion processes coupled through their drifts in such a way that stability of the system depends on the rate of inter-bank borrowing and lending. Systemic risk is characterized by a large number of banks reaching a default threshold by a given time horizon. Our model incorporates a game feature where each bank controls its rate of borrowing/lending to a central bank. The optimization reflects the desire of each bank to borrow from the central bank when its monetary reserve falls below a critical level or lend if it rises above this critical level which is chosen here as the average monetary reserve. Borrowing from or lending to the central bank is also subject to a quadratic cost at a rate which can be fixed by the regulator. We solve explicitly for Nash equilibria with finitely many players, and we show that in this model the central bank acts as a clearing house, adding liquidity to the system without affecting its systemic risk. We also study the corresponding Mean Field Game in the limit of large number of banks in the presence of a common noise.

Pricing Asian options with stochastic volatility

In this paper, we study the Heston stochastic volatility model in a regime where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. We derive a large deviation principle and compute the rate function by a precise study of the moment generating function and its asymptotic. We then obtain asymptotic prices for out-of-the-money call and put options and their corresponding implied volatilities.

Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models

In this paper, we introduce the use of interacting particle systems in the computation of probabilities of simultaneous defaults in large credit portfolios. The method can be applied to compute small historical as well as risk-neutral probabilities. It only requires that the model be based on a background Markov chain for which a simulation algorithm is available. We use the strategy developed by Del Moral and Garnier in (Ann. Appl. Probab. 15:2496–2534, 2005) for the estimation of random walk rare events probabilities. For the purpose of illustration, we consider a discrete-time version of a first passage model for default. We use a structural model with stochastic volatility, and we demonstrate the efficiency of our method in situations where importance sampling is not possible or numerically unstable.

Maturity cycles in implied volatility

Abstract In this paper, we generalize the recently developed dimension reduction technique of Vecer for pricing arithmetic average Asian options. The assumption of constant volatility in Vecers method will be relaxed to the case that volatility is randomly fluctuating and is driven by a mean-reverting (or ergodic) process. We then use the fast mean-reverting stochastic volatility asymptotic analysis introduced by Fouque, Papanicolaou and Sircar to derive an approximation to the option price which takes into account the skew of the implied volatility surface. This approximation is obtained by solving a pair of one-dimensional partial differential equations.