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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… (More)

- J F Clouet, J P Fouque
- 1994

This paper investigates the deformation of an acoustic pulse travelling in a slab of random medium when its width is large compared to the size of the random inhomogeneities of the medium. A limit theorem is shown that explains how the shape of the transmitted pulse can be obtained as a result of a deterministic gaussian convolution of the initial pulse.… (More)

After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is… (More)

- Jean-Pierre Fouque, George Papanicolaou, Ronnie Sircar, Knut Sølna, GEORGE PAPANICOLAOU, Robert Grimmett +2 others
- 2011

Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations.… (More)

- J F Clouet, J P Fouque
- 1997

In the recent years a considerable amount of mathematical work have been devoted to the study of reeected signals obtained by the propagation of pulses in randomly layered media. We refer to 1] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities… (More)

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… (More)

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its time scales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for… (More)

We study the Merton problem of portfolio optimization over a finite horizon when volatility is stochastic and fluctuating on different time scales. We develop a perturbation method for the associated nonlinear PDE and we show how to relate market data implied volatility skews to optimal strategies. Joint work in progress with Ronnie Sircar and Thaleia… (More)

Gaussian copula is by far the most popular copula used in the financial industry in default dependency modeling. However, it has a major drawback — it does not exhibit tail dependence, a very important property for copula. The essence of tail dependence is the interdependence when extreme events occur, say, defaults of corporate bonds. In this paper we show… (More)

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as… (More)