Eric Benhamou

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.

Expansion Formulas For European Options In A Local Volatility Model

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.

A Market Model for Inflation

The various macro econometrics models for inflation are helpless when it comes to the pricing of inflation derivatives. The only article targeting inflation option pricing, the Jarrow Yildirim model (2000), relies on non observable data. This makes the estimation of the model parameters a non trivial problem. In addition, their framework does not examine any relationship between the most liquid inflation derivatives instruments : the year to year and zero coupon swap. To fill this gap, we see how to derive a model on inflation, based on traded and liquid market instrument. Applying the same strategy as the one for a market model on interest rates, we derive no-arbitrage relationship between zero coupon and year to year swaps. We explain how to compute the convexity adjustment and what relationship the volatility surface should satisfy. Within this framework, it becomes much easier to estimate model parameters and to price inflation derivatives in a consistent way.

Analytical Formulas for Local Volatility Model with Stochastic Rates

This paper presents new approximation formulae for European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models [ Int. J. Theor. Appl. Finance , 2010, 13 (4), 603--634] for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data.

Closed Forms for European Options in a Local Volatility Model

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 smooth- ness and it converges with very few terms.

A Comparative Analysis of Basket Default Swaps Pricing Using the Stein Method

Using the Stein numerical method, introduced by El Karoui and Jiao {ElKJ} and El Karoui, Jiao and Kurtz {ElKJK}, we compare, in terms of accuracy and efficiency, the pricing of the basket default swaps (NTDs and CDO Tranches). In the Factor Copula Model framework, we compare the following copula functions: 1 factor and 3 factors Gaussian copula, Clayton copula, Marshall-Olkin copula, Double-t copula and Student copula. Stein numerical method is also compared with the Recursive method of Hull and White, with the Probability Generating Function method (an exact Fourier transform like method) and with the Monte Carlo method.

Is Multi-Factor Really Necessary to Price European Options in Commodity?

The main result of this article is the presentation of the Distribution Match Method. This method applies to a general multi-factor pricing model under assumption of normal law drift. The idea is to find an equivalent one-factor model for European options. The equivalent model admits a weak solution, which has the same one-dimensional marginal probability distribution. Moreover, the one-dimensional distribution can be explicitly calculated under certain condition. This result can consequently induct closed formula for the future price and European option price. We apply these results to two well known commodity models, the Gabillon and the Gibson Schwartz model, to provide the price for the future price and a closed formula for the European options.