A Martingale Result for Convexity Adjustment in the Black Pricing Model

This paper explains how to calculate convexity adjustment for interest rates derivatives when assuming a deterministic time dependent volatility, using martingale theory. The motivation of this paper lies in two directions. First, we set up a proper no-arbitrage framework illustrated by a relationship between yield rate drift and bond price. Second, making ap-proximation, we come to a closed formula with speci…cation of the error term. Earlier works (Brotherton et al. (1993) and Hull (1997)) assumed constant volatility and could not specify the approximation error. As an application, we examine the convexity bias between CMS and forward swap rates.Martingale, Convexity Adjustment, Black and Black Scholes volatility, CMS rates.

Smart Monte Carlo: Various tricks using Malliavin calculus

Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournié et al. (1999), we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ration method of Broadie and Glasserman (1996). We show on numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.Monte-Carlo, Quasi-Monte Carlo, Greeks,Malliavin Calculus, Wiener Chaos.

A Generalisation of Malliavin Weighted Scheme for Fast Computation of the Greeks

This paper presented a new technique for the simulation of the Greeks (i.e. price sensitivities to parameters), efficient for strongly discontinuous payo¤ options. The use of Malliavin calculus, by means of an integration by parts, enables to shift the differentiation operator from the payo¤ function to the diffusion kernel, introducing a weighting function.(Fournie et al. (1999)). Expressing the weighting function as a Skorohod integral, we show how to characterize the integrand with necessary and sufficient conditions, giving a complete description of weighting function solutions. Interestingly, for adapted process, the Skorohod integral turns to be the classical Ito integral.Monte-Carlo, Quasi-Monte Carlo, Greeks,Malliavin Calculus, Wiener Chaos.

Option pricing with Levy Process

In this paper, we assume that log returns can be modelled by a Levy process. We give explicit formulae for option prices by means of the Fourier transform. We explain how to infer the characteristics of the Levy process from option prices. This enables us to generate an implicit volatility surface implied by market data. This model is of particular interest since it extends the seminal Black Scholes [1973] model consistently with volatility smile.Levy process, Fourier and Laplace transform, Smile.

Smart expansion and fast calibration for jump diffusion

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.Comment: in Finance and Stochastics (2009) a paraitr

Pricing convexity adjustment with Wiener chaos

This paper presents an approximated formula of the convexity adjustment of Constant Maturity Swap rates, using Wiener Chaos expansion, for multi-factor lognormal zero coupon models. We derive closed formulae for CMS bond and swap and apply results to various well-known one-factor models (Ho and Lee (1986), Amin and Jarrow(1992), Hull and White (1990), Mercurio and Moraleda (1996)). Quasi Monte Carlo simulations confirm the efficiency of the approximation. Its precision relies on the importance of second and higher order terms