Asymptotic formulae for implied volatility in the Heston model

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.Comment: Presentation in Section 2 has been improved. Theorem 3.1 has been slightly generalised. Figures 2 and 3 now include the at-the-money point

Optimal Fourier Inversion in Semi-analytical Option Pricing

At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities

Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000

Implied volatility asymptotics under affine stochastic volatility models

This thesis is concerned with the calibration of affine stochastic volatility models with jumps. This class of models encompasses most models used in practice and captures some of the common features of market data such as jumps and heavy tail distributions of returns. Two questions arise when one wants to calibrate such a model: (a) How to check its theoretical consistency with the relevant market characteristics? (b) How to calibrate it rigorously to market data, in particular to the so-called implied volatility, which is a normalised measure of option prices? These two questions form the backbone of this thesis, since they led to the following idea: instead of calibrating a model using a computer-intensive global optimisation algorithm, it should be more efficient to use a less robust—hence faster—algorithm, but with an accurate starting point. Henceforth deriving closed-form approximation formulae for the implied-volatility should provide a way to obtain such accurate initial points, thus ensuring a faster calibration. In this thesis we propose such a calibration approach based on the time-asymptotics of affine stochastic volatility models with jumps. Mathematically since this class of models is defined via its Laplace transform, the tools we naturally use are large deviations theory as well as complex saddle-point methods. Large deviations enable us to obtain the limiting behaviour (in small or large time) of the implied volatility, and saddle-point methods are needed to obtain more accurate results on the speed of convergence. We also provide numerical evidence in order to highlight the accuracy of the closed-form approximations thus obtained, and compare them to standard pricing methods based on real calibrated data

Evaluating Discrete Dynamic Strategies in Affine Models

We consider the problem of measuring the performance of a dynamic strategy, rebalanced at a discrete set of dates, whose objective is that of replicating a claim in an incomplete market driven by a general multi-dimensional affine process. The main purpose of the paper is to propose a method to efficiently compute the expected value and variance of the hedging error of the strategy. Representing the pay-off the claim as an inverse Laplace transform, we are able to get semi-explicit formulas for strategies satisfying a certain property. The result is quite general and can be applied to a very rich class of models and strategies, including Delta hedging. We provide illustrations for the cases of interest rate models and Heston's stochastic volatility model.

A robust spectral method for solving Heston’s model

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979)