# Analytical Approximation of Contingent Claims

Sammanfattning: This PhD thesis consists of three separate papers. The common theme is methods to calculate analytical approximations for prices of different contingent claims under various model assumptions. The first two papers deals with approximations of standard European options in stochastic volatility models. The third paper is focused on approximating prices of commodity swaptions in a general model framework. In the first paper, Dynamic Extensions and Probabilistic Expansions of the SABR model, a closed form approximation to prices of call options and implied volatilities in the so called SABR model of Hagan et al. (2002) is derived. The SABR model is one of the most frequently used stochastic volatility models used for option pricing in practice. The method relies on perturbing the model dynamics and approximations are obtained from a second order Taylor expansion. It is shown how the expansion terms can be calculated in a straightforward fashion using the flows of the perturbed model and results from the Malliavin calculus. This technique is further applied to calculate a closed form approximation of the option price in a useful dynamic extension of the original model. The dynamic model is able to match the prices of several options with different maturities and can therefore be used to price path dependents products in a consistent way. In addition, we propose an alternative model specification for the dynamic SABR model where the dynamics of the underlying asset are given by a displaced diffusion. In its non-dynamic version this model has similar properties as the original SABR model but it is more analytically tractable. A closed form approximation of the dynamic version of this model is also derived. The accuracy of the approximations is evaluated in a Monte Carlo study and the method is found to work well for many parameters of interest. The second paper, General Approximation Schemes for Option Prices in Stochastic Volatility Models, further examines the method of approximation employed in the first paper. The method is developed for a general stochastic volatility specification that can generate many commonly employed models as special cases. As an important application the method is used to calculate a second order expansion for the Heston (1993) model. The Heston model is arguably the most often used stochastic volatility model in practice and academic work. A numerical study of the approximations is performed where they are compared to prices and implied volatilities calculated from numerical Fourier transforms. It is found that for several parameters of interest the approximation is very accurate. Relating the proposed method to the existing literature we find that it generalizes the work by Lewis (2005) in several directions. In the case of the Heston model the first order expansion coincides with the approximation proposed in Alos (2006). However, an important advantage of the proposed method is that it can be used to generate higher order terms and it is verified that extending to second order substantially improves the accuracy of the approximation. The third paper, Approximative Valuation of Commodity Swaptions, is concerned with the numerical calculation of prices of European options on commodity swaps. A general approximation scheme for these claims is derived within the model framework of Heath, Jarrow and Morton (1992) extended to include commodity forwards. In models with deterministic volatilities the approach generates a closed form approximation allowing the swaptions to be conveniently priced using Blacks formula. The approximation is evaluated when applied to a Gaussian 2-factor model frequently employed in the literature. A comparison of the approximative prices to Monte Carlo simulations shows that the incurred errors are small for a large set of relevant parameters.

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