Direct inversion for the Heston model
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The Heston stochastic volatility model is commonly used in financial mathematics. While closed form solutions for pricing vanilla European options are available, this is not the case for other exotic options, especially for path dependent ones, where Monte Carlo methods are often applied. In this thesis, we develop an accurate and efficient simulation method for the Heston model, which is then employed in the pricing of options that are computationally challenging. We consider the problem of sampling the asset price based on its exact distribution. One key step is to sample from the time integrated variance process conditional on its endpoints. We construct a new series expansion for this integral in terms of infinite weighted sums of exponential and gamma random variables through measure transformation and decompositions of squared Bessel bridges. This representation has exponentially decaying truncation errors, which allows efficient simulations of the Heston model. We develop direct inversion algorithms combined with series truncations, leading to an almost exact simulation for the model. The direct inversion is based on approximating the inverse distribution functions by Chebyshev polynomials. We derive asymptotic expansions for the corresponding distribution functions to evaluate the Chebyshev coefficients. We also design feasible strategies such that those coefficients are independent of any model parameters, whence the resulting Chebyshev polynomials can be used under any market conditions. Efficiency of our method is confirmed by numerical comparisons with existing methods.