Browsing by Author "Kienitz, Jorg"

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    Bias-Free Joint Simulation of Multi-Factor Short Rate Models and Discount Factor
    (2018) Lopes, Marcio Ferrao; McWalter,Tom; Kienitz, Jorg
    This dissertation explores the use of single- and multi-factor Gaussian short rate models for the valuation of interest rate sensitive European options. Specifically, the focus is on deriving the joint distribution of the short rate and the discount factor, so that an exact and unbiased simulation scheme can be derived for risk-neutral valuation. We see that the derivation of the joint distribution remains tractable when working with the class of Gaussian short rate models. The dissertation compares three joint and exact simulation schemes for the short rate and the discount factor in the single-factor case; and two schemes in the multifactor case. We price European floor options and European swaptions using a twofactor Gaussian short rate model and explore the use of variance reduction techniques. We compare the exact and unbiased schemes to other solutions available in the literature: simulating the short rate under the forward measure and approximating the discount factor using quadrature.
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    Local Stochastic Volatility—The Hyp-Hyp Model
    (2020) Cowen, Nicholas; McWalter, Thomas; Kienitz, Jorg
    Volatility modelling is used predominantly in order to explain the volatility smile observed in the market. Stochastic volatility models are mainly used to capture the curvature of a volatility smile while local volatility models generally model the skew. Jackel and Kahl ¨ (2008) present a hyperbolic-local hyperbolic-stochastic volatility (Hyp-Hyp) model which aims to improve upon existing local and stochastic volatility models such as the stochastic alpha, beta, rho (SABR) and constant elasticity of variance (CEV) models. The advantageous features of the Hyp-Hyp model are corroborated by implementing the model. Jackel and Kahl ¨ (2008) investigate the accuracy of a scaled analytical approximation for implied volatility, based on approximations presented by Watanabe (1987) and Fouque et al. (2000), for the Hyp-Hyp model. They use the approximation to derive an expression for the delta of an option. This dissertation analyses the Hyp-Hyp model, as well as the approximation, by deriving expressions for other sensitivities and by investigating the effect of the Hyp-Hyp model parameters on the volatility smile. The accuracy of the analytical approximation for functional forms other than those defined by the Hyp-Hyp model is explored. A derivation of the approximation is undertaken, presenting corrections to the expressions introduced by Kahl (2007) and used by Jackel and Kahl ¨ (2008).
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    Recursive marginal quantization: extensions and applications in finance
    (2018) Rudd, Ralph; Kienitz, Jorg; Platen, Eckhard
    Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing.