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Title: Numerical methods for foreign exchange option pricing under hybrid stochastic and local volatility models
Author: Cozma, Andrei
ISNI:       0000 0004 6499 0402
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2017
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In this thesis, we study the FX option pricing problem and put forward a 4-factor hybrid stochastic-local volatility model. The model, which describes the dynamics of an exchange rate, its volatility and the domestic and foreign short rates, allows for a perfect calibration to European options and has a good hedging performance. Due to the high-dimensionality of the problem, we propose a Monte Carlo simulation scheme that combines the full truncation Euler scheme for the stochastic volatility component and the stochastic short rates with the log-Euler scheme for the exchange rate. We analyze exponential integrability properties of Euler discretizations for the square-root process driving the stochastic volatility and the short rates, properties which play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a large class of stochastic differential equations in finance, including the ones studied in this thesis. Hence, we prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. Then, we calibrate the model to market data and discuss its fitness for pricing FX options. Next, due to the relatively slow convergence of the Monte Carlo method in the number of simulations, we examine a variance reduction technique obtained by mixing Monte Carlo and finite difference methods via conditioning. We consider a purely stochastic version of the model and price vanilla and exotic options by simulating the paths of the volatility and the short rates, and then evaluating the "inner" Black-Scholes-type expectation by means of a partial differential equation. We prove the convergence of numerical approximations and carry out a theoretical variance reduction analysis. Finally, we illustrate the efficiency of the method through a detailed quantitative assessment.
Supervisor: Reisinger, Christoph Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Options (Finance)--Mathematical models ; convergence ; option pricing ; calibration ; Euler scheme ; Monte Carlo ; conditional Monte Carlo ; integrability