Extending the Levy Processes to Multiasset Products Pricing
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Levy processes have gained great success in pricing single asset options. In this thesis, we introduce a methodology enabling us to extend the single asset pricing technique based on Levy processes to multiasset cases.
In our method, we assume the log-return of each asset as a linear sum of independent factors. These factors are driven by the Levy processes, and the specific Levy process we are studying in this thesis is the Variance Gamma (VG) process. We recover these factors by a signal processing technique called independent component analysis (ICA), from which we get the physical measure (P measure). To price the contingent claims, we still need the risk-neutral measure (Q measure). We bridge the gap between physical measure and risk-neutral measure by introducing the transformation of measure between the P measure and the Q measure. We next write each asset as the linear sum of factors under risk-neutral measure. Thus, we may calibrate the measure change parameters simultaneously through individual listed option data. With the measure change parameters (from P measure to Q measure) being recovered, we're able to price multiasset products by doing Monte Carlo simulation.
In this thesis, we also explore the possibility of extending Levy processes to multiasset product pricing by applying the copula method. Generally speaking, the copula method enables us to introduce the dependence structure for arbitrary marginal distributions. The probabilistic interpretation of copulas is that we may apply the copula method to write the multivariate distributions for any marginal distributions. We consider examples from two different copula families - the elliptical copula family and Archimedean copula family. We studied Gaussian and Clayton one factor copulas as the examples from these two classes. We calibrated the correlation parameters for both methods and found them inconsistent across different strikes and maturities. And like the volatility smile and skew in the Black-Scholes model, we call it the skew and smile effect of correlation for one factor copula method.