Computer Science Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2756
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Item Dependence Structure for Levy Processes and Its Application in Finance(2008-06-06) chen, qiwen; Madan, Dilip B; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this paper, we introduce DSPMD, discretely sampled process with pre-specified marginals and pre-specified dependence, and SRLMD, series representation for Levy process with pre-specified marginals and pre-specified dependence. In the DSPMD for Levy processes, some regular copula can be extracted from the discrete samples of a joint process so as to correlate discrete samples on the pre-specified marginal processes. We prove that if the pre-specified marginals and pre-specified joint processes are some Levy processes, the DSPMD converges to some Levy process. Compared with Levy copula, proposed by Tankov, DSPMD offers easy access to statistical properties of the dependence structure through the copula on the random variable level, which is difficult in Levy copula. It also comes with a simulation algorithm that overcomes the first component bias effect of the series representation algorithm proposed by Tankov. As an application and example of DSPMD for Levy process, we examined the statistical explanatory power of VG copula implied by the multidimensional VG processes. Several baskets of equities and indices are considered. Some basket options are priced using risk neutral marginals and statistical dependence. SRLMD is based on Rosinski's series representation and Sklar's Theorem for Levy copula. Starting with a series representation of a multi-dimensional Levy process, we transform each term in the series component-wise to new jumps satisfying pre-specified jump measure. The resulting series is the SRLMD, which is an exact Levy process, not an approximation. We give an example of alpha-stable Levy copula which has the advantage over what Tankov proposed in the follow aspects: First, it is naturally high dimensional. Second, the structure is so general that it allows from complete dependence to complete independence and can have any regular copula behavior built in. Thirdly, and most importantly, in simulation, the truncation error can be well controlled and simulation efficiency does not deteriorate in nearly independence case. For compound Poisson processes as pre-specified marginals, zero truncation error can be attained.Item Stochastic Volatility with Levy Processes: Calibration and Pricing(2005-11-24) Wu, Xianfang; Madan, Dilip B.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis, stochastic volatility models with Levy processes are treated in parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricing through the partial integro-differential equation (PIDE) approach. First, different models where the underlying log stock price or volatility driven by either a Brownian motion or a Levy process are examined on Standard & Poor's (S&P) 500 data. The absolute percentage errors show that the calibration errors are different between the models. Furthermore, a new method to estimate the standard errors, which can be seen as a generalization of the traditional error estimation method, is proposed and the results show that in all the parameters of a stochastic volatility model, some parameters are well-identified while the others are not. Next, the previous approach to parameter calibration is modified by making the volatility constrained under the volatilty process of the model and by making the other model parameters fixed. Parameters are calibrated over five consecutive days on S&P 500 or foreign exchange (FX) options data. The results show that the absolute percentage errors do not get much larger and are still in an acceptable threshold. Moreover, the parameter calibrating procedure is stabilized due to the constraint made on the volatility process. In other words, it is more likely that the same calibrated parameters are obtained from different initial guesses. Last, for the PIDEs with two or three space dimensions, which arise in stochastic volatility models or in stochastic skew models, it is in general inefficient or infeasible to apply the same numerical technique to different parts of the system. An operator splitting method is proposed to break down the complicated problem into a diffusion part and a jump part. The two parts are treated with a finite difference and a finite element method, respectively. For the PIDEs in 1-D, 2-D and 3-D cases, the numerical approach by the operator splitting is carried out in a reasonable time. The results show that the operator splitting method is numerically stable and has the monotonicity perserving property with fairly good accuracy, when the boundary conditions at volatility are estimated by Neumann conditions.