Computer Science Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2756
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Item Pricing Volatility Derivatives Using Space Scaled Levy Processes(2008-09-02) Prakash, Samvit; Madan, Dilip B; von Petersdorff, Tobias; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The VIX index measures the one-month risk-neutral forward volatility of the S&P500 (SPX) index. While Lévy processes such as the CGMY process can price options on the underlying stock or index, they implicitly assume a constant forward volatility. This makes them unsuitable for pricing options on VIX. We propose a model within the one dimensional Markovian framework for pricing VIX and SPX options simultaneously. We introduce space dependence of volatility by scaling the CGMY process with a leverage function. The resultant process can consistently price options on SPX and VIX of a given maturity. We also perform surface calibrations of options on the two indices separately. We explore the properties of the implied distribution of the SPX from both indices and conclude that the VIX index under-weighs small jumps as compared to large jumps as well as the skewness of the SPX index .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 Pricing variance derivatives using hybrid models with stochastic interest rates(2008-05-02) Smetaniouk, Taras; Madan, Dilip; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis, the research focuses on the development and implementation of two hybrid models for pricing variance swaps and variance options. Some variance derivatives (i.e., variance swap) are priced using portfolios of put and call options. However, longer-term options price not only stock variance, but also interest rate variance. By ignoring stochastic interest rates, variance derivatives utilizing this approach are overpriced. In recent months, the Federal Reserve lowered the funds rate as the equity markets fell. This created correlation between equities and interest rates. Furthermore, interest rate volatility increased. Thus, it is presently crucial to understand how stochastic interest rates and correlation impact the pricing of variance derivatives. The first model (SR-LV) is driven by two processes: the stock return follows a diffusion and the stochastic interest rate is driven by the Hull-White short rate dynamics. Local volatility is constructed with the help of Gyongy's result on recovering a Markov process from a general n-dimensional Ito process with the same marginals. In the solution for the local volatility, the joint forward density of the stock price and interest rate is derived by solving an appropriate partial differential equation. Realized variance can then be computed by Monte Carlo simulation under the forward measure where local variances are collected over each realized path and averaged. Results are presented for different levels of assumed correlation between the stock price and interest rates. Prices obtained are lower than those produced with an options portfolio and this price difference strongly depends on the volatility of the short rate. The second model (SR-SLV) adds one more dimension to the first model. In practice, volatility of a stock may change without the stock price moving. This effect is not captured in SR-LV model, but stochastic local volatility exhibits this trait. In this setting, a leverage function must be calibrated utilizing the joint density of the stock price, interest rate, and a stochastic term governed by a mean reverting lognormal model. By design, the price of variance swaps is the same as under SRLV dynamics. However, variance option prices differ from SR-LV model and are presented for different levels parameters of the new stochastic component. Although this work focuses on pricing variance derivatives, the developed methodology is extended to pricing volatility swaps and options.Item Comparing Regime-Switching Models In Time Series: Logistic Mixtures vs. Markov Switching(2007-05-16) Paliouras, Dimitrios V.; Kedem, Benjamin; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The purpose of this thesis is to review several related regime-switching time series models. Specifically, we use simulated data to compare models where the unobserved state vector follows a Markov process against an independent logistic mixture process. We apply these techniques to crude oil and heating oil futures prices using several explanatory variables to estimate the unobserved regimes. We find that crude oil is characterized by regime switching, where prices alternate between a high volatility state with low returns and significant mean reversion and a low volatility state with positive returns and some trending. The spread between one-month and three-month futures prices is an important determinant in the dynamics of crude oil prices.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.Item Multivariate Levy Processes for Financial Returns(2004-11-10) Yen, Ju-Yi; Madan, Dilip B; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We apply a signal processing technique known as independent component analysis (ICA) to multivariate financial time series. The main idea of ICA is to decompose the observed time series into statistically independent components (ICs). We further assume that the ICs follow the variance gamma (VG) process. The VG process is evaluated by Brownian motion with drift at a random time given by a gamma process. We build a multivariate VG portfolio model and analyze empirical results of the investment.