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Modeling Dependence in Data: Options Pricing and Random Walks


In this thesis, we propose the Markov tree option pricing model and subject it to large-scale empirical tests against market options and equity data to quantify its pricing and hedging performances.

We begin by proposing a tree model that explicitly accounts for the dependence observed in the log-returns of underlying asset prices. The dynamics of the Markov tree model is explained together with implementation notes that enable exact calculation of the probability mass function of the Markov tree process. We also show that the tree model operates in the framework of arbitrage free option pricing.

Next, we show how the discrete Markov tree process can be viewed as a generalized persistent random walk and demonstrate how to approximate it by a mixture of two normals. This derivation enables us to obtain a closed form pricing formula for the European call option allowing for faster calibration using market option data. We then empirically test both the pricing as well as the hedging performance of the Markov tree model against the Black-Scholes and the Heston's stochastic volatility models establishing its superior hedging performance. Additionally, we also analyze different regression based techniques to estimate the parameters in the Markov tree model that obtain increasingly better hedging results. We also lay down statistical procedures to rigorously analyze the hedging performance of any option pricing model.

We then generalize the Markov tree process and explore its relation with the generalized delayed random walk. In doing so, we develop a spectral method for computing the probability density function for delayed random walks; for such problems, the spectral method we propose is exact to machine precision and faster than existing methods. In conjunction with step function approximation and the weak Euler-Maruyama discretization, the spectral method can be applied to nonlinear stochastic delay differential equations. We carry out tests for a particular nonlinear SDDE that shows that this method captures the solution without the need for Monte Carlo sampling.

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