Random Binominal Option Pricing Model And Simulation

The emergence of financial derivatives in 1970s marked a most significant and exciting event in the history of finance. Over the past two decades, financial derivatives remain an exuberant vitality, of which, options play a key role. Considerable structural products such as bull-call warrants, equity linked notes (ELN), break-even notes have the elements of options.Since the emergence of options trading, especially of securities options trading, researchers have been engaged in the studies of options pricing. In 1973, Prof. Fisher Black and Prof. Myron Scholes at the University of Chicago published "The Pricing of Options and Corporate Liabilities", where they presented the well-known Black-Scholes model for options pricing (B-S model for short). Since its birth, the B-S model has received strong responses and universally high opinions. While certain researchers conducted thorough tests on the model's accuracy, many others presented various opinions on the problems in the model and expanded the model for the purposes of improvement and extension. Sharp (1978) first proposed the at-money of call option prices by the up and down of share prices. Cox, Ross, Robinstein (1979) obtained the pricing formula using binomial model, which has found wide applications.This thesis concerns options. Based on a thorough review of the existed options research, we particularly examine the binomial and trinomial approaches for options pricing including the latest fuzzy binomial model. We then explore the binomial model with new ideas, which mainly involve the options pricing formula with binomial model whose parameters are modified from constants to random variables satisfying certain random distribution. In the classical CRR binomial model, the two parameters for shares up and down are fixed, which means the shares fluctuation in the continuous model remains unchanged. Since the shares fluctuation changes indeed, we try to represent the two parameters for shares up and down by random variables and to consider their random distribution close to reality, from which we derive some new results.In the traditional binomial model, the parameters of binomial distribution u and d are fixed parameters known in advance. Since u and d represent the margins for shares up and down respectively, the shares fluctuation would remain unchanged if u and d are unchanged within the entire n time intervals. In other words, the shares fluctuation is fixed in the traditional CRR-RB binomial model for options pricing. To reflect the changeable fluctuation of share yields in the model, we set u and d as random variables. Assume that the up parameter and down parameter are independent random variables satisfying beta distribution and that u, d and r satisfy 3-variable normal distribution, we investigate the analytic solutions of the new model, study the parameters estimation and Markov chain Monte Carlo, and compare the numerical methods and results.Let d/r and r/u be independent random variables satisfying Beta (.;β₁,β₂) and Beta (.;α₁ ,α₂) respectively, we obtain the following options pricing formula:Although A(k) and B(k) are rather complicated, they may be approximately estimated in practical computation since the two series absolutely converge.Under the assumption that X'=...and Y'=... satisfy normal distribution, we obtain the new options pricing formula as follows:Since the above formula is very complicated, computational mathematics methods have to be employed in practical applications. We offer in-depth analysis on the implementation of the formula, adopting Hastings-Metropolis algorithm and Monte-Carlo numerical integration of importance sampling for computing the formula.We finally verify our model using the data of the Five-Grain Liquor call warrant and put warrant in Chinese securities market. The result shows that the B-S model and random binomial model may both well simulate the call warrant, whose simulation results differ little with both being basically identical at majority time points and each having its own advantages at minority time points. In the beginning stage of the call warrant, however, the simulation by random binomial model is good but by B-S model not. At most time points, the simulation of the put warrant by the B-S model and random binomial model are poor, which is due to the product features of put warrants and the crazy speculation by certain Chinese securities traders. Despite the poor simulation, the results by the B-S model and random binomial model are basically identical. The random binomial model is slightly better than the B-S model in simulation results.This thesis has investigated the extension of binomial model for options pricing. The extended model needs to be further improved. In theoretical approach, we have tried several random variables distribution close to reality. It is to be determined which assumption is closer to reality. In model deduction, the derived formula is rather complicated requiring advanced computational mathematics and sophisticated computer programming for practical computation. In model verification, the reliability and credibility are significantly impaired by the facts that China has no options in real sense and that China's warrants are far from mature in few quantity, manipulated prices and low relativity with underlying securities. With the introduction of real options in China, more and better market data will be available benefiting the verification and improvement of our model.