Option Valuation And Investment Risk Under The Finite State Multi-Period Model

One of the most important tasks of the modern mathematical finance is to seek forthe theory and methodology which are beyond the classical Black-Scholes formula. Thosenew models are devoted to supply a gap of the classical Black-Scholes formula, or to buildup the theories and methodologies for the incomplete market. There are some economistswho document that the price processes in the real market are discrete and suggest usingpurely jump processes to describe the dynamics of the underlying asset prices insteadof the continuous geometric Brownian motion adopted by the Black-Scholes framework.Therefore a purely jump process is adopted in this research to describe the dynamics ofthe underlying asset price, called the finite state multi-period model.The financial market under the finite state multi-period model is an incompletemarket, and the theories and methodologies for the fairly pricing of the derivative secu-rities in incomplete market are needed. If the market is a fair market and there is noarbitrage opportunity, there is a measure under the non-arbitrage constrain such thatthe summation of the price process and the cumulated dividends is a martingale underthis measure, which is called equivalent martingale measure. If the market is a completemarket, that is to say any contingent claim is replicable by the portfolio of the underlyingasset and the bond, there is a unique equivalent martingale measure such that the priceof the contingent claim can be expressed by the mathematical expectation under theequivalent martingale measure. However, the set of equivalent martingale measures isnot in general singleton in incomplete market, so some principle is needed to choose oneof the equivalent martingale measures as the pricing measure, such that the price of theclaim can be expressed as the mathematical expectation under this measure.Several criteria have been proposed in the literature, and the principle of minimizingthe relative entropy is regarded as one of the proper criteria. So the pricing measure inthis research is chosen according to this criterion, called the minimal entropy martingalemeasure. The finite state multi-period model for one underlying asset is considered in thisresearch, and the minimal entropy martingale measures are deduced for the underlyingassets with or without dividend payments respectively. The minimal entropy martingale measure is the equivalent martingale measure whichadds the lest information into the natural market measure, so it can incorporate themost historical information into the prediction of the future dynamics, and it ensures thefairness of the market. The pricing methods for the evaluation of European option andAmerican option are proposed, which are based on the Monte Carlo simulation and theMarkov chain Monte Carlo simulation respectively, using the minimal entropy martingalemeasure.The performance of the canonical valuation method, the historical volatility basedBlack-Scholes formula, and the proposed method is compared in a classical artificialBlack-Scholes world. From the results of simulations, it is found that the performanceof the proposed method is better than that of the canonical valuation when the optionsare out-of-the-money and at-the-money. When the options are in-the-money, the per-formance of the proposed method is very well, while that of the canonical valuation isbetter. In the comparison between the proposed method and the historical volatilitybased Black-Scholes formula, it is found that the performance of the proposed methodis satisfaction, although which is a nonparametric method without any knowledge thatthe price process of the underlying asset is a geometric Brownian motion. This is anevidence that the proposed method can collect the information in the historical pricesand apply to the pricing procedure e?ciently.An 8 weeks investment plan is designed to test the performance of the proposedmethod in the real market. The European option is bought under the current marketprice of the underlying asset following the valuation method every day in the investmentplan, and the total gross returns of the whole investment plan is used as the measurementof the e?ciency of the valuation method. In the empirical investigations involving thehistorical price of S&P 500 stock index, it is found that the proposed method is morecautious than the canonical valuation and the Black-Scholes formula, especially for thedeep out-of-the-money European options the lost of the proposed method is the leastamong the three valuation methods.The e?ciency of the valuation method for American option is measured by theperformance of its optimal exercise boundary in the 8 weeks investment plan. In the empirical investigations involving the historical price of S&P 100 stock index, it is foundthat the optimal exercise boundary of the proposed method can help the investor exerciseshis American option at proper time e?ciently.Under the finite state multi-period model, not only the derivative security valuationis discussed, but also the connection between the market tendency and the relationshipof the dividend rate and the riskless interest rate. And investing in the stock marketis proved to be riskier than investing in the fixed income market under the finite statemulti-period model in this research.