Research Of Extended European Options Pricing Models

The Black and Scholes Formula has been one of the most important consequences of the study of continuous time models in finance under the complete financial market assumptions. The Black and Scholes demonstrated Black-Scholes option pricing formula, but it does not match the actual situation. How to expand research for options pricing is one of the important work according to the practice. In the dissertation, using risk neutral martingale measure theory, the principle of no-arbitrage hedge, the technique of change numeraire, as well the option pricing formula, we research the pricing of extended European-style options. In the dissertation, our works are following:1. Generally, the pricing of quanto options are discussed under the floating exchange rate system, but we know that there be a fixed exchange rate system and floating exchange rate system in exchange rate systems. In two different exchange rate systems, with time change, the change of the exchange rate is different form. The fixed exchange rate is not random, and the floating exchange rate is random. Based on these characteristics, this paper presents a systematic framework to derive pricing formulas for different European-style options under different exchange rate systems based on the empirical conclusion and gets closed form solutions of quanto options pricing.2. The extension of the pricing formulations to two-asset exchange options with the quanto feature is considered. Under the floating exchange rate system, the option pricing is three-dimensional in mathematics. We use the numeraire, the equivalent martingale measure, and re-use conclusions of pricing formulations of quanto options and exchange options under two different exchange rate systems, and then we derive the price formulas of quanto exchange options under different exchange rate system.3. In the Black and Scholes Formula, the fitness of the model has been questioned on the basis of the assumption of constant volatility since empirical evidence shows that volatility actually depends on time in a way that is not predictable. Options pricing has been questioned under the assumption of the constant volatility. To study option pricing more comprehensive, we have established a dynamic asset pricing model, so we can study the price of the past how to impact on the price of present and future. First of all, in this article we derive an analogue Kazmerchuk, Swishchuk, and Wu (2007) of Black–Scholes formula for a call option value in the market with stock price having time-delay in the diffusion term. Using stochastic functional differential equations theory and the equivalent martingale measure theory, we further expand to study to option pricing model for dividend-paying stock. We derived option pricing formula for the payment of continuous dividends and discrete dividends. Then, we derive an analogue Arriojas, Hu, Mohhaammed,et a1(2007) of Black–Scholes formula for a call option value in the market with stock price having time-delay in the diffusion term and drift term. We further expand to study to option pricing model for dividend-paying stock too. We derived option pricing formula for the payment of continuous dividends and discrete dividends too.4. In this paper,using principles of economics,presenting demand and supply functions of options, as well as supply and demand adjustment model of option price, get differential equations of the option price. then using a similar cellular neural network modeling approach, according to the characteristics of European options,we obtain a more general with time-delay the European-style option price to satisfy the differential equation model, and we proved that the solutions of option price adjustment model are existence,as well as the solution is globally exponentially stable, our conclusion is consistent with the actual.Generally, financial mathematics is divided into two branches: normative financial mathematics and empirical financial mathematics. The paper mainly uses the research methods of normative financial mathematics in the expansion European type option pricing. The European-style options pricing is the basis of other complex options pricing, so it has a large number of research findings. However, in the paper, expansion research of European-style option pricing is new. Our paper has an important reference values for investors and designers of options, so it is important to theoretical and practical significance.