| The pricing of financial derivatives and estimating the risk of financial products are two important parts in financial mathematics. The analysis of behavior of financial products price is the key to solving the financial mathematics problems, also is the reference of measuring the value of the mathematical models. In real life, financial products are various, the price of them changes at any time. Through the appearance, how to use the best mathematical way to describe the characteristic of behavior of financial products price is the key to our research.In this issue, the author made some work on these aspects:①Tested and verified the behavior of stock price in China's Shanghai and Shenzhen stock exchange market has obvious fractal feature, it can be characterized by fractal theory.②Base on the fractal theory, improved the classic traditional option pricing model-CEV (constant elasticity of variance) model which is base on the efficient market hypothesis. Using fractional Brownian motion to replace geometric Brownian motion, the efficient market hypothesis is generalized as fractal market hypothesis. In this paper, a fractal CEV model is proposed, which is based on the fractal market hypothesis. Meanwhile, the option pricing formula obeying the fractal CEV model was derived. Because it was very difficult to get analytical solutions of the option pricing equation, a Monte Carlo simulation is provided.③In this paper, a new method for estimating the VaR (Value at Risk) of stock option is proposed, which is also based on fractal theory. As the behavior of stock-market price is analyzed theoretically and some exemplifications are given. It shows that the continuously logarithmic returns of stock can be stimulated by fractal distribution. VaR calculating formula of stock option is deduced, and some examples are given. |
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