Analysis, Numerical Methods For Solving The Black-scholes Equation

In today's complex and volatile financial markets, financial derivatives have attracted wide welcome by its powerful leverage and hedging features. Since the Chicago Stock Exchange started the foreign exchange futures transaction in 1972, there come out endless variety of financial derivatives, including futures, forwards, swaps and options. China has also launched a stock index futures and margin recently. Options are used to control risk within a certain range, reducing transaction risk through hedging.In the options study, the most innovative work goes to Black, and Scholes, who derived famous Black-Scholes option pricing equation in 1973. As they made outstanding contribution to option pricing theory, Black and Scholes were awarded Nobel Prize for Economics in 1997. After 1973, many financial experts and mathematicians also did a great deal of work, extending Black-Scholes Equation of the normal distribution to more realistic state of the situation, or constructing a new numerical form to calculate the Black-Scholes equation. Options research methods are mainly binary tree, Monte Carlo methods and finite difference methodExperienced 30 years of reform and opening up, China's financial market are growing to maturity. Besides the general demand for financial products, the need of financial derivatives is also growing. Margin trading and stock index futures, has landed on China's financial derivatives market one after another, and opened the door for financial derivatives' coming. Under such development trend, strengthening the research on the financial derivatives models, especially the option pricing model, is particularly important.Black-Schoels equation and Navier-Stokes equation have many similarities, and they are the binary quadric partial differential equations with variable coefficient. But the difference is that the diffusion coefficient of Black-Scholes equation is negative, if we use the same finite difference method which used to treat conventional Navier-Stokes equations, we may get unstable results. The Black-Schoels equation with variable coefficient has no analytical solution, but we can give its approximate result through numerical method. The main difficulty is the format of the structure and the selection of mesh size and boundary. Different numerical methods may eventually lead to the opposite results. The main work of this paper is to construct a high-precision format to calculate the Black-Schoels equation, give the numerical solution of constant coefficient equations, and compare the results with those of the analytical solution. Besides, it also gives the numerical solution of variable coefficient equation. It also made some preliminary attempts for the use of high-precision lattice in numerical solution of Black-Scholes equation. At the same time, this paper also provides a specific definition for the volatility in Black-Scholes equation and empirical research on the Chinese market, linking the abstract theory and the actual situation of Chinese stock market.