Surveys On Application Of Homotopy Method In Pricing Financial Derivatives

The current financial derivatives which plays a more and more important role in the international financial market is broadly called the core of the capital market. It develops being based on traditional financial instrument including options;futures, forwards, swaps etc.Options is a very extensive kind of financial derivatives used widely in the financial market, by which company and big financing institution usually carry on risk management. Convertible bonds(CBs)is also the complicated financial instrument that used widely in the financial market.As for approaches of option, valuation of American -style options is much more complicated than that of European -style . It depends on a few other factors :the price of assets (for example stock), the time to expiration , strike price, bank interest rate, volatility .For American options, the essential difficulty lies in the problem that it is allowed to be exercised at any time before the expiry and mathematically such an early excise right purchased by the holder of the option, changes the problem into a so-called free boundary problem, since the optional exercise boundary , prior to the expiration of the option is now time-dependent and is part of the solution. As a result of the unknown boundary being part of the solution, the valuation of American options becomes any other free boundary problems. This is very different from the valuation of European options as the latter is only a linear problem of the well-know Back-Scholes equation is solved.It is well known that CBs with combining characteristics of bonds and options are hybrid financial instruments . The characteristic of bonds means CBs is a kind of bonds derived from interest rate, so value is influence by interest rate; while The characteristic of options means it is a kind of stock, value is influenced by stock price. If conversion is allowed at any time prior to expiry, we say that CBs is of American style. When add default risk and bonus policy of issuer to valuation, Valuation of CBs is more complicated than that of option.It is French mathematician Lowis Bachelier who first puts forward valuation of option model in 1900. Later on, Sprekle , Boness, S amuelson... etc. put forward different valuation of option model respectively. But they all can not completely solve. Until two mathematicians, economist Fischer Black and Myron Scholes deduced mathematical formula in 1973, proposed the approaches of option valuation and other similar financial deritvatives. Fischer Black and Myron Scholes deduced valuation of European option model under the assumption condition of risk neuter, complete market and assets obeying Brown exercise etc. making use of the strategy of continuous trade constructed a formula of European put option :It doesn't depend on the hobby of investor and lead all investors to the risk neutral world. Afterward, Merton, Cox, Ross, Ingersoll carried on thorough research and improvement to valuation of option again, and expand it to the stock option, stock option index , monetary option... etc. valuation of financial derivatives which established foundation of option valuation.The problem of valuation of American put option can trace back to McKean and Merton who first suggested that the valuation of American options should be treated as a free boundary problem .this has drawn considerable research interests in this area. The research of option valuation model is the foundation of CBs valuation mode , The theoretical framework for pricing CBs valuation was started by the Ingersoll, Bremanh and the Schwartz , the pricing CBs depends on Company market value, taking no account of interest rate variety, therefore, their model is essentially single factor model. Later, Bremanh and Schwartz investigated the effect of stochastic interest rate and found that the effect of a stochastic term structure on convertible bond prices is so small that it can be ignored for empirical purposes. In 1986, McConnel and Schwartz developed a valuation model, using the stock value as the underlying stochastic variable.There are two types of approximate approaches, numerical solutions and analytical approximations for the valuation of American options, each type has its own advantages and limitations. Of all numerical approaches, there are two subcategories , in the first subcategory, the Black-Scholes equation is directly solved with both time and stock being discretized. Typical approaches are the finite difference method and the finite element method. Approaches in the second subcategory are based on the risk-neutral valuation at each time step. The Binomical method and the Monte Carlo simulation method are the typical examples in this subcategory. Many of these methods still require intensive computation before a solution of reasonable accuracy can be obtained and in some cases, such as the explicit finite -difference scheme, the method may not even converge as pointed by Huangetal. For American options, it can be shown that the solution near maturity is of singular behavior. Naturally, it is difficult for most numerical methods to calculate the option price accurately in the neighborhood of maturity.In comparison with numerical methods, analytical approximations are usually of simple form and can be easily computed, typical methods in this category include the compound-option approximation method, the quadratic approximation method, the capped option approximation, the integral-equation method and the simple approximation through a nonlinear algebraic equation derived by Bunch and Johnson.About two last years, based on Pseulo-steady-state approximation, Doctor Zhu Songping proposed a new analytical approximation formula for the optimal exercise boundary . In the meaning of Financial engineer, his formula seems to be simple and versatile. It must be pointed out that all these analytical approximation methods still require a certain degree of computation at the end. However, unlike the numerical approximation approaches mentioned, various approximations are made in order to reduce the intensity of the final numerical computation.This paper gives a broad overview.of Zhu Songping' s research, who presented a closed-form solution for the valuation of American options by constructing a Taylor's series expansion of the unknown option price and the unknown optimal exercise price based on the homotopy analysis method, The terminology "closed-form" has been used in the literature of financial derivatives' pricing theory in different ways. Here we use the definition given by Gukhal, That is, by being a "closed-form" solution, it is meant that the solution can be written in terms of a set of standard and generally accepted mathematical functions and operations. A solution in the form of an infinite series expansion is certainly an in closed form by this definition. By "explicit", we mean that the solution for the unknown function can be determined explicitly in terms of all the inputs to the problem. At present, as we know, such a solution does not exist in the other literature.The parameter p is obtained as a result of the homotoply deformation constructed when the parameter P is varied continuously in the domain[0, 1] the new series solution id a closed-form exact solution because all the differential equations and boundary conditions can be satisfied exact and it is an explicit analytical solution because not only the option price, but also the optimal exercise boundary are determined explicitly as a function of all the input variables such as risk -free interest rate, the volatility and the time to expiration.Because Doctor Zhu Songping makes use of the homotopy analysis method, the nonlinear problem of valuing American options is analytically solved .It can be expressed explicitly in a closed form in terms of the input parameters such as the risk-free interest rate, the volatility and the time to expiration. This closed-form analytical solution can be used to validat other numerical solutions designed for more complicated cases where no solutions exist.Being the new important result of mathematical field and financial circles, the explicit solution discovered by Zhu Songping have become standard of verify other mathematics models, his method is more useful than that of before, applied to look for explicit solution of other American-style options .As financial mathematician, Pittsburgh university professor Ken Kortanek say, for American put option, find out its accurate explicit solution is a milestone of quantity economy.