Difference Schemes to the Problem of Option Pricing

<p>Financial derivatives are very important for the economy. Interest financial derivatives are distinguished by more complex relationship between the prices and the external conditions.</p><p>Often financial instrument pricing problems are reduced to option pricing problems. Calculation of arbitrage-free prices for financial instruments is the most challenging and up-to-date problem in mathematical economics. One of the main approaches to the pricing problem is to derive a partial differential equation that accurately describes the behavior of the arbitrage-free price of a financial instrument, then to set the initial-boundary value problem, to solve it numerically and, lastly, to evaluate the accuracy of the solution. In this work we use a finite differences method as a numerical method.</p><p>The work considers such financial derivatives as European and American put options, and options, which prices and conditions of exercise depend on multiple parameters. To numerically evaluate option price depending on a single parameter we set the initial-boundary value problem, consider several types of difference schemes (explicit scheme, implicit scheme, with a half-sum scheme, mixed scheme) and describe the various advantages and disadvantages of each of the scheme. Calculated European put option value compared with the analytical solution. Calculated American put option value compared with value which was calculated using the Monte Carlo method.</p><p>The differential equations describing the price behavior were derived for the multifactor option pricing problem. The difference scheme for the equations is given and an algorithms for its solving is presented.</p><p>For all options which were considered in this work we provide the dependency of their prices on various parameters.</p>