Application Of Monte Carlo Methods In Quantitative Finance

The derivatives are important hedging tools in secondary market. To successfully avoid the risk, the derivatives must be priced accurately. Some simple options have analytic solution under severe assumptions. However, we usually use numerical methods to price options. Binomial lattices method and finite difference method calculate in a high speed for low dimensional derivative. However, for exotic options, Monte Carlo simulation may be the only viable approach in practical application.We do research in application of quantitative finance using Monte Carlo simulation. Firstly, we introduce the general principles of the Monte Carlo methods. Be aimed at the disadvantage of large calculation of ordinary Monte Carlo method, we proposed variance reduction techniques and Quasi-Monte Carlo simulation to improve it. In this paper, the variance reduction methods consist of antithetic sampling, control variate method, conditional Monte Carlo simulation and importance sampling, Quasi-Monte Carlo simulation consist of Halton low-discrepancy sequences and Sobol low-discrepancy sequences. Next we introduce quantitative financial problem. We selected to price value of constant mix strategy (CM strategy) multi-period return guarantee. This value is theoretical basis of charging by the issuers who use CM strategy set stop-loss to manage a principal guaranteed fund. The underlying asset is driven by compound Poisson process and Wiener process. This pricing problem embeds path-dependent options. Monte Carlo simulation is good at dealing with such a high dimensional quantitative financial problems. We derive the expression of the present value of CM strategy multi-period return guarantee that based on risk-neutral measurement. Then we use ordinary Monte Carlo method, antithetic sampling, control variate method, conditional Monte Carlo simulation and Halton low-discrepancy sequences combined with Brownian Bridge respectively to derive the simulation formulas of this present value.Numerical solutions of value of CM strategy multi-period return guarantee were calculated under given parameters by the above simulation formula. Results show all fiveMonte Carlo methods can calculate the numerical solution effectively. Then we appraise the accuracy of the first four Monte Carlo methods through the length of the confidence interval under a given level of significance. Results show ordinary Monte Carlo method has the worst accuracy. The accuracy of the three variance reduction techniques has been improved. The best is the conditional Monte Carlo simulation. Then we use conditional Monte Carlo simulation to analyze value of CM strategy multi-period return guarantee on the different parameters range. The results show the two parameters of random jump sizes have unable pricing range. When the two parameters are greater than a certain value, value of CM strategy multi-period return guarantee tends to zero. Four parameters that called periods, Poisson process intensity, ratio of risk assets and initial value of the portfolio have a positive effect on value of CM strategy multi-period return guarantee. Volatility of risk assets also has a positive effect on this value, but there is an inflection point. When the volatility exceeds a certain value, value of CM strategy multi-period return guarantee tends to a constant. When the guaranteed line is less than 1, value of CM strategy multi-period return guarantee is 0. With guaranteed line is equal or greater than 1, this value increases rapidly.