Discretizations Of Heston Model Under Quasi-Monte Carlo Method

Black-Scholes model is one of the most famous and useful model for option pricing. Under Black-Scholes model, volatility of underlying asset is constant. If Black-Scholes model were absolutely true to price options, the implied volatility would be only one value regardless of the asset price of the option. But actually, the smile and skew of implied volatility show that market option prices are not priced under Black-Scholes model although it has been proven that Black-Scholes model is the best way to estimate the volatility. Thus the question is, what the relationship between the implied volatility and the true volatility is. Heston model is one of the most famous stochastic volatility models which was proposed by Heston in 1993. There is lots of evidences show that the volatility is stochastic and the distribution of the returns of risk asset has longer tails than normal distribution. It also shows that Black-Scholes model has too strict assumptions for asset prices so that stochastic volatility model is able to reflect the true market more appropriately. But Heston model is very complex, especially on calculating exact solutions of options even if the option is European. After Heston proposed the closed-form solution, more and more researchers turn to the discretization numerical solutions of the model.The paper does some research in quasi-Monte Carlo method for discretizations of Heston model. We choose Euler Scheme and Milstein Scheme for the discretizations and European call option, Asian call option and Lookback call option for the options choices. We use quasi-Monte Carlo method instead of Monte Carlo method, which was widely used before, to generate more uniformly points. In which, we use Sobol’ sequence and randomized quasi-Monte Carlo method since Sobol’ sequence is much better than Halton sequence and Faure sequence cannot do better than it. Whereas randomized quasi-Monte Carlo method can improve the efficiency of simulation and does good help of the results. In the end of the paper, we prove that quasi-Monte Carlo method has higher efficiency than Monte Carlo method using discretizations which means the variance is reduced under quasi-Monte Carlo method.