The Research On Almost Exact Simulation Of Heston Stochastic Volatility Model

Since Heston stochastic volatility model effectively overcomes the disvantages of the well-known Black-Scholes such as "volatility smile" and so on, it is often used to descirbe dynamics of variance and price process of underlying asset especially for option. When the underlying has only one stock, we can capture the closed-form solution of Heston model, i.e. derivatives can be pricing exactly under such condition. In fact, at the real market, underlyings of derivatives are usually much more complex, in such cases, we can’t obtain the closed-form solution of Heston model, that is to say, derivatives can’t be pricing in these conditions. However, pricing derivatives accurately is the basic premise of risk management in financial market. In this way, researches on numerical schemes of Heston model are becoming increasingly important in computational finance. In this article, I choose a option with one stock which price follows random walk, and propose two modified almost exact discretization schemes to find out the numerical solution for Heston model based on the exist numerical schemes.First of all, we can distinguish the existing discretization schemes of Heston model into two categories based on different principle of dealing with variance process:one is biased Ito-Taylor schemes, the other is almost exact schemes. The former category use a discretization method with a constant estimate of the variance during each time step, conbining with proper intermediate operator, Ito formula and Taylor expansion formula are both introduced to these schemes to lower the decay rate of state variables. And then, initial differential equations are inverted into difference equations directly to roughly estimate the model. Obviously, the advantage of these methods is its small computational cost,in this way, these schemes always have high computing efficiency. However, the disadvantage of these schemes is they rely on model’s parameter setting, if parameters of Heston model violate the famous controlling standard:Feller condition, use these methods for pricing derivatives usually brought large discretization deviations (compared with closed-form solution). The latter is based on sampling from the exact conditional distribution of viarance to get the exact or nearly exact samples of variance process or integrated variance process.One of its advantages is we can obtain the numerical solution which is closely to closed-form solution. Because the exact or almost exact samples are generated by using large quantities of inverse Fourier or the first modified Bessel functions computing, the computation cost of these schemes are always quite expensive. In this way, although elegant, but practitioners rarely adopt direct exact simulation scheme.Secondly, after observing both advantages and disadvantages of existing numerical schemes, I propose ES-QE scheme and QE-IG scheme for discretization of Heston model by combining ES and QE scheme together and the same as QE and IG scheme. And then, analysis and algorithm of both schemes are given. They are the innovations of this article.Then, use matlab to achieve numerical simulation of both new methods and several typical methods in pricing European call and American put option separately to test the efficiency and accuracy of these new algorithms. Here, I proposed modified least square Monte-Carlo-neural network method (MLSM-NN) to pricing American put option.Finally, conclusions prospects for future work are given based on the simulation results.