Research On High-order Compact Difference Schemes For Pricing American Options

Since 1970s, the options market has began to rise. After more than 40 years’rapid development, the options market has become the most important part of the derivatives market. On February 9,2015, Shanghai introduced the Shanghai Stock Exchange (SSE) 50 ETF options, China began to carry out the options trading formally. Though Chinese options market developed rapidly in the last year, there still exists a huge gap with the international mature options market in the scale of the options market due to a late start, that is to say, Chinese options market have a tremendous development space.Accurate pricing is the basic premise for options acting as a risk management tool. But because of it can be performed in advance, the American options pricing models don’t have analytical solution at most times. That is why the American options pricing problem is always one of the most important and popular issues in the discipline of financial mathematics.As the most classical options pricing model, there are many research achievements about B-S model. Among them, D.Y. Tangman proposed to take a front-fixing point transformation to the partial differential equation (PDE) and this changes the unknown boundary into a fixed boundary, and then D.Y. Tangman discretized the PDE by using the High-order compact format.There are two fictitious points P-1j+1 and P-2j+1 in the discrete format. When dealing with the virtual point, D.Y. Tangman assumed that the options price at the fictitious points equaled to the value of transformed payoff function directly. This method lowered the precision of the whole format. At the same time, front-fixing point transformation turned the linear PDE to a nonlinear PDE.Bertram During and Michel Fournie applied High-order compact format to American options pricing under the Heston model. Oosterlee et al. proposed that tightly refining the grids around the strike price by applying Grid-stretching transformation to the B-S model. Though both of the methods have achieved good results, the calculation accuracy was far away from satisfactory.To solve above problems, this paper carries on researches about differential pricing of American options on B-S model and Heston model in three respects.First, different from D.Y. Tangman’s rough treatment to the two virtual points, this paper introduced two-order and three-order finite difference to approximate the virtual points. By doing this, calculation accuracy has promoted and convergence speed has quickened obviously. Meanwhile, when applying the Newton iteration, we only deal with the optimal exercise boundary, thus reducing the computational complexity without losing any accuracy.Second, when dealing with the severe nonlinearity of the PDE after front-fixing point transformation, this paper came up with a new scheme that we applied the High-order compact discrete to the part of PDE which excluded the optimal exercise boundary. This method refrained from using Newton iterations and improved computational precision.Third, to improve the accuracy of Heston pricing model, we applied Grid-stretching transformation to Heston model firstly and then using the High-order compact discrete. In fact, there are two parts among this method. First we take grid-stretching transformation to the underlying asset and then we take grid-stretching transformation to the two dimensions of underlying asset and volatility.The improvement of options pricing method is helpful for enterprises to take more fine risk management and it can also promote the development of options market and improve the liquidity of the spot market. This is just the research meaning of this paper.