Numerical Partial Differential Equations in Finance Explained by Karel in 't Hout

Author:Karel in 't Hout
Language: eng
Format: epub
Publisher: Palgrave Macmillan UK, London

8.1 Explicit Method

First consider application of the forward Euler method. Set for the spatial grid and choose and time steps. Figure 8.1 displays the two obtained fully discrete approximations at . The top graph shows the result for . Clearly, this numerical solution is very poor, with excessively large positive and large negative values. The bottom graph shows the result for . This forms a visually fine numerical solution; compare Figure 1.​2 for the graph of the exact solution. Selecting smaller numbers of time steps than leads to similar incorrect results as in the top of Figure 8.1, whereas taking larger numbers than yields decent results. Hence, there appears to be a critical, minimal number of steps that is required, equal to about . Considering next numerical experiments reveal critical numbers of time steps approximately equal to , respectively. These numbers grow fast and appear to be directly proportional to m 2.

Figure 8.1Fully discrete approximation of call option value function for obtained with the forward Euler method if (top) and (bottom)

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