1D Augmentation Sample Clauses

1D Augmentation. Equation (2.2) states the 2D Black-Scholes formula. A 2D Black-Scholes code comes with the download of FreeFem++ and a more advanced code can be found in the manual for FreeFem++ [7]. This was the starting point for analysis in 2D. Some alterations have been made to the code to change user interaction and visualization. The altered code can be seen in the Appendix. With an understanding of the 1D situation, it was a valuable exercise to see the FreeFem++ interpretation of the 1D problem. To do this we can simply multiply the y component by 0, and take a ρ = 0 (we remember that this coefficient is the correlation between the two stocks). Thus the second stock price will have no effect on the solution. In doing so we see: Figure 5.1: FreeFem++ 1D Error: Value Figure 5.2: FreeFem++ 1D Error: Time Value This solution is as expected. The y axis has no relevance on the solution since we are treating stock two as if it does not exist. We can then compare the values of option at each time step given different values along the x axis. Comparing each time step we see the line of V = 0 pushing closer and closer to the S = 35 with this line ending at time T on S = 35. We see small blips of error along the final iso value color. This is likely due to the fineness of the mesh. If we severely decrease the number of nodes in mesh down to 10 on each border, we see: As can be seen, the majority of the error arises around this final iso value. Similar to the earlier observations, this error decreases as time approaches T . For example, one of the final time images shows 5.5: Figure 5.3: Coarse Mesh 1D, Beginning