June 28th, 2015, 1:37 pm
OK using explicit Euler, i compute the price of a European and after each step back, i compare the ratio of the regressed price at each grid point with that of BlackScholes result - i then take this ratio per grid point and adapt the regressed value i get for the American (before any early exercise conditions are applied). so if the European was 99% of the BS result, i will multiply the regressed price of the American by 100%/99% to "fix" it. I also update the European grid with the BS prices (so that the adjustment ratio at each time step is small and incremental).here I price 2yr put, 100% spot strike, 15% vol and 4% rates, no divs. The numbers shown are abs(ln(price[dt] / price[inf])) where price[inf] are my "true" prices if you ran the thing for ages (these may not be reference accurate prices but it gives you the idea).DT Explicit European Explicit American Adjusted American ADE European ADE American1/5 125.02% 128.78% 8.29% 85.98% 56.41%1/10 73.09% 74.44% 5.45% 24.66% 18.55%1/20 37.79% 36.64% 1.84% 5.61% 4.62%1/30 23.98% 21.84% 0.31% 2.43% 2.08%1/50 12.88% 11.17% 0.45% 0.89% 0.77%1/100 5.66% 5.89% 1.00% 0.25% 0.20%1/250 2.12% 2.18% 0.39% 0.07% 0.02%1/500 1.04% 1.02% 0.14% 0.05% 0.00%1/1000 0.52% 0.48% 0.05% 0.04% 0.01%1/2000 0.26% 0.21% 0.00% 0.04% 0.01%Running the same for 80% spot strike:DT Explicit European Explicit American Adjusted American ADE European ADE American1/5 6.16% 5.51% 5.62% 38.80% 5.83%1/10 26.48% 22.77% 3.27% 3.42% 7.62%1/20 1.86% 0.57% 1.95% 1.36% 1.34%1/30 26.20% 23.99% 1.07% 0.89% 0.19%1/50 1.17% 2.12% 0.43% 0.39% 0.09%1/100 5.98% 5.91% 0.23% 0.10% 0.06%1/250 3.95% 3.97% 0.11% 0.01% 0.00%1/500 0.04% 0.49% 0.06% 0.01% 0.01%1/1000 0.80% 0.87% 0.02% 0.01% 0.02%1/2000 0.12% 0.01% 0.00% 0.01% 0.02%So, basically, it works in its aim of improving the numerical error observed in the explicit euler method quite considerably. It brings about a convergence quicker than ADE for the ATM strike. Although i would argue its fairly limited in scope, like all control variate schemes. The things i price at work have no similar product to use.Cuch, I know "no one uses explicit euler for PDE" - i personally use ADE everywhere i can. However, it was an idea trying out as i hadn't heard of anything analogous to control variates outside of MC.