Black-Scholes is an analytical model for the valuation of stock options, developed by Myron Scholes and Fischer Black, based on the prior work of Robert Merton. Merton and Scholes won the Nobel Prize for their contributions in 1997; Black was ineligible, having passed away in 1995 from cancer. The Black-Scholes model has since been extended to address other sorts of derivatives and is extremely useful because all of the input variables can be observed, which is unusual and convenient, for economics. The Black-Scholes equation can be transformed to the familiar heat equation, which is relatively easy to solve analytically, the most popular methods being Fourier analysis or Green’s Functions.
The Black-Scholes kernel benchmark was supplied by SunGard and Intel in both source form and an executable version (compiled with ICC 9.0). The kernel uses the continued fractions method to approximate a solution to the Black-Scholes equation. The executable takes two inputs: the number of threads to be used, and the number of iterations to be performed. For the benchmark, the number of threads was varied from 1 to 8 by powers of 2, and the number of iterations was fixed at 200,000,000.
Figure 3 – Black-Scholes Kernel
The standard deviations for the Nocona system were 0.025, 0.321, 0.017 and 0.165 seconds for 1, 2, 4 and 8 threads respectively. The standard deviations for Bensley were 0.015, 0.006, 0.015 and 0.02 seconds. The co-efficient of scaling from 1 to 8 threads for the Nocona system was 2.99, while the Bensley system scaled by a factor of 5.40. The Nocona system scaling is simply astounding, and the Bensley system only slightly less incredible. In order to scale by a factor of 3, Hyper-Threading must add around 50% to nearly perfect scaling from adding another physical processor; the most aggressive estimates that Intel has put forth for Hyper-Threading are 20-30%, so this is quite a surprise. The Bensley system also scales perfectly to four physical processors (which is quite an achievement), and then gets a 35% boost from Hyper-Threading. At eight threads, the Bensley system executed the kernel 74% faster.