We study the binomial, trinomial, and Black-Scholes-Merton models of option pricing. We present fast parallel discrete-time finite-difference algorithms for American call option pricing under the binomial and trinomial models and American put option pricing under the Black-Scholes-Merton model. For $T$-step finite differences, each algorithm runs in $O(\left(T\log^2{T}\right)/p + T)$ time under a greedy scheduler on $p$ processing cores, which is a significant improvement over the $\Theta({T^2}/{p}) + \Omega(T\log{T})$ time taken by the corresponding state-of-the-art parallel algorithm. Even when run on a single core, the $O(T\log^2{T})$ time taken by our algorithms is asymptotically much smaller than the $\Theta(T^2)$ running time of the fastest known serial algorithms. Implementations of our algorithms significantly outperform the fastest implementations of existing algorithms in practice, e.g., when run for $T \approx 1000$ steps on a 48-core machine, our algorithm for the binomial model runs at least $15\times$ faster than the fastest existing parallel program for the same model with the speed-up factor gradually reaching beyond $500\times$ for $T \approx 0.5 \times 10^6$. It saves more than 80\% energy when $T \approx 4000$, and more than 99\% energy for $T > 60,000$. Our option pricing algorithms can be viewed as solving a class of nonlinear 1D stencil (i.e., finite-difference) computation problems efficiently using the Fast Fourier Transform (FFT). To our knowledge, ours are the first algorithms to handle such stencils in $o(T^2)$ time. These contributions are of independent interest as stencil computations have a wide range of applications beyond quantitative finance.

翻译：我们研究了二项式、三项式和Black-Scholes-Merton期权定价模型。针对二项式和三项式模型下的美式看涨期权定价，以及Black-Scholes-Merton模型下的美式看跌期权定价，我们提出了快速的并行离散时间有限差分算法。对于$T$步有限差分，在$p$个处理核的贪心调度器下，每个算法的时间复杂度为$O((T\log^2{T})/p + T)$，这显著优于现有最优并行算法的$\Theta({T^2}/{p}) + \Omega(T\log{T})$时间。即使在单核上运行，我们算法的$O(T\log^2{T})$时间在渐近意义上也远小于已知最快串行算法的$\Theta(T^2)$运行时间。实际中，我们算法的实现显著优于现有算法的最快实现；例如，当在48核机器上运行$T \approx 1000$步时，我们的二项式模型算法比现有最快并行程序快至少15倍，且加速比因子随着$T \approx 0.5 \times 10^6$逐渐超过500倍。当$T \approx 4000$时，节能超过80%；当$T > 60,000$时，节能超过99%。我们的期权定价算法可视为利用快速傅里叶变换（FFT）高效求解一类非线性一维模板（即有限差分）计算问题。据我们所知，我们的算法是首个能在$o(T^2)$时间内处理此类模板的算法。这些贡献具有独立价值，因为模板计算在量化金融之外还有广泛的应用。