Abstract

A family of Exponentially Fitted Block Backward Differentiation Formulas (EFBBDFs) whose coefficients depend on a parameter and step-size is developed and implemented on the Black–Scholes partial differential equation (PDE) for the valuation of options on a non-dividend-paying stock. Specific EFBBDFs of order 2 and 4 are applied to solve the PDE after reducing it into a system of ordinary differential equations via the method of lines. The methods are shown to be superior to the well-known Crank–Nicolson method since they are $L$-stable and do not exhibit oscillations usually triggered by discontinuities inherent in the payoff function of financial contracts. We confirmed the accuracy of the methods by initially applying them to a prototype example based on the one-dimensional time-dependent convection–diffusion equation with a known analytical solution. It is demonstrated that the American put can be exercised early by computing the hedging parameter “delta”, which specifies the condition for early exercise of the put option. Although the methods can be used to price all vanilla options, we elect to focus on the put due to its optimality.

摘要

在Black-Scholes偏微分方程（PDE）上开发并实现了一系列系数拟合的块向后微分公式（EFBBDF），其系数取决于参数和步长，用于对非股息支付的股票进行期权估值。通过线法将2阶和4阶特定EFBBDF分解为常微分方程组后，可用于求解PDE。这些方法被证明优于众所周知的Crank-Nicolson方法，因为它们$\mathrm{大号}$-稳定，不会出现通常由金融合同的收益函数固有的不连续性引起的振荡。我们通过将方法最初应用于基于一维时间相关的对流扩散方程的原型示例并使用已知的解析解来确认方法的准确性。事实证明，可以通过计算对冲参数“ delta”来提前行使美国看跌期权，该参数指定了提前行使看跌期权的条件。尽管可以使用这些方法来定价所有优先选择权，但由于其最优性，我们选择着重于看跌期权。