A short history of PDE methods as applied in finance:
It is better to have 5 solutions to 1 problem that 1 solution for 5 problems (each problem has its own peculiarities and recipes may break down and then Plan B is needed). A big problem IMO is that the same old FDM models are being used on all problems (hammer and nail analogy).
18.2 Background and History
The two major categories of numerical methods for time-dependent partial differential equations are called Alternating Direction Implicit (ADI) method (Peaceman (1977), Roache (1998), Craig-Sneyd (1988)) and Locally One-Dimensional (LOD) or (Soviet) Splitting methods (Yanenko (1971), Marchuk (1982), Marchuk, Rusakov, Zalesny and Diansky (2005)). These methods originated in the 1960s in the USA and USSR, respectively and they were used to solve partial differential equations in reservoir engineering, fluid dynamics, heat transfer and nuclear engineering. The methods all have one thing in common: they decompose a multi-dimensional problem into a sequence of simpler one-dimensional problems. The differences are minor and non-essential and it is the author’s opinion that ADI is a special and somewhat clumsier sub-case of Splitting. Notwithstanding, ADI tends to be the term that is used in computational finance literature and applications. Historically, relatively few researchers have had exposure to Splitting method. Most of the original research was written in Russian in journals that were not readily accessible in the West. The author was fortunate in that he did have knowledge of, and access to the relevant sources.
To our knowledge, splitting methods applied to computational finance were first introduced in Duffy (2006). The first production application is discussed in Davidson and Levin (2014) in which the authors use the Marchuk four-cycle modification of Crank-Nicolson splitting method (see chapter 22 where we introduce Marchuk’s method) to value mortgage-backed securities (MBS) under a two-factor Gaussian rate model consisting of a short rate and a slope factor. These factors are uncorrelated. Second-order monotone schemes are produced. Another notable application is discussed in Sheppard (2007) in which splitting is applied to the Heston stochastic volatility model. The author used Duffy exponential fitting for the convection-diffusion term and Yanenko’s method for the mixed derivative term.
In general, we prefer LOD to ADI because it is more robust, is easy to program and it has been successfully applied to various kinds of applications in computational finance (see for example, Sheppard (2007) where LOD is used to price options in the Heston stochastic volatility model and for which ADI didn’t perform as expected). We have more anecdotal examples and for this reason we do not discuss ADI (at least, ADI classic) in this book.
