Why PDE Solvers Can Outperform American Option Approximations
Summary
The document compares analytic approximations for American option exercise boundaries, including Kim’s integral equation and the Whaley approximation, with numerical PDE methods. It argues that PDE solvers can be preferable when pricing requires reliable risk measures, flexible exercise rules, or volatility term structures. Approximation methods may produce directional pricing errors across options in some conditions, and they do not readily accommodate Bermudan exercise or provide Greeks without additional work.
The answer also questions whether approximations save computation in practice: a PDE solver can estimate the early exercise premium without an especially fine grid, and multiple contracts may be efficiently handled together. It cites professional library and trading experience in support, but this is anecdotal rather than a controlled benchmark. The discussion is an opinionated comparison, not a detailed implementation guide, and it does not establish that PDE methods are always faster or more accurate for every contract and use case.
Key ideas
- Analytic American option approximations may have biased errors under some market conditions.
- Greeks needed for trading may require additional derivation or computation when using an approximation.
- Approximations have limited support for volatility term structures and Bermudan exercise conditions.
- PDE solvers can price early exercise premiums and handle multiple contracts in a shared framework.
- The claimed practical advantages are based partly on professional experience rather than controlled comparisons.
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Full text
# What are the downsides of using Kim's integral equation (1990) to determine the exercise boundary of an American option? # What are the downsides of using Kim's integral equation (1990) to determine the exercise boundary of an American option? I'm new to the industry and trying to wrap my head around American options pricing. The integral equation(1) from Kim (1990) doesn't seem to make any strong assumptions, and approximating the integral with the trapezoidal rule and getting a piecewise exponential early exercise boundary seems like a solid approach to me. What are the downside of this, and why might you want to use other methods? What other methods are common? (1) See e.g. https://www.diva-portal.org/smash/get/diva2:510809/FULLTEXT01.pdf References I.J. Kim (1990). "The Analytic Valuation of American Options", Review of Financial Studies 3, 547–572. ## Answer by Brian B (score 2) https://quant.stackexchange.com/a/78785 Note: the most popular approximation in quant libraries I have known is not from Kim (1990) but rather the Whaley approximation. There are several good reasons to prefer numerical PDE solvers over Whaley or other approximations. With the approximations: - Deltas and other risk parameters are usually left as an exercise for the reader, but are crucial for professional options trading. - Their errors are biased, and will be in the same direction for all options under certain market conditions, leading to lopsided positions - They have no way to account for a term structure of volatility (mildly compensated in Bjerksund and Stensland) - They cannot treat altered boundary conditions such as Bermudan exercise - It is quite unclear that, even for a single option price computation, they save any floating point operations over a well-implemented PDE solver. (The numerical PDE solver would be used only to estimate the early exercise premium, so it does not need the tiny grid spacings most people think it does) - In the most common situation of wanting to price multiple option contracts, it is very easy to "stack" the contracts for a PDE solver. With some sorting tricks one can even get extra computational efficiency. It is markedly more difficult to vectorize the approximations, leading to higher code complexity and, again, FLOP counts quite likely higher than the PDE solver. Although almost every professional options pricing library I have seen has one of these approximations coded into it, I haven't seen any professionals use these approximations since 1995 (and that team quickly moved to trinomial trees by early 1996). For these reasons, I consider the analytic approximations to be fun but irrelevant intellectual exercises for the academics and dabblers.
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