American Option Finite Differences: Explicit Projection and SOR

Summary

The discussion distinguishes two numerical approaches for enforcing early exercise in American option pricing. With a fully explicit finite-difference scheme, each backward time step can be computed from values at the preceding step. Afterward, each node’s value can be raised to at least its intrinsic value, implementing the early-exercise constraint through a pointwise maximum.

Partially implicit schemes differ because values at a new time step depend on other values at that same step. The nodes are therefore coupled, so simple post-step projection is not the same procedure; a method such as projected successive over-relaxation can enforce the constraint while solving the coupled system. The answer frames the distinction in terms of the numerical scheme’s dependencies rather than claiming the two algorithms are universally interchangeable. It cites a textbook treatment but supplies no derivation, convergence analysis, or comparison of numerical accuracy.

Key ideas

• For a fully explicit scheme, backward induction permits applying the intrinsic-value floor after each time step.
• Partially implicit schemes couple values across nodes within the same time step.
• Projected successive over-relaxation can handle the early-exercise constraint in a coupled solve.
• The appropriate procedure depends on the finite-difference scheme.

Tags

# Pricing an American derivative with finite differences


# Pricing an American derivative with finite differences












I have a basic fundamental question on pricing an American option in the Black-Scholes (BS) framework: I seem to confuse two different approaches to price any early exercise,

- Write down a linear complimentary problem and use SOR to solve it;

- Solve the Black-Scholes PDE, but at every time step choose the maximum between the intrinsic value and the value from BS solution numerically.

Are these approaches equivalent or what's the difference between the two?

## Answer by LocalVolatility (score 2, accepted)

https://quant.stackexchange.com/a/30087

When you use a fully explicit finite difference scheme, you can simply apply the backward induction step and afterwards ensure that the option price at each node is at least equal to the intrinsic value. This is possible as any value $V_{i + 1, j}$ at time $\tau_{i + 1}$ only depends on the values $V_{i, j}$ at time $\tau_i$.

When you use a scheme that is at least partially implicit, then the values $V_{i + 1, j}$ at $\tau_{i + 1}$ additionally also depend on the other values at the same time step. In this case you use e.g. projected successive over-relaxation.

This issue is explicitly discussed in Chapter 78.9 of Wilmott (2006), pp. 1244ff.

References:

Wilmott, Paul (2006) "Paul Wilmott on Quantiative Finance", John Wiley & Sons

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.