Finite Difference Methods Near Discontinuous Option Payoffs

Summary

The document addresses finite difference pricing when an exotic option has a discontinuous payoff, such as a binary or barrier, and nearby vanilla strikes are used for hedging. A coarse asset-price grid may fail to represent the payoff boundary accurately and can cause substantial valuation error. It lists three ways to address the problem: use a Brownian bridge to estimate the probability of crossing a boundary between grid points, use implicit time stepping to reduce oscillations near discontinuities, and use control variates based on related instruments with known analytic prices.

These are suggested techniques rather than a worked comparison. The document provides no numerical results, implementation details, or guidance on choosing grid spacing, time steps, or a control variate. The methods target different sources of error, so their effectiveness depends on the contract, boundary treatment, and solver setup.

Key ideas

• A coarse asset-price grid can misrepresent discontinuous exotic payoffs and distort valuations.
• A Brownian bridge can estimate the chance of crossing a barrier between grid points.
• Implicit time stepping can reduce oscillations associated with discontinuities.
• Control variates can use related instruments with analytic prices to improve estimates.
• The suggested techniques are not compared quantitatively in the document.

Tags

# What different techniques exist for modeling exotics near payoff discontinuities in Finite Difference method?


# What different techniques exist for modeling exotics near payoff discontinuities in Finite Difference method?












If you are modeling an exotic, like a binary or a barrier, and hedging it with vanillas that have strikes quite close to the exotic's strike, then a large asset step size, for example, $\delta S = \frac {K_{max} -K_{min}}{\beta}$, with $\beta = 1 \space or \space 2$, where $K_{min}$ and $K_{max}$ are the min and max of strikes in the basket, does not allow the payoff shape to be correctly modeled. It introduces substantial error in the valuation. To resolve this sort of problem what approaches using FDM do exist?

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/12676

Some techniques I can think of include

- Use a brownian bridge to get a crossing probability for points near the boundary

- Use implicit stepping in your PDE solver (which increases smoothness) as opposed to explicit stepping (which "rings" near discontinuities)

- Employ control variates, by using the same grid to price related instruments having easy analytic pricing solutions

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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