Centering a Finite-Difference Option Pricing Grid on Spot
Summary
The document considers how to set up a finite-difference method (FDM) when the goal is to price one option at a known initial spot, rather than study the option value across many starting prices. The response recommends centering the price grid on the current spot and making it wide enough to include a sufficient range of likely terminal prices. It specifically notes that the grid should cover the forward price at maturity.
A finite-difference calculation produces option values across its grid at the present time, even when only one spot value is needed. Those neighboring values can also support grid-based estimates of delta and gamma. The response cautions that these estimates may not represent total delta when volatility itself depends on spot. The document gives practical grid-setting guidance but no numerical example, convergence study, or precise rule for choosing the grid width; the required coverage therefore depends on the model and desired accuracy.
Key ideas
- Center the price grid on the known current spot when pricing a single option.
- Choose a grid wide enough to cover a sufficient range of terminal prices, including the maturity forward.
- Finite differences return present-time option values at every grid spot, not only the specified starting spot.
- Neighboring grid values can be used to estimate delta and gamma.
- Grid-based sensitivities may omit effects from spot-dependent volatility.
Tags
Full text
# How are FDE's implemented when one wants one particular price? # How are FDE's implemented when one wants one particular price? Say I want to price a particular call option in the Black Scholes model using finite difference methods. The value process of this option $V(s, t)$ satisfies a PDE. I can use finite difference methods to determine $V(s, t)$ for $s, t$ lying in some discrete grid. Working backwards in time, I can get to $V(s, 0)$ for all $s$ in that grid. This is my price, for all initial spots, as long as those spots are on the grid. But my original goal was to just price 1 call option, and I already know the initial spot. So really I only want $V(s', 0)$ for some fixed $s'$. My question is, how should I then implement my FDM scheme? For example, should I "manually" make sure that my $s'$ will lie on that grid, hence avoiding the need to interpolate from the grid? What other things ought I do? TLDR: Basically my question is what changes should I make to a FDM-scheme when I want one particular price $V(s', 0)$ and am not really interested in the general FUNCTION $V(s, t)$ for all kind of $s, t$. ## Answer by Ezy (score 1) https://quant.stackexchange.com/a/42210 Answer is very straightforward: always center your grid on the current spot value and make sure that it covers sufficiently many standard deviations on each side to cover enough of the terminal distribution at maturity. In particular it should cover the forward at maturity. NB: - By definition when you use a FDM you will get back the grid of all values of the function at the present time. This differs from say Monte-Carlo types of methods where all the random paths would start at the spot value. - Also having the values of V(S,0) for various S allows to calculate delta/gamma "on the grid". Be careful though that this may not be the total delta if say you have a volatility model that depends on spot itself.
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