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Interpolating Finite-Difference Option Values and Greeks

Article Quant Q&A · Author: Jesper Tidblom
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Summary

The document explores how to extract an option’s present value, delta, and gamma from a finite-difference pricing grid. For an American option in the Black–Scholes model, placing the current spot directly on a grid node allows the price and local finite-difference Greeks to be read from nearby nodes. A fixed grid may be preferable when comparing prices across spot values, because rebuilding the grid for each spot can introduce additional discretization differences.

When spot is off-grid, interpolation is needed. The question considers cubic splines in one dimension, bicubic interpolation and sequential one-dimensional interpolation in two dimensions, and alternatives such as B-splines or local surface fits. It raises concern that interpolation smoothness may be insufficient for stable second derivatives, and that sequential interpolation can depend on the order of variables. No preferred method or numerical evidence is supplied; the discussion is a practical question about accuracy and consistency, especially for gamma.

Key ideas

  • A grid node at the current spot gives a direct price estimate and supports finite-difference estimates of delta and gamma.
  • Rebuilding a grid for each spot can make cross-spot comparisons sensitive to discretization differences.
  • Off-grid prices and Greeks require interpolation or fitting of the grid values.
  • Interpolation smoothness and variable order can affect second-derivative estimates in two dimensions.
  • The document raises possible interpolation approaches but gives no recommended standard method.

Tags

OptionsDerivatives pricingStatistics
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# Calculating present value and greeks from finite difference grids, best practice


# Calculating present value and greeks from finite difference grids, best practice












The question I have is rather simple, but I fail to find clear answers. Given a finite difference grid used for valuation of some option say, what is the best practice when it comes to calculating the present value and the greeks, given the values on the discrete grid ?

To make it concrete, let us say we are pricing an American option in the standard Black-Scholes model using some finite difference scheme. We now want the price, the delta and the gamma at the current spot value. There seems to be some conflicted opinions of people on this site regarding if one should put the spot value on the grid or not. There are arguments for both positions: If we put the spot on the grid, then we can simply get the value of our option directly from the value at that node. We can also get the delta and gamma directly by finite difference quotients using nearby nodes.

It seems most people advice against putting the spot on the grid though. The argument here seems to be that if you want to calculate the option value for another spot value, you will have to remake the grid based on that value. Then the results will be based on different grids, which probably will introduces extra discretization errors in the comparison. You will compare apples and oranges so to speak. Am I understanding this correctly or is there some other reason for not putting the spot on the grid? What if you just need the option value for a certain spot value? Are there still any drawbacks of putting the spot on the grid?

If one is not putting the spot on the grid, then what is the best practice when it comes to interpolation of the value and the greeks? A cubic spline seems suitable in one dimension since it is twice continuously differentiable and thereby enough to get a continuous gamma when the spot varies, or does this have any obvious drawbacks?

What about in two dimensions, like in the Heston model? It is possible to do an iterated one dimensional cubic spline procedure giving some decent results. However the result depends slightly on which variable we start with. I have read about bicubic interpolation, but this does, as I understand it, not typically result in a twice differentiable surface, since it is just constructed from local information near the cell. Also there are no conditions on the second derivatives with respect to each variable, which makes this type of interpolation unsuitable for calculation of gamma What about B-splines, or some least squares fitting of a locally cubic surface? I guess there must be some standard method to do it that I am unaware of since this must be a very common issue?

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