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Delta force: option pricing with differential machine learning

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Abstract

We show how and why to use a financially meaningful differential regularization method when pricing options by Monte Carlo simulation, be that in polynomial regression or neural network context.

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Notes

  1. Over the last decade the Bachelier model has had a revival in quant circles. There are several reasons for this: It is born with a negative skew in implied volatilities, things we thought were positive have gone negative (interest rates—even oil prices), and working with constant (rather than proportional) coefficients is mathematically easier—as for instance in Jesper Andreasen and Brian Huge’s work (Andreasen & Huge, 2012) on option pricing via expansions.

  2. Both of the operations are what mathematicians would call formal, i.e. not rigorously justified, but let’s not worry about that here; it can be fixed [(Broadie & Glasserman, 1996, Appendix A), L’Ecuyer, 1995] with the theory of generalized functions—or verified by elementary means for the Bachelier (and the Black–Scholes) model.

  3. In the recurrent example in this paper one can safely use \(w= 1/2\) as prices and Deltas are of the same magnitude. In general we suggest using \(w = sd (D)/ (sd(C) + sd(D))\), where sd denotes standard deviation. This ensures that terms in the “naked” version of Eq. (4)—i.e. when \(\theta = 0\)—have the same variance.

  4. To make dimensions fit snugly, Y starts with a zero-column, so strictly speaking Eq. (5) can’t be used for \(w= 0\).

  5. For a non-infinitesimal hedge frequency, say \(dt =1/52\), we cannot say that the discrete use of the continuous \(\varDelta\) gives the lowest hedge error standard deviation. In fact, as shown by Wilmott (1994) we can do slightly better when rates are non-zero. However, (a) we have results that ensure dt \(\rightarrow 0\)-convergence and (b) rates are zero here, so we ignore that.

  6. We could increase the range parameter d or change to an estimation method that does not explicitly invert matrices.

  7. Handling the square and particularly the sum part of the criterion function efficiently does require thought – but of a different (vectorization) nature.

  8. Here we are skipping a bunch of nasty details regarding matrix multiplication, differentiation and restrictions on input and output dimensions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, DK-2100, Copenhagen Ø, Denmark

    Magnus Grønnegaard Frandsen, Tobias Cramer Pedersen & Rolf Poulsen

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  1. Magnus Grønnegaard Frandsen
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  2. Tobias Cramer Pedersen
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  3. Rolf Poulsen
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Correspondence to Rolf Poulsen.

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Frandsen, M.G., Pedersen, T.C. & Poulsen, R. Delta force: option pricing with differential machine learning. Digit Finance 4, 1–15 (2022). https://doi.org/10.1007/s42521-021-00041-7

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