I was wondering if there exist options or other derivatives that do not have a known closed-form analytic expression (i.e., some sort of Black-Scholes PDE) and are usually priced using Monte Carlo methods, but that could have such an expression? Specifically, I am wondering if it is possible to analyze the price data of some derivative as a function of time and underlying price to discover a PDE using something like symbolic regression or other ML-based techniques?
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3$\begingroup$ You could start with American options. These are exchange traded for single stocks. Should be plenty of data for these. There are plenty of analytical approximations, but no exact analytical / closed-form expression. American options PDE is a free boundary problem. $\endgroup$– FridoCommented May 30, 2023 at 16:07
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2$\begingroup$ An intetest rate swap that either party can cancel on coupon dates, i.e. Bermudan puttable/callable, in practice, is likely to be priced using MC. $\endgroup$– Dimitri VulisCommented May 30, 2023 at 16:14
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$\begingroup$ @Frido, thanks, I might try to look at some data for American options and see what happens. Do you know of any options whose actual PDE (not just solution) is not known? It seems that this unilateral extinguisher one suggested by Dimitri might be one, but I cannot find a lot of literature on it. $\endgroup$– Physics PenguinCommented May 30, 2023 at 20:40
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1$\begingroup$ The PDE, or equivalently the SDE that drives the underlying(s), is usually known because the model that is used is known. What stops us from solving explicitly is usually that the payoff is complicated (American option, arithmetic Asian option in BS), or that the PDE/SDE has non-constant coefficients such as local volatility $\sigma(x,t)\,.$ $\endgroup$– Kurt G.Commented May 31, 2023 at 5:42
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$\begingroup$ @KurtG. I see, so it is more so a computational problem of solving the known PDE/SDE, which might not even have an analytic solution, so approximate solutions are sought after instead? $\endgroup$– Physics PenguinCommented May 31, 2023 at 15:06
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1 Answer
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A unilateral extinguisher is a simple example, as it is easy to describe.
Counterparties A and B have a portfolio of swaps between them, e.g. interest rate swaps or cross-currency swaps. For similicity, you can even consider just one swap. For simplicity, assume no margin, collateral, or netting agreements.
The portfolio has some mark to market value V.
If CDS-like credit event happens to credit C then: if V>v for some contractually specified strike v, then the portfolio "extinguishes", i.e. all the swaps in the portfolio are canceled, but some recovery may be paid out. Else the portfolio lives on.
In practice, the recovery could depend not only on V, but on the time and other non-trivial parameters.
It is easy to write a closed-form pricer for a zero-recovery bilateral extinguisher, which just extinguishes with no recovery irrespective of the value of V. But I am not aware of a methodology other than MC to price a unilateral one.
answered May 30, 2023 at 16:44