Pricing Compounded In-Arrears Rate Options with PDEs

Summary

The document asks whether an option on a compounded-in-arrears overnight rate must be priced with Monte Carlo simulation. Its answer describes a backward-pricing alternative for Asian-style payoffs: add the accumulated rate to the model’s state variables, then solve a partial differential equation. The same state augmentation can be used in a lattice approach, though the document specifically points to PDE literature. This method represents the path-dependent accumulation needed by the payoff while retaining a backward valuation framework.

The answer also notes a model-specific exception: caplets on risk-free rates in the Hull–White model have closed-form solutions. No derivation, numerical example, or comparison of computational performance is supplied, so the note does not establish when a PDE or closed form is preferable to Monte Carlo in practice. Results depend on the payoff and rate model, and the closed-form observation is limited to the stated caplet setting.

Key ideas

• A path-dependent Asian-style payoff can be priced backward by adding its accumulated rate as a state variable.
• A PDE can then account for the evolving underlying rate and its accumulated value.
• The answer points to literature on PDE methods for Asian options but provides no derivation or numerical comparison.
• Hull–White caplets on risk-free rates are noted as a special case with closed-form solutions.

Tags

# Compounded in-arrears payoff with backward pricing method


# Compounded in-arrears payoff with backward pricing method












RFR's will be compounded in-arrears, if I have an option on this rate am I forced to use Monte Carlo or are there techniques in the literature that will price path-dependent trades with a backwards pricing method (PDE/lattice etc)?

## Answer by piterbarg (score 5, accepted)

https://quant.stackexchange.com/a/59178

Yes you can price Asian-style options in PDEs, by introducing an extra state variable which is the accumulated Asian variable (in your case the RFR rate). It is well covered in literature, a quick Google search on "pricing Asian options in PDE" should bring plenty of results. For example this, equation (3) shows the PDE to solve. Notwithstanding your comment, for caplets on RFR rates in the Hull-White model there are closed-form 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.