Malliavin Calculus for Hedging American Swaptions

Summary

The document sets out a question about computing the delta of an American swaption. The author prices the European-style payoff under a Black model in which the swap rate follows a diffusion, then estimates an early-exercise boundary by Monte Carlo simulation. An adjustment to the principal amount, linked to a loan contract, motivates the chosen exercise criterion. The author wants a hedge sensitivity but is concerned that finite differences of Monte Carlo prices may be unhelpful.

Malliavin calculus is raised as a possible way to estimate Greeks, with a paper on Bermudan options offered as a possible starting point. The central issue is whether that approach can be adapted to an American swaption and how to hedge the contract. The document provides no answer, derivation, implementation, numerical comparison, or evidence that the proposed adaptation works. It is therefore a problem statement that identifies a pricing setup and a research direction, rather than a complete hedging method; any adaptation would need to address the exercise feature and the specific swaption model.

Key ideas

• The author prices an American swaption using a Black-model payoff based on the swap rate.
• Monte Carlo simulation is used to estimate an early-exercise boundary.
• The author seeks a delta estimate and considers Malliavin calculus as an alternative to finite differences.
• A Bermudan-option treatment is suggested as a possible reference, but the document does not establish that it applies directly.
• No hedge construction, implementation details, or performance evidence are supplied.

Tags

# American Swaption Heding with Malliavin Calculus


# American Swaption Heding with Malliavin Calculus












Hedging American Swaption

Hello, I priced an American swaption using Black model with swap rates diffusion to find the european (call) price at t.

$$ C_t = (\delta \sum_{j=n+1}^{M+1} Z_t^{T_j})[R(t,T_n,T_m) - \hat{R}]^{+} $$

$$ dR(t,T_n,T_M) = \sigma R(t,T_n,T_M) dW_t $$

Then I found the early exercice boundary via MC Simulation, with this method.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION

The choice of this method rely on the need to have an explicit criterion for the optimal exercise, and because I had to add a depreciaton factor on the the principal amount (contract linked to a loan).

As you may know, it is not very interesting to compute greeks with finite difference of a Mc Price.

Now I want to hedge this american swaption, so I am trying to calculate the $\Delta$ of the contract.

I know that Malliavin calculus can give good results in this domain, but I can't find any paper for the implementation of the method for an american swaption.

I found this paper for american option, but I am not sure to be able to adapt it.

Applications of Malliavin calculus to the pricing and hedging of Bermudan options

Do you think this approach could be generalised to American Swaption ? What would be your approach to hedging an American Swaption ?

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