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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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Could you define delta, Z and R^hat, please? –  Konsta Jul 9 at 20:15
    
$\Delta$ is the derivative of the value with respect to the underlying R, Z is the Bdiscount factor, $\hat{R}$ is the strike rate. –  lmorin Jul 9 at 21:31
    
Sorry, I meant the other delta in front of the summation sigma in C. It might be just a constant? Hm, maybe it is not a delta at all... So the hat does not denote some kind of averaging but R^hat is simply a constant, correct? –  Konsta Jul 9 at 21:58
    
Yes this is a constant representing one cash flows. 1/Tenor if we took an amount of 1 or K/Tenor if we took an amount K. –  lmorin Jul 9 at 22:03

1 Answer 1

Hopefully this is marginally helpful. The delta is simply the ratio of the Malliavin derivatives. By chain rule,

$$\mathscr{D}_t C_t = {\partial C_t \over \partial R_t } \mathscr{D}_t R_t$$

Thus

$$ \Delta_t = {\mathscr{D}_t C_t \over \mathscr{D}_t R_t } $$

You should be golden if you can work out what those MDs are for whatever model you're in.

Edit: Expansion

First, regarding

I can't find any paper for the implementation of the method for an american swaption.

It quite possibly does not exist.

As a practical add-on to the above it's also worth noticing that

$$ {\mathscr{D}_t C_t \over \mathscr{D}_t R_t } = {C_t \mathscr{D}_t \log C_t \over R_t \mathscr{D}_t \log R_t } $$

this observation helps to maintain positivity while conducting the simulation.

Because you have already solved for the early exercise boundary, you do not have to worry about doing the backward induction, you already have the sample paths of the option and the underlying. There should also be no requirement to compute a Skorohod integral either.

See page 911 of this for a great resource in getting at the Malliavin derivatives. Just write any stochastic process that you have in integral form, then take the malliavin derivatives using the techniques shown in the appendix of this article.

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This is a path wise method suggested when we already have a pricing method and it seems to work. I was looking for a full pricing and hedging Malliavin method (ie using representation theorems). –  lmorin Jul 8 at 23:42

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