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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

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