# How to price Swaptions with short rate models?

I have specified a (Lognormal) short-rate model (non-affine) under the Risk-Neutral measure $Q$ as a shifted exponential vasicek:

$r(t) = e^{y(t)} + \phi(t)\\ \text{with} \quad dy(t) = \kappa(\theta - y(t))dt + \sigma_y dW(t)$

where $\phi(t)$ is a shift based on parameters in $y$.

I can compute ZC Bond prices with the short-rate model, from which I can obtain forward rates. I want to use this to price financial derivatives, in particular Swaptions.

1) Following Privault Proposition 14.6, the price of a European Swaption is given by $P(t,T_i, T_j) \mathbb{\hat{E}}_{i,j} \Big[ (S(T_i, T_i, T_j) - K)^+ \vert \mathcal{F}_t \Big]$ i.e. the expected payoff under the $(T_i, T_j)$ - Forward measure, where $S(\cdot)$ is the Swap Rate. Is there a way I can use MC simulations, to simulate this price? Or is pricing via Black the only option (since we simulate Lognormal distributed Rates)

2) I understand that the $T$-Forward Measure takes ZC Bonds as numeraire and by Girsanov Theorem the drift of the short-rate model changes under the $T$-Forward Measure. How would it be possible to specify my short-rate model under $T$-Forward Measure?

• For one-factor models you can use Jamishidian's trick. You can't use Privault, because you don't have a nice expression for the dynamics of the swap rate. The measure you would want to use in 2) would be the one that makes the dynamics of the swaprate driftless, which, again, you don't have an expression for. – Olaf Jan 12 '17 at 16:23
• Thanks for the comment. So if I get things right: We can rewrite the price of a swaption as the sum of ZCB options. These however need to be priced using Black's formula - for which we need the bond price volatility as input. Since we have no analytical ZCB in our model, is there a way we can we overcome this problem? – reteip Jan 12 '17 at 22:22
• @reteip You can use numerical methods to compute each ZCB option (like MC). Jamshidian only states your swaption can be written into sum of ZCB option, it doesn't require Black's formula. – SmallChess Jan 14 '17 at 1:09