# 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? 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. Jan 14 '17 at 1:09

I know its been a while but I would like to answer this question for all the people that arrives from now on. I hope that is okay.

Let's divide the problem in two main parts. The first one is the computation of the zero coupon bond $$P(t, T)$$. In this case, you are using a short rate model given by the factor dynamics $$dy(t)$$ and the short rate dynamics $$r(t)$$. As we know, the zero coupon bonds are given by:

$$P(t, T) = \mathbb{E}_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right].$$

This expectation and, consequently, the zero coupon bond $$P(t, T)$$ can be solved analytically for many short rate models. This is usually accomplished by solving an underlying Riccati system of ordinary differential equations. I would have to check if this is the case for your particular short rate model. However, if this is not the case, you could always simulate the dynamics of $$y(t)$$ using a Monte Carlo simulation and compute the expectation given above numerically, but that doesn't make much sense since the main motivation for short rate models is that they provide analytical expressions for zero coupon bonds, avoiding the need of Monte Carlo on top of Monte Carlo simulations.

Now, once we have the zero coupon bonds $$P(t, T)$$, let's price a European Swaption. Please, notice that $$P(t, T)$$ could be obtained using a different model, such as the Libor Market Model or the HJM framework.

Since a European Swaption gives the holder a right, but not an obligation, to enter a Vanilla Swap at a future date, it is important to first compute the price of a Vanilla Swap (the word Vanilla is used since I am considering a the simplest swap, i.e., notional equal to one, contiguous time intervals, etc). The present value of this contract is given by:

\begin{align} V_s(t) &= \mathbb{E}_t^Q \left[ \sum_{i=1}^N D(t, T_{i+1}) \cdot \tau_i \cdot (L(T_i, T_i, T_{i+1}) - k) \right] \end{align}

where $$T$$ describes the tenor structure of the fixings and payments, i.e. $$0 \leq T_1 \leq T_2, \dots, T_N$$, $$\tau_i = T_{i+1} - T_i$$, $$D(t, T)$$ is the discount factor and $$L$$ is the Libor rate. Let's recall that the forward Libor rate is a martingale under a specific measure:

$$L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau) \right] \quad \text{with } t \leq T.$$

Now, performing a change of measure in the swap valuation and using the result given above, we get:

$$V_s(t) = \sum_{i=1}^N P(t, T_{i+1}) \cdot \tau_i \cdot (L(t, T_i, T_{i+1}) - k).$$

Please, notice that the price of a swap at time $$t$$ (valuation date or current date) can be valued at time t using only the term structure of interest rates observed on that date. In particular, swap values are not affected by the dynamics of rates, only they current levels.

Now, suppose that in the European Swaption the holder has the right to enter the previous Swap in $$T_1$$. Its value at time $$t = T_1$$ is given by:

$$V_{es}(T_1) = \max(V_s(T_1), 0) = \left( V_s(T_1) \right)^+.$$

Then, its value at time $$t < T_1$$ is given by:

$$V_{es}(t) = \mathbb{E}_t^Q \left[ D(t, T_1) \cdot V_{es}(T_1) \right]$$

Now, this expectation can be solved numerically using the results of a Monte Carlo simulation and the results of the short rate model for the zero coupon bonds $$P(t, T)$$.

On the other hand, the Jamshidian trick could be used at this point where you get that the Swaption payoff is given by $$N+1$$ put options on zero coupon bonds. However, since the expectation over this payoff cannot be tractable analytically you have to solve it numerically or make an approximation. I can elaborate on this if it is wanted.

Hope this helps, thanks!

• I don't understand the distinction between zero coupon bonds, which you have denoted by %P(t,T)% and modelled using short rate dynamics and discount factors %D(t, T_{i+1})% used when computing the value of a vanilla swap. Aren't these two terms similar? And secondly, when you say, "performing a change of measure in the swap valuation", what measure are you now working with? - I'm assuming the forward measure using $D_{.,T_{i+1}}$ as numeraire? Jun 14 '21 at 18:43
• sorry for the delay! no, the zero coupon bond is the expected value of the discount factor, which are not the same thing when the short rate or interest rate is stochastic. On the other hand, the change of measure ends up using $P(t, T_{i+1})$ as numeraire (not the discount factor but the discount bond or zero coupon bond). Thanks! Jun 18 '21 at 13:04