Calibrating time-dependent volatility to swaption prices

We have seen in a previous post how to fit initial discount curves to swap rates in a model-independent way. What if we want to control the volatility parameter to match vanilla rates derivatives as well? Just as we found for vanilla calls and puts, we will need to chose a model, for example the Hull-White extended Vasicek (HWeV) model that we’ve seen before (there are a few reasons why this isn’t a great choice, discussed later).

    \[dr = \Bigl( \theta(t) - ar(t) \Bigr) dt + \sigma(t) dW_t\]

Our next choice is which vanilla rates options we want to use for the calibration. A common choice is the interest rate swaption, which is the right to enter a swap at some future time T with fixed payment dates \{T_i\} > T and a strike X. These are fairly liquid contracts so present a good choice for our calibration. A ‘payer swaption’ is one in which we pay fixed and recieve floating, and a ‘receiver swaption’ is the opposite. For simplicity, for the rest of this post we will assume all payments are annual, so year fractions \tau are ignored.

A reciever swaption can be seen as a call option on a coupon-paying bond with fixed payments equal to X at the same payment dates as the swap. To see this, consider the price of a swap discussed before:

    \[S(t,\{T_i\}, X) = P(t,T_N) - P(t,T_0) + \sum^N_{i=1} X \cdot P(t,T_i)\]

where t is the time now. This is exactly the same as the price of buying a unit of bond at time T_0 for $1, receiving the fixed coupons X at each payment period, and receiving the original notional back at time T_N.

So, the price of a swaption is an option on receiving a portfolio of coupon payments, each of which can be thought of as a zero-coupon bond paid at that time, and the value of the swaption is the positive part of the expected value of these:

    \[C(t, T_0, T_N, K, X) = P(t,T_0) \cdot {\mathbb E}\Bigl[ \Bigl( K - \sum^N_{i=1} X \cdot P(t,T_i) \Bigr)^+ \Bigr]\]

where the cash payments have been replaced by the strike of the option (the only ‘notional’ payment that will be paid)

This isn’t readily tractable, but we make use of Jamshidian’s decomposition (I won’t go further here – this is worth it’s own post!) to re-write this with the max inside the summation:

    \[C(t, T_0, T_N, K, X) = P(t,T_0) \cdot {\mathbb E}\Bigl[ \sum^N_{i=1} \Bigl( K_i - \cdot P(t,T_i) \Bigr)^+ \Bigr]\]

where K_i is the price of a ZCB at time T_0 expiring at T_i if rates at T_0 make the value of the coupon-bearing bond equal to X.

Looking at this expression, we see that each term is simply the present value of an option to buy a ZCB at time T that expires at one of the payment dates T_i with strike K_i. So the price of a swaption has been expressed entirely as the price of a portfolio of options on ZCBs!

For the HWeV model, these are deterministic and depend only on the initial rate, and calibrated time dependent parameters in the model. Since rates are gaussian in HWeV this can be done analytically. Calculating these for time-varying parameters is algebra-intensive and I leave it for a later post, but for constant parameters the calculation is described in Brigo and Mercurio pg 75-76 and gives a price of

    \begin{align*} ZBC(t,T_0,T_i,K) &= P(t,T_0) {\mathbb E}\Bigl[ K - X \cdot P(t,T_i) \Bigr]^+ \nonumber \\ &= P(t,T_i) \cdot \Phi(h) - X \cdot P(t,T_0) \cdot \Phi(h-\sigma_p) \nonumber \end{align*}


    \begin{align*} \sigma_p &= \sigma \sqrt{\frac {1-e^{-2a(T_0 - t)}} {2a}} B(T_0,T_N) \nonumber \\ h &= {\frac 1 {\sigma_p}} \ln{\frac {P(t,T_N)} {X \cdot P(t,T_0)}} + {\frac {\sigma_p} 2} \nonumber \\ B(T_0,T_N) &= {\frac 1 a} \Bigl[ 1 - e^{-a(T_N - T_0)}\Bigr] \nonumber \end{align*}

We can see how we could use the above to calibrate the volatility parameter to match a single market-observed swaption price. When several are visible, the challenge becomes to choose a piecewise continuous function to match several of them. In HWeV this can be done analytically, but for more general models some sort of optimisation would be required.

Since these contracts have an exercise date T when the swap starts and the swaps themselves will have another termination date T_N which define a 2-dimensinal space \{T,T_N\}, it will not be possible to fit all market-observable swaptions with a one factor model. Many alternatives are discussed in the literature to deal with this concern, but the general procedure is the same. Practically, we should choose the most liquid swaptions and bootstrap to these, and only a few (5Y, 10Y etc) will practically be tradable in any case.