Rates quants need no introduction to the stochastic alpha, beta, rho (SABR) model.
Widely used to price European swaptions, the model has a useful expansion that allows users to calibrate it to the market-implied volatility surface analytically. The ease with which this can be done has made the model very popular, despite the fact it produces prices that can be arbitraged.
The implied volatility expansion – commonly called the Hagan expansion – doesn’t really capture the dynamics of the swaptions at the high and low strikes, especially for longer maturities, making mispricing increasingly likely.
One way to obtain arbitrage-free prices is by using numerical techniques to calibrate the SABR model by solving a partial differential equation (PDE). However, this is cumbersome, and does not produce prices quickly enough for real-time trading.
In this month’s technical paper, Discrete time stochastic volatility, Thomas Roos, a London-based derivatives consultant, proposes a quicker model to obtain arbitrage-free prices for swaptions.
The quant creates a joint distribution of the volatility and the underlying forward rate, which allows one to solve the pricing formula using a one-dimensional integral.
This is important, because numerical methods use discretisations, or a large number of different values of underlying factors to find the output – a process that is computationally very intensive. The computational time for directly solving the SABR model numerically, which involves a two-dimensional PDE, increases as a function of N to the power of 3, where N is the number of points in the underlying, volatility and time discretisations. On the other hand, the computational time taken by Roos’ model, which uses a one-dimensional integral, depends linearly on N.
The approach can also be extended to price some simple types of constant maturity swaps (CMS) and forward-starting options faster.
One key advantage of the model is that the parametrisation allows much better control of implied volatility at the higher strikes or the wings.
“Because it’s numerical, there really are no constraints about what that functional form might look like. That gives quite a bit of flexibility, and especially it gives one the ability to control what the high wing looks like,” says Roos.
The advantage of being able to control the high wing separately is very important for interest rate modelling where we have to price constant maturity swapsThomas Roos, derivatives consultant
This is especially important for CMS products, which are much more sensitive to rates at the high wing.
“The advantage of being able to control the high wing separately is very important for interest rate modelling where we have to price constant maturity swaps,” says Roos. “The problem is that the payout of these CMS products grow at the square of the swap rate for large rates as opposed to standard European payouts, which grow linearly in the rate. And this dependence on high rates makes them very sensitive to the high wings of the volatility smile, where there really are no liquid quotes you can calibrate to. So you are completely at the mercy of your extrapolation, and the prices generated by your smile model may be well off market.”
The swaptions market is not new to pricing issues. When European rates plunged into negative territory in 2012, the SABR models used to price swaptions started producing nonsensical values – a phenomenon that was observed even at low positive rates. This triggered many valuation disputes and pushed quants to come up with techniques to solve the negative rates problem.
More recently, the industry decided to shift to a new pricing methodology to bring European cash-settled swaption prices in line with the underlying swaps – a mismatch that had been ignored for years until the difference started to get magnified by falling rates.
While the breakdown of models under negative rates is a more recent problem, the issue of arbitrageable prices has always existed.
To fix this, most firms typically apply some kind of crude adjustment to their existing implementation of the SABR model, which reduces the arbitrage instead of eliminating it completely.
“Most places use some kind of SABR variant usually with proprietary adjustments to mitigate the arbitrage, and there are quite a variety of approaches there. For example, some people might patch on a different low wing tail from a different distribution,” says Roos. “Other people just try to make ad hoc adjustments to the implied volatility formula itself. Quite a few of these methods mitigate the arbitrage rather than removing it completely and people typically just live with whatever is left over.”
While Roos’ model may be slower than the popular analytical way of solving the SABR model, a little extra computational pain can help eliminate arbitrage completely, which, given the size of the derivatives market, seems like a fair trade-off.