Bruno Dupire’s local volatility model gained popularity over the past two decades for being able to uniquely map volatility to spot and time, and therefore calibrate exactly to the volatility smile implied from option prices.
The local volatility model requires the derivative of the option price with respect to both maturity and strike as inputs. This is pretty straightforward for equity and forex derivatives modelling, but when it comes to interest rates, things get tricky.
Swaption prices, for instance, are only available when the expiry date is the swap fixing date, and hence the derivatives with respect to time and strike cannot be obtained from market prices.
“In Dupire’s approach, we need option prices for a wide range of expiries and a wide range of strikes. For example, a five-year into 10-year swap – we only have the one option in the market for that underlying swap. We can look at a four-year into 10-year swaption in the market, but it has a different underlying swap compared with the five-year into 10-year. That means we cannot apply Dupire’s approach as it is. This is a technical problem,” says Kenjiro Oya, an executive director in the exotic rates team at Nomura Securities in Tokyo.
Many have tried to fix this by modelling the price process of a rolling maturity swap, a spot-starting interest rate swap with constant maturity, instead of a standard swap underlying.
The advantage of this is that swaptions on rolling maturity swaps as the underlying are available for a range of tenors. While more market quotes are available for calibration, modelling this is still challenging, because Dupire’s formula would have to go through a cumbersome transformation to be able to calibrate to interest rate smiles of multiple tenors. In addition, modelling multiple underlying swaps can get numerically intensive.
In The swap market model with local stochastic volatility, Nomura’s Oya proposes a local volatility model that can skip the transformation and directly calibrate to interest rate smiles of multiple tenors and multiple underlying swap processes in a non-parametric manner.
In his model, Oya uses the rolling swap rate itself as the main state variable – previous competing models did not use the rolling swap rate as the state variable.
“In past research, they used the local volatility function information for other models, where spot starting rolling swap rates are not the modelling objects or modelling bases, so they need to reinterpret that local volatility function using the time of their state variables,” says Oya.
By applying Dupire-style local volatility stripping, the swap market model smile calibration quality as well as computational efficiency can be enhanced substantiallySenior quant at a European bank
The quant obtains Dupire’s local volatility function of the rolling swap rate and simulates it using a simple Brownian diffusion process. This step eliminates the need for any kind of transformation. The resulting model can calibrate Dupire’s local volatility function to multiple rates tenors and can calibrate, for instance, a five-year 10-year swaption in two seconds.
“By applying Dupire-style local volatility stripping, the swap market model smile calibration quality as well as computational efficiency can be enhanced substantially,” says one senior quant at a European bank.
Because of their sensitivity to both rates and volatility, swaptions have always brought modelling challenges. For instance, until recently, calibration of the stochastic alpha, beta, rho (SABR) model to market-implied swaption prices required time-consuming numerical techniques to produce arbitrage-free prices. Faster, analytical techniques on the other hand were prone to mispricing at the higher and lower strikes.
The advantage of using a local volatility function to calibrate swaption prices is that they produce a much better fit to the implied volatility surface compared with stochastic volatility models such as SABR, although at the cost of poor spot-volatility dynamics. Stochastic volatility models, on the other hand, are really good at capturing spot-volatility dynamics.
To make the best of both worlds, quants started developing local stochastic volatility models over a decade ago that combine elements of both types of models. Since Oya’s model is a local stochastic volatility model, it has the additional advantage of being able to capture dynamics as well as it can calibrate to the implied volatility surface. But Oya goes a step further by removing a cumbersome transformation step, thereby allowing the model to calibrate to multiple tenors and underlyings.
This opens up new possibilities – for instance, exotic rates products such as Bermudan swaptions can be priced more consistently for different strikes, which would be tricky with other models, given the model parameters might change for different strikes.