Hedging rate exotics, Bergomi-style

Banks typically hedge their positions in exotic rate derivatives, such as Bermudan swaptions, using vanilla interest rate swaps.

By doing so, they hedge their portfolio imperfectly. They can delta-hedge the first-order profit and loss risks, using swaps to mimic the exposure to movements in rates or volatility. But second-order P&L risks from gamma and theta – the exposure to the rate of change of the underlying, and exposure to time decay of an option, respectively – can be more difficult to manage.

Inefficient hedging and unwanted exposure are problems banks have so far lived with, as a satisfactory solution has not yet been proposed.  

In this month’s technical, The swap market model with Bergomi stochastic volatility, Kenjiro Oya, executive director of the fixed income desk in the global market quant team at Nomura, proposes a theoretical framework to make the hedging of these products more rigorous.

He takes inspiration from how exotic equity derivatives traders manage their volatility exposures. This is challenging, because it involves jointly modelling spot prices and implied volatilities, and the correct approach is still an open question.

However, a solution to this issue has been proposed by Lorenzo Bergomi as part of his award-winning series of papers on volatility modelling. His so-called forward variance model is a stochastic volatility approach that allows for the pricing of exotic equity products.

In exotic rates markets, however, a corresponding solution is still missing. Oya’s idea is to adapt part of the theoretical framework that works for equity exotics to rate exotics.

“The novelty of the paper is to propose an interest rate model with stochastic volatility where the total gamma-theta P&L is explicit, while both swaption and interest rate swaps are used as the hedging instruments. This Bergomi model approach is not found anywhere in interest rate modelling literature,” says Oya.

In essence, he borrows the idea behind the Bergomi stochastic volatility model to develop his swap market model, which looks at the joint dynamics of swaptions and interest rate swaps.

More specifically, he focuses on co-terminal swaptions – products with the same expiry date – as they are the most natural choice for hedging Bermudan swaptions, with the minimum number of hedging instruments.

The novelty of the paper is to propose an interest rate model with stochastic volatility where the total gamma-theta P&L is explicit, while both swaption and interest rate swaps are used as the hedging instruments

Kenjiro Oya, Nomura

“This is an important paper, further developing the new approach to interest rate derivatives, which may be called ‘swap rate à la stock’,” says Dariusz Gatarek, professor of financial engineering at the Polish Academy of Sciences, who co-wrote the widely used Libor market model.

The development of the swap market model was inspired by the progress made in data-driven approaches, which Oya hopes to apply to help generate more reliable datasets for training neural networks.

“I came up with this idea when I played around with the deep hedging paper,” reveals Oya, referring to the recent and already influential work by JP Morgan’s Hans Buehler that proposes a deep learning approach to the hedging of vanilla equity options.

But vanilla equity options are highly liquid products with relatively short maturities. Using deep hedging for exotic rate products, where maturities are much longer and liquidity may be lower, is more complex. A large number of scenarios need to be generated to train the model, and, if both swaptions and interest rate swaps are used as the hedging instruments, the no-arbitrage condition is important for the stability of the results.

Decades-long scenarios cannot be built just on historical observations. It is possible, however, to generate them using classic derivatives theory.

“For interest rate exotics, we need to have market scenarios for training that last 10 years or more, so we would need to rely on a market scenario generator to have a sufficient training dataset,” says Oya.

The hedging strategy for those scenarios can be developed by applying deep hedging, and here is where Oya’s model steps in. Deep hedging uses neural networks, which need to be trained with good data to deliver a robust output.

Oya’s model can generate scenarios of interest rate swaps and swaptions with the no-arbitrage condition. Scenarios obtained this way would constitute a powerful training dataset for the neural networks, potentially expanding the applicability of deep hedging to long-dated interest rate derivatives.