The financial crisis highlighted the importance of modelling dependence between assets. Correlation is one common way to measure that dependence, but as the number of assets gets larger and derivatives structures get more complex, dependence becomes complicated to capture.
One example of this is in the pricing of foreign exchange correlation swaps. These swaps work by paying out the realised correlation between two currency pairs in exchange for a fixed correlation – the strike. The two currency pairs could, for example be US dollar/sterling and euro/sterling.
Banks typically use forex correlation swaps to hedge correlation risk in their exotics books. It is intuitively understood that at times of stress, the correlations between the volatilities of the forex rates should, in theory, become stronger. Because correlations are generally complex to factor into models, dealers typically price the swaps by fixing the correlations between the volatilities of the three rates close to 1. However, these correlations are significant drivers of the swap prices.
“Usually, either for simplicity or to achieve better numerical stability, people hardcode the correlation between the different volatilities in the model to 1. But it turns out the correlation between the volatility processes is the most important thing,” says Alvise De Col, head of forex money markets, equities and commodities model validation at UBS in Zurich.
In this month’s first technical, Foreign exchange correlation swap: problem solver or troublemaker?, De Col and his co-author, Patrick Kuppinger, a quantitative analyst within the UBS business solutions team, show that not modelling these correlations appropriately can cause forex correlation swaps to be mispriced.
The swaps are typically priced using local volatility models, with no means to describe the volatility correlation structure, or stochastic volatility models, where the correlations between the volatilities of the forex rates are fixed at one. In their paper, the quants use a so-called stochastic local volatility model, where this restriction is lifted – that is, it allows correlations between the forex rate volatilities to take on any value.
Stochastic volatility models are usually good at capturing the dynamics of the spot in relation to volatility but do not calibrate well to prices, and local volatility models do the exact opposite. The combination of the two is called a stochastic local volatility (SLV) model, and it possesses the positive features of both.
SLV models are used by many banks now, but applying them to more complex, multi-asset products is tricky. What the UBS quants do is control the level of mixing of the stochastic and local parts of the model together with the correlations between the forex rate volatilities to explore the swap price range.
This is an important result, because it means there is significant model-dependence on the value of such a key quantity as the fair value of correlationPeter Austing, Citadel
“To price the swaps correctly, and even to risk-manage them correctly, and for the desk to find the right hedges for these positions, they need to have a view on these correlations, because their impact on the price can be material. Then you need to have a suitable model to free out these terms,” says De Col.
On comparing their model’s prices with that of existing methods such as Black-Scholes, local correlation and other SLV models, the authors find the price can vary by up to 25 correlation points or percentage points.
“It is really material, also considering the typical notionals at stake. For a swap priced at 50 correlation points, that will be from 30 to 55, so it’s a large range,” says De Col.
Others agree the additional dynamics captured is essential in pricing the products. “This is an important result, because it means there is significant model-dependence on the value of such a key quantity as the fair value of correlation,” says Peter Austing, a quantitative researcher at Citadel in London. “In addition to hammering home this important point through empirical studies, the authors provide a very nice and useful set of approximations showing how the fair correlation is impacted by the various key model drivers.”
Such results make the models much more useful, since they provide an insight into the pricing to trading teams, says Austing.
The paper highlights the importance of the phenomenon of correlation smiles – or the dependence of implied correlation on the moneyness of the trade – in the pricing of derivatives. Because of the complexity associated with modelling correlations, serious research in factoring them into prices only came about in the last decade in the equity space and the forex space.
The UBS quants’ paper takes a further step, by exploring this effect on forex correlation swaps. Given the significant effect the correlation smile can have on the prices of these products, these dynamics shouldn’t be ignored just because of convenience.