This series of Learning Curves examines the ability of interest rate models to capture relevant market information, in particular the observed market-implied volatility surface. Part I investigates the interest rate derivatives available in the market.
The interest-rate market provides two main sources of information, the yield curve and interest-rate volatility. This information can be extracted from traded instruments. However, it is best to use only liquidly traded instruments so that the price reflects the market expectations and is not affected by other factors like liquidity premium.
The yield curve is generally defined through a combination of cash/money market rates, futures prices, and Libor swap rates. Interest-rate volatility information can be obtained from quoted interest-rate options prices. The liquidly traded options normally used for this are vanilla options on Eurodollar futures, caps and floors, and European swaptions.
Let us examine the volatility aspect in further detail. The interest-rate options mentioned above can be priced using the Black-Scholes model. This model assumes that the underlying rate, Ft , is lognormally distributed
with being the implied volatility of the rate. The underlying rate is a forward rate (for example, three-month LIBOR) for options on Eurodollar futures, caps and floors, or a swap rate in the case of swaptions. The Black-Scholes model gives closed-form pricing equations for these vanilla options, whereby the price is given as a function of the forward curve and the implied volatility.
In fact, in the market, these options are not quoted as a price but rather as an implied volatility, and the closed-form Black-Scholes equations are used as a means to quote and convert this volatility to a price.
If the Black-Scholes model provided an accurate description of the underlying rate (and therefore the market), then the implied volatility surface for all these liquid options would be flat, and equal to the implied volatility . That is, as the strike and maturity of the option changes, the quoted implied volatility should not change. It is a well-known observation that this is not the case in the market, and that the implied volatility does vary with the strike and maturity of the option.
Consider the cap and floor implied volatility surface, illustrated in the figures below for the Japanese yen and euro market. It is clear from this information that the surfaces exhibit a term structure (maturity dependency), as well as an implied volatility smile or smirk structure (strike dependency).
JPY cap/floor implied volatility skew
EUR cap/floor implied volatility smile
In particular, the Japanese yen market implied volatility has a strong skew behaviour, whereas the euro (and also the U.S. dollar) markets exhibit a smile structure. Historically, this has not always been the case. In fact, it is only recently that the U.S. dollar volatility structure changed from a skewed environment to curve with a slight smile for out-of-the-money caps.
This deviation from the Black-Scholes model is because the distribution of the underlying ,ø(FT ,T) is not actually lognormal, as the model assumes. Consider the value of a caplet, C(K,T) struck at K and with a maturity of T. The value is given in terms of a general distribution as the integral
It can easily be shown that the distribution can be inferred by assuming a continuum of caplet prices at that maturity through the following relationship:
From the implied volatility surfaces of yen and euro caps, it therefore can be inferred that the distributions do indeed deviate from lognormal. The yen skew curve implies that the distribution has a fatter left tail and a thinner right tail, while the euro and U.S. dollar smile structure shows that the market distribution is kurtotic, having fatter left and right tail. A fatter tail implies the market places a higher probability than the lognormal model on rates moving there.
These resulting distributions are primarily due to supply and demand effects in the market. However, additional factors also include risk and position limits within an organization, as well as small premium effects.
Using the Black-Scholes model effectively means that for each option, a new implied volatility must be used (which is interpolated from the implied volatility surface), and therefore each option is priced using a different model (i.e. different underlying distribution). This is not an issue for the vanilla market options, like caps/floors and swaptions, because this is just a model to allow price quoting through volatility. However, a problem arises when pricing more structured derivatives because it is no longer clear which of these models should be used, and in general, none of them are suitable. Also, any resulting volatility hedge that the model calculates would not be accurate as the model only correctly prices one of the possible hedging instruments. More importantly, any delta hedge could be incorrect because the possible moves in the implied volatility surface that result from interest rate moves are not incorporated within the model.
When managing portfolios of interest rate derivatives, one should ideally have a single model consistent with the market and capable of capturing all the market factors that affect the portfolios value. This makes relative value calculations possible, as well as consistent pricing of the derivative and its hedging instruments. Such a model, with all its factors and related computational overhead, would not be viable in a trading system. However, a practical approach is still to employ a single model capable of pricing all the derivatives, which captures the main factors that affect the portfolio value.
Within such a model the instruments that are used to build the yield curve and calibrate the interest rate model are generally the liquidly traded instruments used to hedge the portfolio. Determining which instruments to use in the hedge depends upon whether the hedge is a delta or vega hedge--yield curve instruments are used for a delta hedge, and the options for a vega hedge. With the correct determination, the model will price the derivative instrument consistently with the instruments used in its hedge.
This week's Learning Curve was written by Douglas Long, European head of quantitative research and development at Principia Partnersin London.