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On Pricing and Hedging in the Swaption Market On Pricing and Hedging in the Swaption Market: How Many Factors, Really?∗ Rong Fan† Anurag Gupta‡ Peter Ritchken§ October 1, 2001 ∗The authors would like to thank seminar participants at presentations made at the Eleventh Annual Deriv- atives Securities Conference in New York, The 2001 European Financial Management Association Meetings in Paris, and the Seventh Annual US Derivatives and Risk Management Congress in Boston. †Case Western Reserve University, WSOM, 10900 Euclid Ave., Cleveland, OH 44106-7235, Phone: (216) 368—3849, Fax: (216) 368—4776, E-mail: [email protected] ‡Case Western Reserve University, WSOM, 10900 Euclid Ave., Cleveland, OH 44106-7235, Phone: (216) 368—2938, Fax: (216) 368—4776, E-mail: [email protected] §Case Western Reserve University, WSOM, 10900 Euclid Ave., Cleveland, OH 44106-7235, Phone: (216) 368—3849, Fax: (216) 368—4776, E-mail: [email protected] On Pricing and Hedging in the Swaption Market: How Many Factors, Really? Abstract This article examines how the number of stochastic drivers and their associated volatility struc- tures affect pricing accuracy and hedging performance in the swaption market. In spite of the fact that low dimensional one and two-factor models do not reflect historical correlations that exist among forward rates, we show that they are capable of accurately pricing swaptions as well as higher order multifactor models, across all expiry dates and over all underlying swap matu- rities. Effective out-of-sample pricing is necessary but not sufficient for good hedging. Indeed, regarding hedging, we show there are significant benefits in using multifactor models. This is true even if one accounts for the fact that fewer hedging instruments are required when single factor models are used to hedge swaptions. Our empirical findings have strong implications for the modeling and risk management of an array of actively traded derivatives that closely relate to swaptions. JEL Classification: G12; G13; G19. Keywords: Interest Rate Derivatives; Swaptions; Volatility Structures; Pricing; Hedging. According to the Option Clearing Corporation, the notional amount of derivatives held by US commercial banks is about $40 trillion, with interest rate derivatives contracts accounting for 80% of this total. Over-the-counter contracts account for over 90% of the total notional amount, with exchange-traded contracts making up the rest. In this large market, the primary option contracts are caps and floors on interest rates and swaptions, which are options to enter or cancel swaps. The notional amount of these over the counter contracts exceeds 5 trillion dollars, making them amongst the most important interest rate claims that trade. Based on the size of this market, it is not surprising that significant effort has been placed on developing pricing models for these claims.1 Despite the importance of caps and swaptions, there is still wide divergence of opinion on how to best value these claims. It is widely believed that since the term structure of interest rates is driven by multiple factors, interest rate claims should be valued using multifactor models. Standard arbitrage arguments of Heath, Jarrow and Morton (1992) have shown that for pricing purposes, models differ according to the assumptions imposed on the volatility and correlation structures of forward rates. The exact specification of the volatility structures for forward rates, and the appropriate number of factors to be used, are considered to depend on the particular application. For example, Rebonato (1999) argues that while a one-factor model, in which forward rates are instantaneously perfectly correlated, might suffice for the pricing of caps, it is very unlikely to be useful for pricing swaptions, since they depend heavily on the correlation among forward rates.2 Recent advances in modeling methodology have made it possible to use multi-factor models to price even complex interest rate claims, like American swaptions.3 This has led to a deeper discussion on how many factors are really necessary to model interest rate claims such as caps and swaptions, as well as more complex claims like Bermudan and American swaptions. Several studies have attempted to address these issues and, as we discuss in the next section, the results are mixed and somewhat confusing. We attribute much of the confusion to the fact that some studies focus on pricing issues, while others focus on hedging. Actually, the answer depends on whether the model is to be used for pricing alone, or whether it is to be used for hedging. 1Indeed, given the liquidity of caps and swaptions, traders often demand that models for interest rate exotics, such as Bermudan swaptions and resettable caps, have the property that they price these claims at, or at least very close to, their market values. 2A cap consists of a portfolio of caplets, each caplet representing an option on an individual forward rate. In contrast,aswaptioncanbeviewedasanoptiononaportfolioofforwardrates.Asaresult,therelationship between caps and swaptions is determined largely by the correlation structure among the forward rates. Rebonato claims that since in a one-factor model forward rates are perfectly correlated, such models will tend to overprice swaptions. 3Recent studies by Andersen (2000), Carr and Yang (1999), Longstaff, Santa-Clara and Schwartz (2001b), Longstaff and Schwartz (2000) and Pedersen (1999), for example, contribute to the literature by developing methodology that allows contracts such as Bermudan swaptions to be numerically priced, relative to a given array of European swaptions, and consistent with the current term structure. 1 As we discuss later on, if models are evaluated solely on pricing performance, then lower order models might be acceptable. However, if models are evaluated based on hedging performance, then a more demanding standard is established, and lower order models might be unacceptable. Good pricing performance is a necessary condition for a useful model; good hedging performance, however, is sufficient! In this article we investigate a broad range of one, two, three and four-factor models with different volatility structures, that incorporate varying degrees of level and maturity dependence to pick up skewness and volatility hump effects. We closely examine how these models perform in pricing and hedging swaptions. Typical data sets in this market consist of at-the-money con- tracts, with an array of expiry dates and maturity dates of the underlying swaps. Characterizing the biases in prices produced by the models along these two dimensions is important, and has not been well documented by empiricists to date.4 A good model should have the property that all expiry and maturity effects are well explained. Of course, if we had data on the prices of away-from-the-money swaptions, then skew effects could also be examined. While such data is not available to us, we are fortunate to have price data on caps with multiple strikes, which we use to explore skew effects. In the first part of the paper we investigate the impact of adding additional factors on pricing swaptions. To measure pricing effectiveness, we calibrate a model using known swaption and term structure data. Once the parameters are estimated we can price claims in the future contingent on the future term structure. If the “out-of-sample” residuals are “small” and have no biases, then the model is viewed positively. In our analysis, the “out-of-sample” tests are conducted one, two, three, and four weeks after calibration. We replicate some of the results of Longstaff, Santa-Clara and Schwartz (2001a), hereafter LSS. They consider models with up to eight factors, and show that a four-factor model is necessary to price swaptions accurately. LSS argue that the large improvements obtained by adding factors is due to the fact that low dimensional models are unable to produce realistic correlations among forward rates and getting these correlation effects right is crucial for pricing swaptions. We provide an alternative explanation to their results, and identify specific one and two-factor models that can price swaptions as effectively as their four-factor model. Indeed, for the purpose of pricing swaptions, one-factor models may suffice. In the second part of the paper we examine hedging effectiveness. Given a model, we can establish a hedge position in bonds. Then, if the model is correct, holding the hedge position for a short time increment should lead to small price changes relative to the unhedged position, regardless of the future term structure. The models can then be evaluated based on their hedging errors. We show that models that price well do not necessarily hedge well. In particular, multiple 4An exception is Jaganathan, Kaplin and Sun (2001) who characterize the pricing performance of their models according to the expiry dates of the swaption. 2 factors play a bigger role here. Our one-factor model, which was competitive with a four-factor model for pricing, is much less precise when viewed from the hedging perspective. For hedging swaptions, multi-factor models are desirable! Given these results for vanilla swaptions, it is clear that hedging products such as Bermudan swaptions, and other exotics, which typically are priced relative to a core set of swaptions, will be more effective with multi-factor models. The paper proceeds as follows. In the first section we review the literature, sort through the current set of confusing empirical results, and highlight the contributions that this article makes to the literature. In the second section we discuss the set of 18 different models that we evaluate. In the third and fourth sections we discuss our data and model implementation. In section five we discuss our experimental design for examining pricing accuracy and hedging precision. In section six we closely examine a nested set of principal component based models. In section seven we compare the performance of our best principal component based model with alternative lower order models.
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