113_Cap_Smiles_JF_2007 - THE JOURNAL OF FINANCE VOL. LXII,...

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THE JOURNAL OF FINANCE VOL. LXII, NO. 1 FEBRUARY 2007 Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile? ROBERT JARROW, HAITAO LI, and FENG ZHAO ABSTRACT Using 3 years of interest rate caps price data, we provide a comprehensive documenta- tion of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed-form formula for cap prices. We show that although a three-factor stochastic volatility model can price at-the-money caps well, significant negative jumps in inter- est rates are needed to capture the smile. The volatility smile contains information that is not available using only at-the-money caps, and this information is important for understanding term structure models. T HE EXTENSIVE LITERATURE ON MULTIFACTOR DYNAMIC term structure models (here- after, DTSMs) of the last decade mainly focuses on explaining bond yields and swap rates (see Dai and Singleton (2003) and Piazzesi (2003) for surveys of the literature). The pricing and hedging of over-the-counter interest rate derivatives such as caps and swaptions has attracted attention only recently. World-wide, caps and swaptions are among the most widely traded interest rate derivatives. According to the Bank for International Settlements, in re- cent years, their combined notional value exceeds 10 trillion dollars, which is many times larger than that of exchange-traded options. The accurate and efficient pricing and hedging of caps and swaptions is therefore of enormous practical importance. Moreover, because cap and swaption prices may contain information on term structure dynamics not contained in bond yields or swap rates (see Jagannathan, Kaplin, and Sun (2003) for a related discussion), Dai and Singleton (2003 p. 670) argue that there is an “enormous potential for new insights from using (interest rate) derivatives data in model estimations.” The extant literature on interest rate derivatives primarily focuses on two issues (see Section 5 of Dai and Singleton (2003)). The first issue is that of Jarrow is from the Johnson Graduate School of Management, Cornell University. Li is from the Stephen M. Ross School of Business, University of Michigan. Zhao is from the Rutgers Busi- ness School, Rutgers University. We thank Warren Bailey, Peter Carr, Fousseni Chabi-Yo, Jefferson Duarte, Richard Green (the editor), Pierre Grellet Aumont, Anurag Gupta, Bing Han, Paul Kupiec, Francis Longstaff, Kenneth Singleton, Marti Subrahmanyam, Siegfried Trautmann, an anony- mous referee, and seminar participants at the Federal Deposit Insurance Corporation, Rutgers University, the 2003 European Finance Association Meeting, the 2004 Econometric Society Winter Meeting, the 2004 Western Finance Association Meeting, Bank of Canada Fixed-Income Confer- ence, the 15th Annual Derivatives Conference, and the Financial/Actuarial Mathematics Seminar at the University of Michigan for helpful comments. We are responsible for any remaining errors.
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This note was uploaded on 10/31/2010 for the course NBA 5550 at Cornell University (Engineering School).

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113_Cap_Smiles_JF_2007 - THE JOURNAL OF FINANCE VOL. LXII,...

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