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Interest Rate Derivatives Pricing with Volatility Smile

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Handbook of Computational Finance

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Abstract

The volatility “smile” or “skew” observed in the S&P 500 index options has been one of the main drivers for the development of new option pricing models since the seminal works of Black and Scholes (J Polit Econ 81:637–654, 1973) and Merton (Bell J Econ Manag Sci 4:141–183, 1973). The literature on interest rate derivatives, however, has mainly focused on at-the-money interest rate options. This paper advances the literature on interest rate derivatives in several aspects. First, we present systematic evidence on volatility smiles in interest rate caps over a wide range of moneyness and maturities. Second, we discuss the pricing and hedging of interest rate caps under dynamic term structure models (DTSMs). We show that even some of the most sophisticated DTSMs have serious difficulties in pricing and hedging caps and cap straddles, even though they capture bond yields well. Furthermore, at-the-money straddle hedging errors are highly correlated with cap-implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. These findings strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets. Third, we develop multifactor Heath–Jarrow–Morton (HJM) models with stochastic volatility and jumps to capture the smile in interest rate caps. We show that although a three-factor stochastic volatility model can price at-the-money caps well, significant negative jumps in interest rates are needed to capture the smile. Finally, we present nonparametric evidence on the economic determinants of the volatility smile. We show that the forward densities depend significantly on the slope and volatility of LIBOR rates and that mortgage refinance activities have strong impacts on the shape of the volatility smile. These results provide nonparametric evidence of unspanned stochastic volatility and suggest that the unspanned factors could be partly driven by activities in the mortgage markets.

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Notes

  1. 1.

    The nonparametric forward densities estimated using caps, which are among the simplest and most liquid OTC interest rate derivatives, allow consistent pricing of more exotic and/or less liquid OTC interest rate derivatives based on the forward measure approach. The nonparametric forward densities can reveal potential misspecifications of most existing term structure models, which rely on strong parametric assumptions to obtain closed-form formula for interest rate derivative prices.

  2. 2.

    Andersen and Benzoni (2006) show the “curvature” factor are not significantly correlated with the yield volatility and it is true in this paper as well, therefore the volatility effect here is not due to the “curvature” factor.

  3. 3.

    The differences between parameter estimates with and without the correction term are very small and we report those estimates with the correction term B k .

  4. 4.

    In the empirical analysis of Li and Zhao (2006), the QTSMs are chosen for several reasons. First, since the nominal spot interest rate is a quadratic function of the state variables, it is guaranteed to be positive in the QTSMs. On the other hand, in the ATSMs, the spot rate, an affine function of the state variables, is guaranteed to be positive only when all the state variables follow square-root processes. Second, the QTSMs do not have the limitations facing the ATSMs in simultaneously fitting interest rate volatility and correlations among the state variables. That is, in the ATSMs, increasing the number of factors that follow square-root processes improves the modeling of volatility clustering in bond yields, but reduces the flexibility in modeling correlations among the state variables. Third, the QTSMs have the potential to capture observed nonlinearity in term structure data (see e.g., Ahn and Gao 1999). Indeed, ADG (2002) and Brandt and Chapman (2002) show that the QTSMs can capture the conditional volatility of bond yields better than the ATSMs.

  5. 5.

    We acknowledge that with jumps in LIBOR rates, both the historical and instantaneous covariance matrix of LIBOR rates contain a component that is due to jumps. Our approach implicitly assumes that the first three principal components from the historical covariance matrix captures the variations in LIBOR rates due to continuous shocks and that the impact of jumps is only contained in the residuals.

  6. 6.

    Many empirical studies on interest rate dynamics (see, for example, Andersen and Lund; Ball and Torous; Chen and Scott) have shown that correlation between stochastic volatility and interest rates is close to zero. That is, there is not a strong “leverage” effect for interest rates as for stock prices. The independence assumption between stochastic volatility and LIBOR rates in our model captures this stylized fact.

  7. 7.

    Our interpolation scheme is slightly different from that of Han (2007) for the convenience of deriving closed-form solution for cap prices.

  8. 8.

    For simplicity, we assume that different forward rates follow the same jump process with constant jump intensity. It is not difficult to allow different jump processes for individual LIBOR rates and the jump intensity to depend on the state of the economy within the AJD framework.

  9. 9.

    The market prices of interest rate risks are defined in such a way that the LIBOR rate is a martingale under the forward measure.

  10. 10.

    In order to estimate the volatility and jump risk premiums, we need a joint analysis of the dynamics of LIBOR rates under both the physical and forward measure, as in ?, Pan (2002), and Eraker (2004). In our empirical analysis, we only focus on the dynamics under the forward measure. Therefore, we can only identify the differences in the risk premiums between forward measures with different maturities. Our specifications of both risk premiums implicitly use the 1-year LIBOR rate as a reference point.

  11. 11.

    Andersen and Brotherton-Ratcliffe (2001) and Glasserman and Kou (2003) develop LIBOR models with stochastic volatility and jumps, respectively.

  12. 12.

    Due to the wide range of moneyness and maturities of the difference caps involved, there could be significant differences in the prices of difference caps. Using percentage pricing errors helps to mitigate this probelem.

  13. 13.

    The LIBOR forward curve is constructed from weekly LIBOR and swap rates from Datastream following the bootstrapping procedure of LSS (2001).

  14. 14.

    Throughout our discussion, volatilities of LIBOR rates refer to market implied volatilities from cap prices and are different from volatilities estimated from historical data.

  15. 15.

    See Han (2002) for more detailed discussions on the impact of time varying correlations for pricing swaptions.

  16. 16.

    RMSE of a model at t is calculated as \(\sqrt{{u}_{t }^{{\prime} }\left (\hat{\theta } \right ) {u}_{t } \left (\hat{\theta } \right ) /M}\). We plot RMSEs instead of SSEs because the former provides a more direct measure of average percentage pricing errors of difference caps.

  17. 17.

    We would like to thank Pierre Grellet Aumont from Deutsche Bank for his helpful discussions on the influence of MBS markets on OTC interest rate derivatives.

  18. 18.

    While the prepayments rates were also high in later part of 2002 and for most of 2003, they might not have come as surprises to participants in the MBS markets given the two previous special periods.

  19. 19.

    See, for example, Deuskar et al. (2003).

  20. 20.

    See,,,, and many others.

  21. 21.

    See Jaffee (2003) and Duarte (2008) for excellent discussions on the use of interest rate derivatives by Fannie Mae and Freddie Mac in hedging interest rate risks.

  22. 22.

    While the slope factor can have nontrivial impact on prepayment behavior, the volatility factor is crucial for pricing interest rate options.

  23. 23.

    We thank the referee for the suggestion of examining the effects of ARMs origination on the forward densities.

  24. 24.

    In results not reported, we find that the nonlinear dependence of the forward densities on the volatility factor remain the same as well.

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Authors and Affiliations

  1. Stephen M. Ross School of Business, University of Michigan, Ann Arbor, MI, 48109, USA

    Haitao Li (Professor of Finance)

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  1. Haitao Li
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Correspondence to Haitao Li .

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Editors and Affiliations

  1. , Risk Management Institute, National University of Singapore, 21 Heng Mui Keng Terrace, Level 4, Singapore, 119613, Singapore

    Jin-Chuan Duan

  2. L.v.Bortkiewicz Chair of Statistics, C.A.S.E. Centre f. Appl. Stat. & Econ., Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, 10099, Germany

    Wolfgang Karl Härdle

  3. , Department of Computational and Data Sci, George Mason University, University Drive 4400, Fairfax, 22030, Virginia, USA

    James E. Gentle

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Li, H. (2012). Interest Rate Derivatives Pricing with Volatility Smile. In: Duan, JC., Härdle, W., Gentle, J. (eds) Handbook of Computational Finance. Springer Handbooks of Computational Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17254-0_7

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