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2015 | OriginalPaper | Chapter

86. Unspanned Stochastic Volatilities and Interest Rate Derivatives Pricing

Author : Feng Zhao

Published in: Handbook of Financial Econometrics and Statistics

Publisher: Springer New York

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Abstract

This paper first reviews the recent literature on the unspanned stochastic volatilities (USV) documented in the interest rate derivatives markets. The USV refers to the volatility factors implied in the interest rate derivatives prices that have little correlation with the yield curve factors. We then present the result in Li and Zhao (Journal of Finance, 62, 345–382, 2006) that a sophisticated DTSM without USV feature can have serious difficulties in 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 that the unmodified dynamic term structure models, assuming the same set of state variables for both bonds and derivatives, are seriously challenged in capturing the term structure volatilities. We also present a multifactor term structure model with stochastic volatility and jumps that yields a closed-form formula for cap prices from Jarrow et al. (Journal of Finance, 61, 341–378, 2007). The three-factor stochastic volatility model with Poisson jumps can price interest rate caps well across moneyness and maturity. Last we present the nonparametric estimation results from Li and Zhao (2009). Specifically, the forward densities depend significantly on the slope and volatility of LIBOR rates, and mortgage market activities have strong impacts on the shape of the forward densities. 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. These findings reinforce the claim that term structure models need to accommodate the unspanned stochastic volatilities in pricing and hedging interest rate derivatives.

The econometric methods in this chapter include extended Kalman filtering, maximum likelihood estimation with latent variables, local polynomial method, and nonparametric density estimation.

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Another issue is the relative pricing between caps and swaptions. Although both caps and swaptions are derivatives on LIBOR rates, existing models calibrated to one set of prices tend to significantly misprice the other set of prices. For a more detailed review of the literature, see Dai and Singleton (2003).

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.

Andersen and Benzoni (2006) show the “curvature” factor is 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.

The affine models include the completely affine models of Dai and Singleton (2000), the essentially affine models of Duffee (2002), and the semi-affine models of Duarte (2004). Other DTSMs include the hybrid models of Ahn et al. (2003), the regime-switching models of Bansal and Zhou (2003), and models with macroeconomic jump effects, such as Piazzesi (2001), and many others.

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.

The difference cap is the difference of the caps with subsequent maturities and the same strike prices. Instead of having caplets ranging from as early as 6 months, the difference cap only has caplets of a small maturity region.

We acknowledge that with jumps in LIBOR rates, both the historical and instantaneous covariance matrices 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 capture the variations in LIBOR rates due to continuous shocks and that the impact of jumps is only contained in the residuals.

Many empirical studies on interest rate dynamics (see, e.g., Andersen and Lund 1997; Ball and Torous 1999; Chen and Scott 2001) 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.

Our interpolation scheme is slightly different from that of Han (2002) for the convenience of deriving closed-form solution for cap prices.

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.

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

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 measures, as in Chernov and Ghysels (2000), 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.

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

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 problem.

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

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

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

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

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.

While the prepayment 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.

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

See Andersen and Lund (1997), Ball and Torous (1999), Brenner et al. (1996), Chen and Scott (2001), and many others.

See Jaffee (2003) and Duarte (2006) for excellent discussions on the use of interest rate derivatives by Fannie Mae and Freddie Mac in hedging interest rate risks.

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

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

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

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.

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Title: Unspanned Stochastic Volatilities and Interest Rate Derivatives Pricing
Author: Feng Zhao
Publisher: Springer New York
Book: Handbook of Financial Econometrics and Statistics
Print ISBN: 978-1-4614-7749-5

Electronic ISBN: 978-1-4614-7750-1

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4614-7750-1_86