Hedging performance of the Libor market model: the cap market case

Abstract

This article investigates the hedging performance of the Libor Market Model (LMM) as well as the need to use models that explicitly incorporate Volatility Specific Factors (VSF) to better the hedging results. We compare the hedging performance of a standard LMM to that of a Constant Elasticity of Variance (CEV) LMM and find that, although the volatility risk is not completely removed by a hedge portfolio composed only of bonds, using a standard LMM is adequate to obtain high hedging performance in the cap market.

Keywords:

• interest rate caps
• Libor market model
• constant elasticity of variance
• hedging

JEL Classification::

• G12
• G13
• G19

Acknowledgements

We would like to thank Patrice Poncet and Radu Tunaru as well as participants at the AFFI 2004 December conference and the FMA 2005 European conference for helpful comments. Any remaining errors are our own.

Notes

¹ The authors test two hedge strategies: factor hedging and bucket hedging. For factor hedging, the hedge instruments are the same regardless of the option maturity. The number of these hedge instruments depends on the number of factors in the model. Whereas, for bucket hedging, the hedge instruments are different for each option and correspond to all cash flow dates. The number of the instruments are independent of the number of factors used in the model. Driessen et al. (2003) find, in the case of bucket hedging, similar pricing performance results between single-factor and multi-factor models.

² In their terminology, this feature is called unspanned stochastic volatility.

³ The LMMs have elicited wide interest among academics and practitioners.

⁴ Previous interest rate models used, for example the spot rate, the long rate and the instantaneous forward rate as underlying variables. See, for instance, Brennan and Schwartz (1982), Hull and White (1990) and HJM (1992).

⁵ Practitioners rely on the Black (1976) model to price caps and floors.

⁶ Under the spot measure the associated numeraire is denoted B^S (t) with B^S (t) = . Thus, B^S (t) is the price process obtained by rolling over one-period zero-coupon bonds (see Jamshidian (1997) for further details). Moreover, the underlying rate can also be written under other measures such as (i) the terminal measure which associated numeraire is B(t, T_{N +1}) or (ii) the forward measure where the associated numeraire is B(t, T_{i +1}). We point out that under this latter probability measure, the forward Libor rate L(t, T_i ) is a martingale.

⁷ That is, η(t) is the smallest integer such that t ≤ t_{η ( t )}.

⁸ See Protter (2004).

⁹ See Cox and Ross (1976).

¹⁰ The skew is very pronounced for small values of α.

¹¹ See Appendix A in Andersen and Andreasen (2000) for a proof.

¹² All the data are from Bloomberg.

¹³ In contrast, many authors rely on a parametric form of the correlation matrix. However, the proposed forms provide a too simple link between forward rates and do not account for the particular shapes that can exist between short, intermediate and long rates as illustrated in Fig. 1. For example, one correlation function used is

where ρ^{i , j} is the correlation coefficient between the two forward Libor rates L(t, T_i ) and L(t, T_j ) and ζ is a positive constant.

¹⁴ If the number of factors is equal the number of forward rates (N), we write

for 1 ≤ i, p ≤ N.

¹⁵ See Martellini et al. (2001) for further details.

¹⁶ Since the data do not include the prices of caplets over their lives but prices of new options, we need to extrapolate the price for a given caplet 1 week after its inception. This is performed as follows. In week i we have the quoted caplet volatility σ_i of maturity T_i . 1 week later, we interpolate, from available caplet volatilities, the value of σ_i , which maturity date is T_i less 1 week.