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Pricing and hedging volatility smile under multifactor interest rate models

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Abstract

The paper extends Amin and Morton (1994), Zeto (2002), and Kuo and Paxson (2006) by considering jump-diffusion model of Das (1999) with various volatility functions in pricing and hedging Euribor options across strikes and maturities. Adding the jump element into a diffusion model helps capturing volatility smiles in the interest rate options markets, but specifying the mean-reversion volatility function improves the most. A humped volatility function with the additional jump component yields better in-sample and out-of-sample valuation, but level-dependent volatility becomes more crucial for hedging. The specification of volatility function is more crucial than merely adding jumps into any model and the effect of jumps declines as the maturity of options is longer.

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Notes

  1. For example, Ritchken and Chuang (1999) test a three-state variable Markovian HJM model using a humped volatility structure of forward rates, but only using price data for ATM caplets. Longstaff et al. (2001) use a string model framework to test the relative valuation of caps and swaptions using ATM cap and swaptions data. An exception is Jarrow et al. (2007) who study a multifactor term structure model with stochastic volatility and jumps to capture volatility smile of caps.

  2. See the discussion of Gupta and Subrahmanyam (2005) and Kuo and Lin (2007).

  3. Das (2002), Johannes (2004), and Piazzesi (2005) show that jumps are important for capturing interest rate dynamics.

  4. Specification 4 is very much a two-factor Gaussian model, where one factor has a degenerate volatility and the other factor controls the T − t, as pointed out by the anonymous reviewer. The interactions between the two factors generate the hump structure.

  5. If market prices are not available, the HJM model can be calibrated using Monte Carlo simulation (see Kramin et al. (2008)).

  6. Since the forward rate process in the HJM model is non-Markov, the newly developed efficient method such as Kramin et al. (2005) cannot be used.

  7. Since our option formula is highly nonlinear, the final parameter value may be subject to the initial input for each parameter. We adopted the common approach used in practice, where the initial value of each parameter on date t is given by the date t − 1. Since the one-factor model is nested within the two-factor model, to achieve the best estimate, volatility parameters for two-factor models are provided by the corresponding one-factor model. Hence, for two-factor models, μ, γ, and λ are estimated.

  8. Only the two-factor EXP model is shown in Fig. 3 because we later show that this model outperforms other models in in-sample estimation and out-of-sample prediction.

  9. All models, except the one-factor ABS model, are specified with several parameters in their volatility functions so that the volatility of term structure of forward rates cannot be obtained directly. To examine the stability of volatility of forward rates, it can only be obtained from the one-factor ABS model.

  10. ARE is computed by the difference between market and model prices over the market price and RAE is the absolute value of ARE.

  11. Euribor puts are used to hedge against the rise of interest rates, but the corresponding calls are used against a decline in rates.

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  1. Department of Finance, Tunghai University, 181, Taichung-Kan Road, Taichung, 407, Taiwan

    I.-Doun Kuo

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Correspondence to I.-Doun Kuo.

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Kuo, ID. Pricing and hedging volatility smile under multifactor interest rate models. Rev Quant Finan Acc 36, 83–104 (2011). https://doi.org/10.1007/s11156-010-0172-5

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