Interest Rate Model Calibration and Risk-Management Using Semidefinite Programming

  • ABSTRACT: In the original Black & Scholes (1973) model, there is a one-to-one correspondence between the price of an option and the volatility of the underlying asset. In fact, options are most often directly quoted in terms of their Black & Scholes (1973) implied volatility. In the case of options on multiple assets such as basket options, that one-to-one correspondence between market prices and covariance is lost. The market quotes basket options in terms of their Black & Scholes (1973) volatility but has no direct way of describing the link between this volatility and that of the individual assets composing the basket. Today, this is not yet critically important in equity markets where most of the trading in basket options is concentrated among a few index options, we will see however that it is crucial in interest rate derivative markets where most of the volatility information is contained in a rather diverse set of basket options.

Indeed, a large part of the liquidity in interest rate option markets is concentrated in European caps and swaptions. In the first chapter of this work we will show how one can express the price of swaptions (and caplets) as that of an option on a basket of zero-coupon bonds in one approach, or a basket of forward Libor rates in another. This basket option representation is exact in the first case and we will show how it provides an excellent pricing approximation in the second. In particular, this will allow us to reduce the problem of pricing swaptions in both the Gaussian H.J.M. model (see El Karoui & Lacoste (1992), Duffie & Kan (1996) or Musiela & Rutkowski (1997)) and the Libor market model (see Brace et al. (1997), Miltersen, Sandmann & Sondermann (1995) or Miltersen, Sandmann & Sondermann (1997)) to that of pricing swaptions in a multidimensional Black & Scholes (1973) lognormal model. The second chapter is then focused on finding a good pricing approximation for basket calls in this generic model. We derive price expansion where the first term is computed as the usual Black & Scholes (1973) price with an appropriate variance and the second term can be interpreted as the expected value of the tracking error obtained when hedging with the approximate volatility.

Besides its radical numerical performance compared to Monte-Carlo methods, the formula we obtain has the advantage of expressing the price of a basket option in terms of a Black & Scholes (1973) covariance that is a linear form in the underlying covariance matrix. This sets the multidimensional model calibration problem as that of finding a positive semidefinite (covariance) matrix that satisfies a certain number of linear constraints. In other words, the calibration becomes a semidefinite program. Recent advances in optimization (see Nesterov & Nemirovskii (1994) or Vandenberghe & Boyd (1996)) have led to algorithms that solve these problems very efficiently, with a complexity analysis that is comparable to that of linear programs (see Nesterov & Todd (1998)). This means that the general multidimensional market covariance calibration problem can be solved very efficiently.

Finally, we show that these same semidefinite programming techniques provide key sensitivity and risk-management results together with the calibration solution. For instance, we show how all sensitivities of the solution matrix to changes in market conditions can be directly obtained from the optimal solution of the dual calibration problem.

Numerical instability has a direct cost in both poor hedging and incomplete risk description. By reducing the amount of numerical noise in the daily recalibration process and improving the risk-management of interest rate models, we hope these methods will significantly reduce hedging costs and improve the reliability of risk-management computations.