INTRODUCTION
Last week's Learning Curve covered "functional Greeks" for instruments valued using interest rate models with curve- and surface-valued parameters. This week's article will apply this technique to the problem of providing a unified description of the volatility exposure of interest rate options.
Simple interest rate options are generally valued using variants of the Black-Scholes approach. For example, LIBOR rates, swap rates, or bond prices may be taken as analogous to equity prices in the Black-Scholes model. The Black-Scholes formula then gives the value of the option. It is customary to describe the sensitivity of the option value with respect to the parameters of the formula by taking partial derivatives, giving delta, vega, rho, theta and so forth. Of course, since the underlying quantity that the Black-Scholes formula is applied to varies according to the particular instrument being priced, these Greeks cannot be simply added together to describe the exposure of the entire position. To add, for example, deltas for swaptions to deltas for caps in a meaningful way, an adjustment factor based on the formula that expresses swap rates in terms of simple forward rates would have to be computed. The situation is even bleaker for vegas, since the common Black-Scholes models used for caps and swaptions are mathematically inconsistent with each other. Thus, it is not possible to normalize things by a simple change of coordinates. Add to this mix exotic derivatives valued using short-rate models that are inconsistent with the Black-Scholes models, and it is easy for vega analysis to turn into a Tower of Babel.
The use of functional Greeks leads to a solution to the problem of not being able to add the Greeks from different instrument classes together. The idea is to, instead of using multiple instrument-specific valuation models with a few scalar parameters, use a single comprehensive valuation model with functional parameters that can be applied to all instruments. The resulting generalized Greeks can be meaningfully added across instrument classes to provide a unified description of portfolio exposure to interest rate and interest rate-volatility risk.
UNIFIED VEGAS VIA HJM
The goal is to provide a consistent picture of volatility sensitivity for interest rate derivatives. It appears that the Heath-Jarrow-Morton model of the evolution of interest rates is a particularly good place to start. There are two main reasons for this. First, most of the models commonly used to value interest rate derivatives can be represented within the HJM framework. Second, as will be seen below, the HJM formulation admits a very clean separation between term-structure risk and volatility risk.
In the HJM model, it is assumed that the instantaneous forward-rate curve is known at the anchor timet = 0 and its future evolution is modeled out to the time horizon T. We denote the instantaneous forward rate for (absolute) maturity T at time t by f (t, T) . For notational simplicity, a one-factor HJM model will be considered, however the generalization to multi-factor models is straightforward.
Under the single-factor HJM model, the risk-neutral evolution of instantaneous forward rates is given by
where, f 0( T ) f (0 ,T ) is the initial forward-rate curve, * ( t, T ) is the (absolute) volatility of f ( t, T ) and
By standard risk-neutral pricing arguments, the value at time 0 of a derivative with discounted payout X ( f ) at time T >= 0 is given by
where f(f0 ,*, W) denotes the solution of equation 1 and EW denotes expectation over a standard Brownian motion. Last week's Learning Curve illustrated, as an example, the functional vega with respect to the volatlity surface * for a bond option valued under the HJM model.
The beauty of the HJM model for our purpose is that the initial forward curve and the volatility surface * give a complete and intuitive coordinate system for market conditions in the HJM world. This is in sharp contrast to mean-reverting short-rate models such as Hull-White and Black-Karasinski, where the marginal distribution of future rates depends on mean-reversion parameters in addition to the volatility parameters. Taking partial derivatives with respect to the volatility parameters in these models may give a very incomplete and misleading picture of sensitivity to assumptions about the future distributions of interest rates.
STOCHASTIC-VOLATILITY MODELS
There is a potential speed bump in our program. The idea is to represent the various models used for different instrument classes as HJM models so that their, now functional, vegas can be meaningfully added together. The problem is that most of the models used in practice have HJM representations with stochastic volatility. So the problem is: how do we make sense of the partial derivative with respect to a stochastic parameter?
Fortunately, there is a good solution to this problem. Suppose an alternative term-structure model parameterized in terms of a deterministic volatility surface *1 has a stochastic-volatility HJM representation with HJM volatility *(*1,W). For example, consider an "* model" where *(t, T)= f*(t, T)**(t, T), for some deterministic function **(t, T) and real-valued constant *. Since the driving Brownian motion is independent of the volatility function, the vega in the stochastic-volatility case is defined simply by taking the derivative inside the expectation in equation 2:
To relate this to the "traditional" vega with respect to **, a standard change-of-coordinates formula is used to rewrite it as
wheref*(f0,**,W) gives the sample value of f under the * model. The meaning of this formula is that D*v applied to the deterministic-HJM perturbation * is equal to the expected value of the pathwise vega with respect to **, D**X, applied to the stochastic *-model perturbation ** = f -** that corresponds to * for that path. Thus, the punch line is that, while it is not possible to reconcile inconsistent volatility models by a simple change of coordinates, it is possible by a stochastic change of coordinates.
This week's Learning Curve was written by Alvin Kuruc, a senior v.p. at Infinity, a Sungard Company.
