Modern risk management and derivatives pricing requires complicated models of interest rate movements. In any given market, changes in the term structure of interest rates account for most of the risk of sovereign and highly rated corporate bonds and almost all of the risk of interest-rate derivatives, like caps/floors, swaptions, and so forth. A variety of models are being used by practitioners ranging from the simple Vasicek model to multi-dimensional Heath-Jarrow-Morton models. The degree of complexity of each model depends on the number of stochastic variables (also known as factors or dimensions) driving the dynamics of the term structure of rates. The more driving factors the model has, the richer the dynamics and the better the model's ability to fit prices of liquid interest-rate options or to match the historical covariance structure between rates of different maturities. However, the complexity of model implementation also increases with the number of factors, while the speed of pricing typical American options decreases significantly. This phenomenon is known as "the curse of dimensionality". Two-dimensional models offer reasonable compromise between complexity, robustness and speed and this article provides an introduction to these models.
EQUATIONS & PRICING ALGORITHM
Here we consider the case of a two-factor Hull-White model [1]. This model has a lot of analytical tractability for European options on zero-coupon bonds and plain-vanilla instruments like cap/floors, which is useful for testing model implementation. The model short interest-rate dynamics (r(t) below) is described by the following system of stochastic differential equations (SDEs):
These equations describe a short rate that is stochastic (driven by Brownian motion dz1) and is reverting (at speed a) to a mean level that is itself stochastic, mean-reverting to 0-(t)/a (at speed b) and driven by a second Brownian motion dz2, which is correlated with dz1. The deterministic function 0-(t) is determined by the term structure of interest rates. These SDEs lead to the following partial differential equation (PDE) for the option price:
where C(t) is the time-dependent rate of cashflows and subscripts denote partial derivatives.
Option models with stochastic volatility lead to similar two-dimensional PDEs, and the method described here can be easily applied for a variety of other two-factor models. This PDE is solved on a two-dimensional (r,u) grid backwards in time starting from option expiry date T, where the payoff P(r,u,T) is usually known. Then we take small steps backwards in time, going from P(r,u,t) to P(r,u,t-dt), exercising any options along the way. For example, if we deal with an American option on a callable bond, then at each time step we have to apply the following logic:
If P(r,u,t) is greater than "current call threshold" then P(r,u,t) = "current call price."
The straightforward implementation (also called explicit scheme) is equivalent to a trinomial tree approach, and is easy to implement. The drawbacks are that it requires tiny timesteps in order to achieve stability while accuracy of such schemes is only first order in time.
A much faster way to solve the above PDE is by combining alternating-direction-implicit (ADI) and Crank-Nicholson (CN) methods. The CN method was designed to be unconditionally stable, but it takes twice as much time per time step as the explicit method. However, it allows much larger timesteps and is a second order accurate in time, so for a given level of pricing accuracy it can be 10 or more times faster than a tree or an explicit scheme.
The CN scheme can only be applied to one-dimensional problems. The ADI method comes to the rescue here, since it is specifically designed to split multi-dimensional problem into one-dimensional components. This allows much larger time steps and has faster convergence to the true solution.
The idea behind the ADI method is to split the diffusion process in each time step into two sub-processes, each process acting in only one direction. Then the time stepping is done in two sweeps: one sweep diffuses P(r,u,t) in the u-direction, and another in the r-direction. A minor inconvenience is that the ADI method requires the correlation term--third from the right in the equation (1)--to be zero, but equation (1) can be easily transformed by a simple rotation and stretching of variables r and u, whereby the correlation term is eliminated.
The model described above becomes a one-factor Hull-White model if we assume that u(t)=0, so we can easily see what advantages we get by adding an extra dimension.
BENEFITS OF EXTRA FACTORS
The graph above shows the historical term structure of volatility, or TSV (thick solid line). It is the diagonal of the covariance matrix of the US term structure spot rate changes between 1980 and the present time. We can now try to match the observed and the model TSV by adjusting model parameters via some optimization procedure like Levenberg-Marquardt or quasi-Newton method. The graph shows the best-fit TSVs for the one-factor (dashed line) and the two-factor (thin solid line) models. The two-factor model reproduces the observed peak in spot rate volatility at around four-year maturity rather well, while the one-factor model matches only the general trend for the volatility to decrease with maturity. Moreover, the overall misfit error for the two-factor model is five times smaller than for the one-factor model. If we price an American call option (callable in one year) on a 10-year bond with, say, a 6% coupon using the two models, we would find a price difference of USD1.00 on a USD100 principal, which can make the bond look cheap in one model and rich in another.
Matching a historical covariance matrix is a backward looking procedure and most traders interested in hedging would favor a forward-looking one. The market perception of how volatile spot rates will be in the future is contained in prices of actively traded cap/floors and swaptions. For trading applications, the fitting procedure should be modified to fit a set of those prices, rather than the historical TSV. Again, we would find the two-factor model to be superior in its ability to match market prices, therefore making it a better hedging and pricing tool.
REFERENCE:
1 J. Hull and A. White. "Hull-White on derivatives", Chapter 19, RISK publications, 1996
This week's Learning Curve is by Nick Baturin, a fixed income strategy researcher at Barclays Capitalin New York.
