Recently there has been lots of reported interest in the interest rate model of Brace, Gatarek, and Musiela (1997) (BGM), but anecdotal evidence suggests that it is proving difficult to implement. In this article we show how to implement simulation techniques for pricing a wide range of interest rate exotic instruments within the BGM model.
The main advantages of the BGM framework are that forward rates of interest are positive, caps and floors are priced consistently with the Black model, with swaptions reducing to the Black formulae after adding an extra assumption, and that the user has the same flexibility in choosing the volatility functions (either historically from principal components analysis or implied from market prices of interest rate options) as in the HJM framework.
Before pricing interest rate exotics, users first want to calibrate their models to market data. Calibration of the BGM model to caps and swaptions is relatively straightforward. We calibrate a two-factor time-homogeneous BGM model with piecewise flat volatility segments to a set of at-the-money caps U.S. dollar data from the beginning of June 1996. Figure 1 shows the resulting volatility functions (sigma1 and sigma 2). These two curves and the original discount function that we derive from Eurodollar rates and bootstrapped swap rates are the inputs into the BGM model.
Once the model has been calibrated to market data, interest rate exotics can be priced. Unlike the original BGM paper, it is better to model the observed discount function directly and not the discrete forward rates--discount functions are already calculated by many banks, the drift term for the price process is much easier to simulate than the forward rate process, and for swaps the swap rate is recoverable directly from the discount function. Our methodology reduces to working with the following equation;
where P(t,T) is the price of a discount bond at time t for maturity at T, r(t) is the short rate, v1(t, T) represent the n independent volatility functions for discount bond price returns--chosen to be equivalent to the lognormal structure for forward LIBOR rates represented in the BGM article--with the dz terms representing the increments in the uncertainty process.
Tables 1 and 2 show the results of applying our simulation techniques. Table 1 shows prices, over a range of maturities from two to 10 years of resettable, lookback and average rate caps. All caps are assumed to have quarterly resets and 10,000 simulations have been used with the antithetic control variate technique. A simple cap is also priced and compared against the Black price to show our technique returns the prices of the calibrating instruments. For the lookback cap the payoff at the end of each caplet is determined as the maximum level of LIBOR achieved during the capping period minus the observed rate; for a resettable cap the strike price is set at the beginning of the caplet; and for the average rate cap, the strike is set as the average of LIBOR over the capping period. All prices are quoted in basis points with standard errors in brackets.
Table 2 shows the prices of the five-year caps as we increase the number of simulations from 2,000 to 10,000 to illustrate the convergence of the technique.
The results here are obtained without implementing control variates. For pricing American style options (e.g. Bermudan swaptions), the simulation techniques are supplemented with Markovian short rate trees in order to approximate the early exercise strategy.
This week's Learning Curve was written by Les Clewlowand Chris Strickland. Both hold research positions at the Financial Options Research Centre Warwick Business School, UK, and the School of Finance and Economics University of Technology, Sydney, Australia. They are also directors of Lacima Consultants, a derivatives consultancy.
