Mispricing of Discount Bond Options in the Black-Derman-Toy Model Calibrated to Term Structure and Cap Volatilities: An Empirical Study

Abstract

The Black-Derman-Toy short rate model, calibrated to a downward sloping initial term structure of volatilities, incorporates mean reversion by having future short rate volatilities decay with time. This would have a decaying effect on future term structure of volatilities and option prices on discount bonds as well. This effect is empirically examined by comparing discount bond option prices from the BDT tree to that from a HJM lognormal forward rate model fitted with the same initial volatilities. In this case, the BDT tree is found to price options with maturity up to six months within one or two percent of the accurate values. But it severely underprices longer maturity options, with the degree of underpricing increasing sharply with option maturity. The magnitude of underpricing turns out to be independent of discount bond maturities. When the BDT tree is calibrated using six-monthly cap prices, options of all maturities on six-month discount bonds are priced correctly, but options of all maturities on longer maturity bonds are overpriced. This overpricing is found to be independent of option maturities.