research stock | currency converter | forex trading training

These option pricing formulas are the same as those given for the Vasicek model in equations (23.10) and (23.11). They are also equivalent to using Black's model as described in Section 22.2. The variable o> is the standard deviation of the logarithm of the bond price at time T, and the volatility for the bond used in Black's model is <r P /<jT. AS explained in Section 22.3, an interest rate cap or floor can be expressed as a portfolio of options on zero-coupon bonds. It can therefore be valued analytically using the Hull-White model.

The volatility structure in the Hull-White model is determined by both a and a. The model can represent a wider range of volatility structures than Ho-Lee. The volatility at time t of the price of a zero-coupon bond maturing at time T is

a The instantaneous standard deviation at time t of the zero-coupon interest rate maturing at time T is

and the instantaneous standard deviation of the T-maturity instantaneous forward rate is ae~ a/xT ~'\ These functions are shown in Figure 23.5. The parameter a determines the short rate's instant aneous standard deviation. The reversion rate parameter, a, determines the rate at which bond price volatilities increase with maturity and the rate at which interest rate standard deviations decline with maturity. When a = 0, the model reduces to Ho-Lee, and zero-coupon bond price volatilities are a linear function of maturity with the instantaneous standard deviations of both spot and forward rates being constant

Interest Rate Derivatives: Models of the Short Rate

OPTIONS ON COUPON-BEARING BONDS

In Section 23.4 we showed how, when the Vasicek equilibrium model is used, we can express an option on a coupon-bearing bond as a portfolio of options on zero-coupon bonds (see Example 23.1). This section shows how we can do the same for the Ho-Lee and Hull-White models.

In the case of Vasicek's model, we calculated the value of the short rate, r = r K , for which the coupon-bearing bond price equaled the strike price. We then argued that the option on the coupon-bearing bond was equivalent to a portfolio of options on the zero-coupon bonds comprising the coupon-bearing bond. The strike price of each option is the value of the corres ponding zero-coupon bond when r = r K .

We could follow exactly the same procedure for the Ho-Lee and Hull-White models as for the Vasicek model. But it is more convenient to work with the <5f-period rate, R, than with the instantaneous short rate, r. We then never need to calculate partial derivatives of P(0, t) with respect to t. w A convenient choice for St is the time between the maturity of the option and the first subsequent coupon on the underlying bond.

We calculate a value of R for which the coupon-bearing bond's price equals the strike price. Suppose this is R K . The option on the coupon-bearing bond is equivalent to a portfolio of options on the zero-coupon bonds comprising the coupon-bearing bond. The strike price of each option is the value of the corresponding zero-coupon bond when R = R K .