Express fashion coupon

Over the past few years, a wide array of interest rate derivative products has been introduced. Interest rate contingent claims are probably the most complex derivative securities to value, since their prices depend on the dynamics of the whole term structure of interest rates. Several pricing models that rest on theories of the term structure have been proposed. Examples are Brennan and Schwartz (1979), Courtadon (1982), Cox, Ingersoll, and Ross (1985), Dothan (1978), Longstaff (1989), and Vasicek (1977). These models price all interest rate dependent products in a consistent fashion, but they involve several unknown parameters--including the utility-dependent market price for risk and they fail to provide a perfect fit to the initial term structure of interest rates.

Arbitrage-free models avoid many of the limitations of their predecessors. These models take the current term structure as given and only admit future changes in the term structure that preclude arbitrage opportunities. This approach was first proposed by Ho and Lee (1986) in the discrete time framework. Black, Derman, and Toy (1988) extended Ho and Lee's original idea with a model that provides an exact fit to the initial term structure, as well as to the current volatility of all spot interest rates. Heath, Jarrow, and Morton (1990) introduced a, more general framework which admits one or more factors, and which includes the Ho and Lee model as a special case. Hull and White (1990) derived extensions of Vasicek's (1977) and Cox, Ingersoll, and Ross's (1985) models, making them consistent with both the current term structure of interest rates and the current term structure of volatility.

Arbitrage-free models are very appealing from a conceptual point of view, because they ensure that all interest rate contingent claim prices are consistent with the current term structure of interest rates and of volatilities. However, arbitrage-free models require the estimation of the initial yield curve and of the spot and/or forward rate volatility structure, in addition to the parameter estimates of earlier models based on theories of the term structure. This approach is also extremely challenging from an implementation point of view, because most arbitrage-free models developed to date do not have closed-form solutions, and thus involve complex numerical procedures which are computationally very demanding.

A simple alternative to the pricing of interest rate contingent claims, which enjoys a great deal of popularity among practitioners for its simplicity and userfriendliness, employs the bond price as the state variable. Assuming that bond prices follow a geometric diffusion process, with constant volatility and constant short rate, one obtains the Black and Scholes (1973) model. Ball and Torous (1W3) retained the constant volatility assumption, but managed to produce a price process that forces the bond to equal its face value at maturity, heir approach allows bond yields to become infinite at maturity. Schaefer and Schwartz (1987) avoided this complication by postulating a process which enables the bond's volatility to decrease over time and reach zero at maturity. Their pricing formula expresses bond volatility as a linear function of its duration.

As far as we know, this paper constitutes the first empirical investigation of the Schaefer and Schwartz model. We tested this model in the Canadian context on the Government of Canada bond options traded at the Montreal Exchange. The paper is organized as follows. In the initial section, we present the model and discuss its assumptions. Subsequently, we examine biases detected in implied standard deviations from transactions prices, using our analytic approximation. In the penultimate section, we test a specific trading rule which attempts to capture potential arbitrage profits detected by the model. Then we present our conclusions.

A TIME-DEPENDENT VARIANCE MODEL

The Schaefer and Schwartz model allows the dynamics of the underlying bond to change as it approaches maturity, and is based on the assumption that the only state variable governing the behaviour of the option is the price of the underlying bond. Schaefer and Schwartz made the usual assumptions--that capital markets are frictionless (i.e., that there are no taxes, no transactions costs, no short sales restrictions, etc.), that trading takes place continuously, and that the instantaneous rate of interest, r, is a known constant. Furthermore, they postulated a simple diffusion process for changes in yield to maturity on any bond, y:

Equation 1 = dy = mdt + sigma sub y dz,

where m is the instantaneous expected change in yield to maturity, sigma sub y , is the instantaneous standard deviation in yield change, and z is a standard Gauss-Wiener process such that dz = epsilon sq. root of dt and epsilon = = N(O,1). From (1), Schaefer and Schwartz showed that the instantaneous standard deviation of return on a bond, sigma(B,t), may be expressed as the product of a constant of proportionality, k, and the bond's Macaulay (1938) duration D(B,t):

Equation 2 = sigma = kD (B, t).

Writing the value at time t of an option on a default-free coupon bond with price B as F(B,t), the Black and Scholes hedging argument yields the PDE governing the dynamics of the option price:

Equation 3 = 1/2 F sup 2 sub BB sigma sup 2 (B,t) + F sub B (rB - delta) - F sub t - rF = 0,

where sigma(B,t) is as defined earlier, r is the nonstochastic instantaneous rate of interest, and delta is the continuous coupon for the underlying bond. This PDE is isomorphic to Equation (44) in Merton (1973), describing the dynamics of a European option on a dividend paying stock under constant short rate, except that here a is a function of B and t. Once the value taken by this parameter for each time t is known, (3) can be solved numerically, subject to the appropriate boundary conditions.

Evidence supporting the linear relationship between duration and volatility was reported by Brennan and Schwartz (1983), Schaefer and Schwartz (1987), and, more recently, by Gagnon (1990), for the Canadian context.

One can incorporate stochastic interest rates in this framework by treating the bond option as an option to exchange a bond for another bond (see Hull & White, 1989; Margrabe, 1978; Merton, 1973). Furthermore, since the duration of a long term bond will change very little over the life of a typical exchange-traded option, we can safely assume that duration is a known function of time during the option's life and, as in Merton (1973, p. 166), express a2 as the underlying bond's mean instantaneous variance over the option's life:

(Equation 4 omitted)

where T' denotes the underlying bond's time to maturity, T represents the. option's time to maturity, Do is the bond's current duration, and D sub T' - T corresponds to the bond's duration on the option's expiry date. This assumption in turn enables us to solve the PDE to obtain the well-known Merton (1973) continuous dividend-yield model (see also Black & Scholes, 1973).

In order to capture the early exercise premium associated with Canadian bond options, we resorted to the efficient analytic approximation derived by Barone-Adesi and Whaley (1987) and Macmillan (1986). Hence, we may express the price of an American call option with time to maturity T, and exercise price X, on a bond with price B, paying proportional coupons at a rate 6 as follows:

Equation 5 = C (B,T) = c (B,T) + A sub 2 (B/B*)q sup 2 for B < B*

C (B,T) = B - X for B >= B*,

where

c (B,T) = Be sup (b - r)T N (d sub 1 ) - Xe sup -rt N (d sub 2

A sub 2 = (B*/q sub 2 ) [1 - e sup (b - r)T N [d sub 1 (B*)]]

q sub 2 = [-(beta - 1) + ((beta - 1) sup 2 + 4 alpha/K) sup 1/2 ]/2

d sub 1 = [ln (B/X) + (b + 0.5 (kD)2)T]/(kD sq. root of T)

d sub 2 = d sub 1 - kD sq. root of T

beta = 2r/(kD) sup 2

alpha = 2b/(kD) sup 2 ,

b = r - delta represents the cost-of-carry term, B* is the bond's critical price. The pricing function for the American put option is

Equation 6 = P (B,T) = p (B,T) + A sub 1 (B/B**) sup q1 for B > B**

p (B,T) = X - B for B <= B**,

where

P (B,T) = Xe sup -rt N (-d sub 2 ) -Be sup (b - r)T N (-d sub 1 )

A sub 1 = -(B**/q sub 1 ) [1 - e sup (b - r)T N [-d sub 1 (B**)]]

q sub 1 = [- (beta - 1) - ((beta - 1 ) sup 2 + 4alpha/K) sup 1/2 ]/2,

B** is the critical price, and alpha, beta, d sub 1 , and d sub 2 are defined above.

For the purpose of this study, we measured 6 as the continuously compounded coupon yield. This measure was obtained by calculating the continuously compounded rate corresponding to the ratio of the bond's annual coupon to its current price. We investigated other alternative measures, such as yield to maturity, but our empirical results remained insensitive to reasonable estimates for the continuous coupon rate.