Ball and Torous (1983) Bond Price Dynamics and Options

Summary

Ball and Torous (1983) model default-free pure discount bond price dynamics as a Brownian bridge process, exploiting the terminal constraint that bond prices must equal face value at maturity. Using this specification as input to Merton's (1973) preference-free contingent-claim framework, they derive a closed-form formula for European call and put options on default-free bonds — the first preference-free, analytically tractable result of its kind. A simple chi-squared volatility estimator is also provided.

Key Claims

Brownian bridge bond price model: Excess log holding return $\eta(t;m) = \xi(t;m) - \mu(m)t$ is "tied down" at $t=0$ and $t=m$ , yielding a Brownian bridge. The standardized bridge $W$ satisfies $dW = -W/(1-t)\,dt + dZ$ ; its total variance $E[W^2(t)] = t(1-t)$ is non-stationary, peaking at $t=\tfrac{1}{2}$ and returning to zero at maturity.
Bond price stochastic differential equation (SDE): $dP/P = a_1(P,t)\,dt + \sigma_1\,dZ$ , where $\sigma_1$ (instantaneous standard deviation, SD, of excess holding return) is constant for a given maturity — a direct consequence of the Brownian bridge structure.
Preference-free closed-form call option: $C(K,\tau) = P(\tau;1)N(h_1) - P(0;\tau)KN(h_2)$ where $v^2 = \sigma_1^2 + \sigma_\tau^2 - 2\rho\sigma_1\sigma_\tau$ , $h_1 = [\ln(P(\tau;1)/K) - \ln P(0;\tau) + (v^2/2)\tau]/(v\sqrt{\tau})$ , $h_2 = h_1 - v\sqrt{\tau}$ . Put-call parity gives $\Pi(K,\tau) = -P(\tau;1)N(-h_1) + P(0;\tau)KN(-h_2)$ .
Key innovation: Using the bond price (tradeable) as state variable — rather than the interest rate (non-tradeable) — makes the option formula preference-free and closed-form. Brennan-Schwartz (1977, 1982) and Courtadon (1982) use interest rates, yielding preference-dependent solutions requiring numerical approximation.
Volatility estimation: $\hat{v}^2 = \frac{1}{r}\sum_{i=1}^r(X_i - W_i)^2$ is unbiased; $r\hat{v}^2/v^2 \sim \chi^2(r)$ , providing exact confidence intervals for $v^2$ and hence for option prices.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"Default-free pure discount bond prices were posited to follow a non-standardized transformed Brownian bridge process. This specification implicitly incorporates the terminal constraint that the price of a default-free pure discount bond equal its face value at maturity."

"Investors may not agree upon a theory of the term structure, but they will necessarily agree on equilibrium option values."

My Take

The Brownian bridge insight is elegant: the terminal constraint on bond prices is not a side condition but the defining feature of the price process, and the bridge is the natural model for it. The preference-free formula is a significant practical advance over Brennan-Schwartz and Courtadon. The chi-squared estimator is simple and easily implemented. The main limitation is that the constant- $\sigma_1$ assumption (required for analytical tractability) may be empirically restrictive — subsequent stochastic-volatility and multi-factor term structure literature addresses this.