Ball and Torous (BT) [ 4 ] propose an equilibrium methodology to value contingent claims on risk-free zero coupon bonds . The resulting closed-form valuation formula is independent of investor preferences and eliminates the need for numerical estimations of utility-dependent factors.
The underlying state variable is the risk-free zero coupon bond directly. Its price is assumed to follow a Brownian Bridge process, ensuring that it converges to the face value at maturity. Also, since this underlying state variable is a tradable security, a preference-free, closed-form valuation formula for European options may be derived.
Let P ( t, m ) be the time t price of a risk-free zero coupon bond with maturity m . P ( m, m ) = 1. Define ¾ ( t, m ) to be the t -period log return on the zero coupon bond:
Since
and
we see that ¾ ( t, m ) is constrained at t = 0 and t = m .
The yield-to-maturity ¼ ( m ), is defined as the continuously compounded rate of return (per unit time) earned if the bond is bought at time t = 0 and held until maturity t = m . Hence:
Therefore, if the investor commits to holding the bond until maturity he earns a log return of ¼ ( m ) t after t units of time, where t ‰ m . However, if the bond is not held until maturity the return diverges from the deterministic yield-to-maturity and is stochastic. Let · ( t, m ) be the excess log return earned on the risk-free bond over time t . Therefore:
As illustrated by (6.2), the bond return may be decomposed into a deterministic and a stochastic component. Holding the bond for time t , a deterministic return of ¼ ( m ) t is earned. Selling the bond prior to maturity introduces the stochastic component · ( t, m ), which reflects the changing market conditions.
Since:
the stochastic part of the bond return is constrained at t = 0 and t = m .