5.5 Conclusion

5.5 Conclusion

Langetieg's model allows the incorporation of an arbitrary number of economic factors into the bond pricing equation. The short- term interest rate is specified as a linear combination of the economic factors, which are assumed to follow a joint elastic random walk. Hence the bond price becomes a function of this linear combination of economic factors. The ability to find a closed-form solution for the bond price depends on the type of process assumed for the economic factors. The elastic random walk results in a normal distribution which makes such a solution possible.

This model makes a theoretical rather than a practical contribution to interest rate modelling. Its contribution lies in the theoretical framework it provides for the incorporation of multiple economic factors. The model parameters are left unspecified, so application of the model requires the specification of the number and type of economic factors determining the short-term interest rate, the estimation of the parameters of the joint elastic random walk process followed by the economic factors and the estimation of the associated market prices of risk. This is a rather challenging exercise giving rise to many estimation complexities.

Chapter 6: The Ball and Torous Model

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.

6.1 Holding period returns

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:



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.


the stochastic part of the bond return is constrained at t = 0 and t = m .

6.2 Brownian bridge process


where { µ ( s ): s ˆˆ [0, 1]} is a Brownian Bridge process and ƒ is the instantaneous standard deviation of the excess return.

{ µ ( s ): s ˆˆ [0, 1]} is a standardised Brownian Bridge process if:

  • P ( µ (0) = 0) = 1,

  • the process is Gaussian, where


  • P ( µ (1) = 0) = 1.

Hence the standardised Brownian Bridge process is an augmented standardised Brownian motion [1] with the added requirement that it takes on the value zero at time s = 1. The definition of · ( t, m ) in (6.3) satisfies the requirements that · (0, m )= · ( m, m ) = 0 and hence it may be specified as a Brownian Bridge process.

Our assumption that markets are efficient implies that all currently known information is included in current market prices, or market yields. Under this assumption, the return on the bond can only vary from the deterministic return ¼ ( m ) t , by unanticipated information becoming known. Had this unanticipated information been known, market yields would have adjusted to accommodate it. Hence we may conclude that:

It is appropriate to represent the unexpected returns by means of a Gaussian process, since they are a result of random economic events.

Let { Z ( s ): s ˆˆ [0, ˆ )} be a standardised Brownian motion. The properties of a standardised Brownian motion process, for t 0 and 0 where 0 t + < 1, imply [2] :

Define a diffusion process W ( t ), as follows :


  • P ( W (0) = 0) = 1,

  • P ( W (1) = 0) = 1

  • and

which implies that { W ( t ): t ˆˆ [0, 1]} is a standardised Brownian Bridge process.

If { X ( t ): t ˆˆ T } is a diffusion process, where T is some index set, it may be represented as

where ¼ X ( x, t ) and ƒ X ( x, t ) are respectively, the instantaneous mean and variance of the diffusion process. More specifically , we have:

Hence to determine the process describing the evolution of the Brownian Bridge we need to calculate the instantaneous mean and variance, ¼ W ( w,t ) and ƒ 2 W ( w,t ) respectively. Using the specifications in (6.5) above, we have:



and so:


We have determined the instantaneous mean and variance of the standardised Brownian Bridge process to be and ƒ 2 W ( w,t ) = 1 respectively. Hence the standardised Brownian Bridge process is characterised as:

The standardised Brownian Bridge is subject to a restoring force, pulling it back towards zero. The instantaneous variance remains constant, but the total variance, as at time t , is non-stationary and expressed as [ W 2 ( t )] = t (1 ˆ’ t ). This non-stationarity is due to the imposed terminal constraint. Since the excess return is assumed to have the functional form of (6.3), it is modelled as:

The economic interpretation of the time t return ¾ ( t, m ), is given by (6.1) and (6.2); hence:

The discount bond price dynamics are determined by application of Ito's Lemma:

[1] The stochastic process { Z ( s ): s ˆˆ [0, ˆ )} is a standardised Brownian motion if:

  • P ( Z (0) = 0) = 1,

  • the process is Gaussian with


[2] Consider a Brownian motion B t starting at x ,thatis P ( B = x ) = 1. Then the following are true [ 43 ]:


and translating this analysis into the required notation, we may calculate