Analytical Bounds for Treasury Bond Futures Prices

Abstract

The pricing of delivery options, particularly timing options, in Treasury bond futures is prohibitively expensive. Recursive use of the lattice model is unavoidable for valuing such options, as Boyle (1989) demonstrates. As a result, the main purpose of this study is to derive upper bounds and lower bounds for Treasury bond futures prices.

This study employs a maximum likelihood estimation technique presented by Chen and Scott (1993) to estimate the parameters for two-factor Cox-Ingersoll-Ross models of the term structure. Following the estimation, the factor values are solved for by matching the short rate with the cheapest-to-deliver bond price. Then, upper bounds and lower bounds for Treasury bond futures prices can be calculated.

This study first shows that the popular preference-free, closed-form cost of carry model is an upper bound for the Treasury bond futures price. Then, the next step is to derive analytical lower bounds for the futures price under one- and two-factor Cox-Ingersoll-Ross models of the term structure. The bound under the two-factor Cox-Ingersoll-Ross model is then tested empirically using weekly futures prices from January 1987 to December 2000.

Appendix

From Theorem 1, we have:

$$ \begin{array}{c}{E}_t^Q\left[\delta \left(t,T\right)X(T)\right]={E}_t^Q\left[\delta \left(t,T\right)\right]{E}_t^{F(T)}\left[X(T)\right]\\ {}=P\left(t,T\right){E}_t^{F(T)}\left[X(T)\right]\end{array} $$

(71.37)

where δ is strictly less than 1. Due to the risk-neutral pricing result we have, the LHS must equal X(t), and hence:

$$ X(t)=\frac{E_t^{F(T)}\left[X(T)\right]}{P\left(t,T\right)} $$

(71.38)

Note that the forward measure is maturity dependent. Clearly, the Radon-Nikodym derivative (RND) is

$$ \eta \left(t,T\right)=\frac{\delta \left(t,T\right)}{P\left(t,T\right)} $$

(71.39)

Since the measure is T-dependent, so should be the RND (usually, RND is just η(t)). Let the interest rate process be

$$ dr(t)=\widehat{\mu}\left(r,t\right) dt+\sigma \left(r,t\right)d{W}^Q(t) $$

(71.40)

Applying Ito’s lemma,

$$ \begin{array}{c}0= \ln \kern0.5em P\left(T,T\right)= \ln \kern0.2em P\left(t,T\right)+{\displaystyle {\int}_t^T\frac{1}{P\left(u,T\right)}}{P}_u\left(u,T\right) du+{P}_r\left(u,T\right) dr+\frac{1}{2}{P}_{rr}\left(u,T\right){(dr)}^2d\widehat{W}(u)-{\displaystyle {\int}_t^T}\frac{1}{2}{\left[\frac{\sigma \left(r,u\right){P}_r\left(u,T\right)}{P\left(u,T\right)}\right]}^2 du\\ {}= \ln \kern0.5em P\left(t,T\right)+{\displaystyle {\int}_t^T\frac{1}{P\left(u,T\right)}}\left[{P}_u\left(u,T\right) du+{P}_r\left(u,T\right)\widehat{\mu}\left(r,u\right)+\frac{1}{2}{P}_{rr}\left(u,T\right)\sigma {\left(r,u\right)}^2\right] du\\ {}+{\displaystyle {\int}_t^T}\frac{1}{P\left(u,T\right)}{P}_r\left(u,T\right)\sigma \left(r,u\right)d\widehat{W}(u)-{\displaystyle {\int}_t^T\frac{1}{2}{\left[\frac{\sigma \left(r,u\right){P}_r\left(u,T\right)}{P\left(u,T\right)}\right]}^2 du}= \ln \kern0.5em P\left(t,T\right)+{\displaystyle {\int}_t^Tr(u) du}+{\displaystyle {\int}_t^T\frac{1}{P\left(u,T\right)}}{P}_r\left(u,T\right)\sigma \left(r,u\right)d\widehat{W}(u)-{\displaystyle {\int}_t^T\frac{1}{2}}{\left[\frac{\sigma \left(r,u\right){P}_r\left(u,T\right)}{P\left(u,T\right)}\right]}^2 du\end{array} $$

(71.41)

Letting:

$$ \theta \left(t,T\right)=-\frac{\sigma \left(r,t\right){P}_r\left(t,T\right)}{P\left(t,T\right)} $$

(71.42)

and moving the first two terms to the left:

$$ \begin{array}{l}-{\displaystyle {\int}_t^Tr(u) du}- \ln \kern0.5em P\left(t,T\right)={\displaystyle {\int}_t^T-\theta \left(u,T\right)d\widehat{W}(u)}-{\displaystyle {\int}_t^T\frac{1}{2}}\theta {\left(u,T\right)}^2 du\\ {}\frac{\delta \left(t,T\right)}{P\left(t,T\right)}=\eta \left(t,T\right)= \exp \left({\displaystyle {\int}_t^T-}\theta \left(u,T\right)d\widehat{W}(u)-{\displaystyle {\int}_t^T\frac{1}{2}}\theta {\left(u,T\right)}^2 du\right)\end{array} $$

(71.43)

This implies the Girsanov transformation of the following:

$$ \begin{array}{c}{W}^{F(T)}(t)={W}^Q(t)+{\displaystyle {\int}_t^T\theta (u) dt}\\ {}={W}^Q(t)-{\displaystyle {\int}_t^T\sigma \left(r,u\right)}\frac{P_r\left(u,T\right)}{P\left(u,T\right)} du\end{array} $$

(71.44)

The interest rate process under the forward measure henceforth becomes:

$$ dr(t)=\left[\widehat{\mu}\left(r,t\right)+\sigma {\left(r,t\right)}^2\frac{P_r\left(t,T\right)}{P\left(t,T\right)}\right] dt+\sigma \left(r,t\right)d{W}^{F(T)}(t) $$

(71.45)

Note that the forward measure is quite general. It does not depend on any specific assumption on the interest rate process.