Technical Note on Local Time Risk Premiums in Parameterized Models of Interest-Rate Claims
34 Pages Posted: 27 Dec 2019 Last revised: 17 Dec 2020
Date Written: December 6, 2019
This technical note serves to establish proofs for the list of statements in Bakshi, Crosby,
and Gao (2019). We maintain their notation. Equation numbers not prefixed by letters refer to equations in Bakshi, Crosby, and Gao (2019).
Section I studies the quantitative implications of the Vasicek (1977) model, which imparts two
takeaways. First, it indicates that Ct[ℏ, k] is negligible (Ct[ℏ, k] is defined in Bakshi, Crosby, and
Gao (2019, equation (21)) and the sense in which Ct[ℏ, k] is negligible is as made in the statement
of their Result 1). Second, when Ct[ℏ, k] is negligible, then (in contrast to the negative average
return of OTM calls observed in the empirical data the expected excess return to holding an outof-the-money (OTM) call (respectively, put) option is positive (respectively, negative) when the bond futures risk premium is positive.
Section II derives the pattern of the expected excess return to holding an option on bond
futures when the spot interest-rate is a one-dimensional mean-reverting Gaussian process. This is
a closed-form characterization of Case 1 in Bakshi, Crosby, and Gao (2019, Section 4.3).
Section III and Section IV, respectively, derive the pattern of the expected excess return to holding an option on bond futures corresponding to Case 2 (quadratic term-structure model; Leippold
and Wu (2002) and Campbell, Sunderam, and Viceira (2017)) and Case 3 (rare disasters model,
Wachter (2013)) of Bakshi, Crosby, and Gao (2019, Section 4.3).
Section V presents the background steps leading to the derived pattern of the expected excess
returns to holding an option on bond futures in a long-run risks model (as in Bansal and Yaron
(2004) and pursued in Zhou and Zhu (2015)). These steps establish the statement of Case 5 of
Bakshi, Crosby, and Gao (2019, Section 4.3).
Section VI derives a backbone result, using characteristic functions, which links (i) the expected
excess return of options on bond futures and (ii) the bond futures risk premium. Pertinent here is
Lemma VI.1, which is used in our proofs in this technical note.
Keywords: Expected return of options on Treasury bond futures, unspanned components of pricing kernel, interest-rate models, Tanaka’s formula, local time risk premiums
JEL Classification: G10, G12
Suggested Citation: Suggested Citation