Technical Note on Local Time Risk Premiums in Parameterized Models of Interest-Rate Claims

Abstract

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.