Libor Market Model of interest rate options

Abstract

The prices of Libor options are obtained for the quantum finance Libor Market Model. The option prices show new features of the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated. Black's caplet formula for quantum finance is given an exact derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived for the quantum finance Libor Market Model. The approximate Libor option prices are derived using the volatility expansion developed in Section 3.14. The BGM–Jamshidian expression for the Libor interest rate caplet and swaption prices is obtained as the limiting case when all the Libors are exactly correlated. Introduction The Libor option prices are obtained from the Libor zero coupon bonds B L (t, T) – obtained from the Libor ZCYC curve Z L (t, T) discussed in Section 7.9 – and the benchmark three-month Libor L ( t, T ). For notational convenience, Libor zero coupon bonds B L (t, T) will be denoted by B ( t, T ). All the options are defined to mature at future calendar time T 0 , with present time given by t 0 = T − k ; the notation of present being denoted by t 0 is used to simplify the notation. It is natural for these options to choose B ( t, T 0 ) as the forward bond numeraire. In other words, the forward bond numeraire is B ( t, T I +1 ) with I = −1 and Libor drift is calculated for this numeraire. Libor calendar and future time are shown in Figure 6.1 and Libor times t 0 = T − k , T 0 , and T n are shown in Figure 8.1.