Stochastic Duration and Fast Coupon Bond Option Pricing in Multi-Factor Models

Abstract

Abstract. Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond.

Pricing Swaptions within the Affine Framework

“…For example, Singleton and Umantsev [2001] propose an approximation for coupon bond options by approximating the exercise boundary with a linear function of the state variables (i.e., a hyperplane). They show that their technique dominates the speed and accuracy of the stochastic duration approach developed by Wei [1997] and Munk [1999]. They report that it takes approximately 1.4 seconds to estimate the price of a swaption in a two-factor CIR model with an absolute pricing error of ~ (5 ϫ 10 -4 ).…”

Energy Options in an HJM framework

“…In order to cope with this problem, Munk (1999) suggests how to approximate 3 More precisely, g is a C 1 (R, R+) function and h is a C(R, R+) function.…”

“…We shall stick to the nomenclature in Munk (1999) and denote the relative diffusion coefficients of the futures price process as s F j (u, T ), j = 1, . .…”

“…We omit most details in this section; in most cases they can be found in section 1.2 in Munk (1999) in the case of coupon bonds. The stochastic duration measure has the following proper-4 Yet another candidate was suggested in Munk (1999), namely to incorporate explicitly the drift and look…”

“…Our approach to cope with the flow feature is to view the contract as an indivisible portfolio of periodical (weekly or daily) single-delivery forwards. This makes it possible to price an option on the flow contract which is an option on a sum of n single-delivery forwards as a multiple of an option, which is written on a single-delivery forward, following the approach 2 of Jamshidian (1989) 1 and, especially, the extension for multi-factor models by Munk (1999). This approach to option pricing is only an approximate pricing method.…”