Path dependent options on yields in the affine term structure model

Abstract

Abstract. We give analytical pricing formulae for path dependent options on yields in the framework of the affine term structure model. More precisely, European call options such as the arithmetic average call, the call on maximum and the lookback call are examined. For the two last options approximate formulae using the law of hitting times of an Ornstein-Uhlenbeck process are proposed. Numerical implementation is also briefly discussed and results are given in the case of the arithmetic average option.

Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

“…A study of the hitting time T 0 , in the more general case in which 0 is replaced by any fixed level a, has already been made [12]. It is used in several applications; for example, in [11] the distribution of the hitting time was needed to find a pricing formula for interest rate pathdependent options in financial models. A survey on the use of the hitting time of an OU process can be found in [1].…”

A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary

“…The derivation relies on a change of measure approach and delivers an explicit formula. This formula is an amended expression of the result given in Leblanc and Scaillet (1998). It corresponds to the formula given by a time substitution approach when the boundary level coincides with the mean of the invariant measure.…”

Term structure movements implicit in Asian option prices

“…In this work, in the Gaussian model the maximum likelihood estimation procedure described in Chen and Scott (1993) is extended to deal with options 28 . The following bond yields are observed along H different days: rb t (1/252), rb t (21/252), rb t (63/252), rb t (126/252), rb t (189/252), rb t (1) and rb t (1.5)…”

“…Longstaff (1995) use the Vasicek (1977) model to analyze the properties of options on average interest rates. Leblanc and Scaillet (1998) present theoretical results on the pricing of interest rate Asian options under the Vasicek (1977) and CIR (1985) models, based on Laplace transforms. Cheuk and Vorst (1999) adopt a Hull and White (1990) model.…”