Pricing path-dependent securities by the extended tree method

This paper presents a discrete-time method (ET method) for pricing path-dependent securities by the supplementary variable technique and examines the ET method from the point of view of Arrow–Debreu event tree. In particular, this paper identifies sufficient conditions on supplementary variables under which the ET method yields the same price for a path-dependent security as a valuation method based on a comparable Arrow–Debreu event tree. Two examples are provided to illustrate the ET method. The first example is a valuation of collateralized mortgage obligations (CMOs), where the collateral of a CMO is modeled as a pool of mortgage loans with heterogeneous prepayment costs. The second example is a valuation of American average options where the average is computed over a moving period with a fixed length. In addition, this paper presents a measure for the computational size of the ET method and illustrates numerical advantages of the ET method with examples.