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.