Recursive Lower- and Dual Upper-Bounds for Bermudan-Style Options

European Journal of Operational Research, Forthcoming

31 Pages Posted: 21 Oct 2014 Last revised: 14 Jul 2019

See all articles by Alfredo Ibañez

Alfredo Ibañez

Comillas Pontifical University

Carlos Velasco

Universidad Carlos III de Madrid - Department of Economics

Date Written: July 12, 2019

Abstract

Although Bermudan options are routinely priced by simulation and least-squares methods using lower and dual upper bounds, the latter are hardly optimized. In this paper, we optimize recursive upper bounds, which are more tractable than the original/nonrecursive ones, and derive two new results: (1) An upper bound based on (a martingale that depends on) stopping times is independent of the next-stage exercise decision and hence cannot be optimized. Instead, we optimize the recursive lower bound, and use its optimal recursive policy to evaluate the upper bound as well. (2) Less time-intensive upper bounds that are based on a continuation-value function only need this function in the continuation region, where this continuation value is less nonlinear and easier to fit (than in the entire support). In the numerical exercise, both upper bounds improve over state-of-the-art methods (including standard least-squares and pathwise optimization). Specifically, the very small gap between the lower and the upper bounds derived in (1) implies the recursive policy and the associated martingale are near optimal, so that these two specific lower/upper bounds are hard to improve, yet the upper bound is tighter than the lower bound.

Keywords: American and Bermudan options, optimal stopping-times, lower-bounds and upper-bounds, Monte Carlo methods, barrier options

Suggested Citation

Ibañez, Alfredo and Velasco, Carlos, Recursive Lower- and Dual Upper-Bounds for Bermudan-Style Options (July 12, 2019). European Journal of Operational Research, Forthcoming. Available at SSRN: https://ssrn.com/abstract=2512659 or http://dx.doi.org/10.2139/ssrn.2512659

Alfredo Ibañez (Contact Author)

Comillas Pontifical University ( email )

Alberto Aguilera 21
Madrid, Madrid 28015
Spain

Carlos Velasco

Universidad Carlos III de Madrid - Department of Economics ( email )

Calle Madrid 126
Getafe, 28903
Spain
+34-91 6249646 (Phone)
+34-91 6249875 (Fax)

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