Pricing American Options: A Duality Approach

By Martin B. Haugh, Leonid Kogan. Operational Research, 2018

480 citations recorded by the source corpus.

We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worst-case performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.

Subfields: Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/27fb670dfb049453ce8f5fa48a8c20e54684cca6

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Citations in the corpus

MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE‐EXERCISE OPTIONS (2004, unknown): “This is an extension of the single-exercise case (n = 1) where the dual problem, found in Haugh and Kogan (2001) and Rogers (2002), is given by V ,10 = infM∈H0 Et [ max 0≤t T (Zt − Mt) ] .”
Pricing Asset Scheduling Flexibility using Optimal Switching (2008, methodology): “Several canonical choices have been proposed, including the Laguerre polynomials in the original paper of Longstaff and Schwartz (2001), the indicator functions of a partition of the domain of (Xt) in Gobet et al. (2005), and the logistic bases in Haugh and Kogan (2004).”
TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON‐NESTED MONTE CARLO (2009, methodology): “Rogers (2002) and independently Haugh and Kogan (2004) developed a dual method which provides an upper bound for the Bermudan price, given an approximation of the Snell envelope.”
Interpretable Optimal Stopping (2018, background): “ Lagrangean duality. This approach yields upper bounds on the optimal value and can also be used to derive high quality stopping policies. Examples include Rogers (2002), Andersen and Broadie (2004), Haugh and Kogan (2004), Chen and Glasserman (2007), Brown et al. (2010), Desai et al. (2012b) and Go”
A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options (2007, background): “Upper bounds based on the dual problem are derived in [15, 23].”
A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation (2018, background): “Problems of this nature arise naturally in many settings, most notably in the pricing of financial derivatives (Andersen and Broadie, 2004; Haugh and Kogan, 2004; Desai et al., 2012).”
Monte Carlo and Quasi-Monte Carlo Sampling (2009, methodology): “Another family of methods that produce high-biased estimators is the dual approach introduced independently in [351] and [153].”
DGM: A deep learning algorithm for solving partial differential equations (2017, unknown): “The Longstaﬀ-Schwartz algorithm has been further analyzed by [45], [23], and others.”
Multivariate convex regression with adaptive partitioning (2011, methodology): “was generated using the dual martingale methods of Haugh and Kogan (2004) from value functions generated using polynomial basis functions based on the mean of the assets, (1,Yt ,Y 2 t ,Y 3 t ,h(Yt)), where Yt = 1/M ∑i=1 Xi(t), with 2,000 samples.”
Efficient Control Variates and Strategies for Bermudan Swaptions in a LIBOR Market Model (2005, unknown): “Following Haugh and Kogan [2004] and Andersen and Broadie [2004], we refer to the right-hand side of (1) as the primal problem and define the dual problem in (1) as the right hand side of:”
A PRIMAL–DUAL ALGORITHM FOR BSDES (2013, methodology): “resentation in Theorem 3.1 can be interpreted as a dual minimization problem in the sense of information relaxation. This dual approach was introduced for Bermudan option pricing by Rogers (2002) and Haugh and Kogan (2004), and was further developed for discrete time stochastic control problems by B”
Monte Carlo valuation of American options (2002, unknown): “A benchmark example in this subject appears to be the Bermudan n-max-call, studied in Broadie and Glasserman (1997b), and used as a test example by Haugh and Kogan (2001), Fu et al. (2001) , and Andersen and Broadie (2001).”
A New Learning Algorithm for Optimal Stopping (2009, background): “_ x(t) = h(x(t);y); y2Rs; ( 19 ) are well-posed and have a unique globally asymptotically stable equilibrium (y) for each y with ( ) continuous, and,”