Pricing American Stock Options by Linear Programming

Abstract

We investigate numerical solution of finite difference approximations to American option pricing problems, using a new direct numerical method: simplex solution of a linear programming formulation. This approach is based on an extension to the parabolic case of the equivalence between linear order complementarity problems and abstract linear programs known for certain elliptic operators. We test this method empirically, comparing simplex and interior point algorithms with the projected successive overrelaxation (PSOR) algorithm applied to the American vanilla and lookback puts. We conclude that simplex is roughly comparable with projected SOR on average (faster for fine discretizations, slower for coarse), but is more desirable for robustness of solution time under changes in parameters. Furthermore, significant speedups over the results given here have been achieved and will be published elsewhere. Copyright Blackwell Publishers Inc 1999.

Delayed Purchase Options in Single-Leg Revenue Management

“…Another approach is to apply approximate dynamic programming to give lower and upper bounds on the value of the option (Longstaff and Schwartz, 2001;Tsitsiklis and Roy, 2001;Haugh and Kogan, 2008). The pricing problem is also approximated by solving linear programming models (Dempster and Hutton, 1999). We refer the reader to Trigeorgis (1996) for the review of pricing models.…”

Vulnerable options in supply chains: Effects of supplier competition

“…, (13) where W 0 (τ 2 , X) is the time 0 value of a compound call option 6 with expiration date τ 2 , a strike price X, and a payoff based on the value of option V . The curve in Figure 2 is the graph of…”

“…For example, see [6], [8], [7], [30], [13], [37], [12], and references therein. 9 See [24], [21], [23], [15].…”

“…Using expressions (31) and (34), we can rewrite the manufacturer's time 0 problem (12) as (13)). Hence, the manufacturer should order from the slow supplier the following quantity:…”

Valuation of American options by the gradient projection method

“…Moreover, Dempster and Hutton [7] studied American option pricing problem using linear programming approach and Jaillet et al [16] presented variational inequality formulation of American option pricing problem. In this paper, we will construct an extremal problem equivalent to the variational inequality formulation and discuss the gradient projection method for the extremal problem.…”