This article establishes the Poisson optional stopping times (POST) method by Lange et al. (2020) as a near-universal method for solving liquidity-constrained American options, or, equivalently, penalised optimal-stopping problems. In this setup, the decision maker is permitted to â€œstopâ€, i.e. exercise the option, only at a set of Poisson arrival times; this can be viewed as a liquidity constraint or â€œpenaltyâ€ that limits access to optionality. We use monotonicity arguments in function space to establish that the POST algorithm either (i) finds the solution or (ii) demonstrates that no solution exists. The monotonicity of POST carries over to the discretised setting, where we additionally show geometric convergence and provide convergence bounds. For jump-diffusion processes, dense matrix factorisation may be avoided by using a suitable operator-splitting method for which we prove convergence. We also highlight a connection with linear complementarity problems (LCPs). We use the POST algorithm to value American options and compute early-exercise boundaries for Kouâ€™s jump-diffusion model and Hestonâ€™s stochastic volatility model, illustrating the breadth of application and numerical reliability of the method

Hessing, Jean-Claude; Lange, Rutger-Jan; Ralph, Daniel

This article establishes the Poisson optional stopping times (POST) method by Lange et al. (2020) as a near-universal method for solving liquidity-constrained American options, or, equivalently, penalised optimal-stopping problems. In this setup, the decision maker is permitted to â€œstopâ€, i.e. exercise the option, only at a set of Poisson arrival times; this can be viewed as a liquidity constraint or â€œpenaltyâ€ that limits access to optionality. We use monotonicity arguments in function space to establish that the POST algorithm either (i) finds the solution or (ii) demonstrates that no solution exists. The monotonicity of POST carries over to the discretised setting, where we additionally show geometric convergence and provide convergence bounds. For jump-diffusion processes, dense matrix factorisation may be avoided by using a suitable operator-splitting method for which we prove convergence. We also highlight a connection with linear complementarity problems (LCPs). We use the POST algorithm to value American options and compute early-exercise boundaries for Kouâ€™s jump-diffusion model and Hestonâ€™s stochastic volatility model, illustrating the breadth of application and numerical reliability of the method

Jean-Claude Hessing, Rutger-Jan Lange and Daniel Ralph
Additional contact information
Jean-Claude Hessing: Erasmus Universiteit Rotterdam
Daniel Ralph: Cambridge Judge Business School

No 22-007/IV, Tinbergen Institute Discussion Papers from Tinbergen Institute

Keywords: Optimal stopping; Penalty method; HJB equation; Contraction; Fixed Point; Operator Splitting; Implicit Explicit; Linear Complementarity Problem (search for similar items in EconPapers)
JEL-codes: C44 C61 G13 (search for similar items in EconPapers)
Date: 2022-01-28
New Economics Papers: this item is included in nep-his
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