Least squares Monte Carlo methods for numerical evaluation of multi-period costof- capital values

Least squares Monte Carlo methods for numerical evaluation of multi-period costof- capital values

Added in AFIR / ERM / RISK 2019 JUL | IME 2019

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In a previous article published in Insurance: Mathematics and Economics [2], we explored a multiperiod cost-of-capital valuation of a liability cashflow subject to repeated capital requirements. The resulting liability value is given by a backward recursion, mathematically similar to that of dynamic risk measures described in for instance [1]. In a Markovian setting, the value at each time is given by some function of the state variables. Evaluating this function is a problem that suffers from the curse of dimensionality: If the state space is too high-dimensional, direct calculation becomes unfeasible, even when allowing for discretization of continuous state variables. To tackle this problem, we seek to approximate the value as a function of state variables via a linear regression on basis functions in each step of the backward recursion. This approach is somewhat famously applied in [3] for the purpose of valuing complex American options, where this method is referred to as least squares Monte Carlo. In this talk, based on ongoing research, we explore the use of least squares Monte Carlo approach to calculate the cost-of-capital value described in [2], where we will provide some convergence results as well as numerical results.

[1] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. and Ku, H. (2007), “Coherent multiperiod risk adjusted values and Bellman's principle.” Annals of Operations Research, 152, 5-22.

[2] Engsner, H., Lindholm, M. and Lindskog, F. (2017), “Insurance valuation: A computable multiperiod cost-of-capital approach.” Insurance Mathematics and Economics, 72, 250-264.

[3] Longsta_, F.A. and Schwartz, E.S. (2001), “Valuing American Options by Simulation: A Simple Least-Squares Approach.” The Review of Financial Studies, 14 (1),113-147.

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