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Real-Option Valuation in Multiple Dimensions Using Poisson Optional Stopping Times

  • Rutger-Jan Lange, Daniel Ralph and Kristian Støre

Abstract

We provide a new framework for valuing multidimensional real options where opportunities to exercise the option are generated by an exogenous Poisson process, which can be viewed as a liquidity constraint on decision times. This approach, which we call the Poisson optional stopping times (POST) method, finds the value function as a monotone sequence of lower bounds. In a case study, we demonstrate that the frequently used quasi-analytic method yields a suboptimal policy and an inaccurate value function. The proposed method is demonstrably correct, straightforward to implement, reliable in computation, and broadly applicable in analyzing multidimensional option-valuation problems.

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Corresponding author

*Lange (corresponding author), lange@ese.eur.nl, Erasmus University Rotterdam Econometric Institute; Ralph, d.ralph@jbs.cam.ac.uk, University of Cambridge Judge Business School; and Støre, kristian.store@nord.no, Nord University Business School.

Footnotes

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1

We thank two anonymous referees, Jan van Casteren, Dick van Dijk, Øystein Gjerde, David Mauer (a referee), and Coen Teulings for helpful comments. We also thank Hendrik Bessembinder (the editor) for his helpful guidance.

Footnotes

References

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Abel, A. B., and Eberly, J. C.. “Optimal Investment with Costly Reversibility.” Review of Economic Studies, 63 (1996), 581593.
Abramovitz, M., and Stegun, I.. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Vol. 55, 10th ed. Washington, DC: National Bureau of Standards (1972).
Adkins, R., and Paxson, D.. “Reciprocal Energy-Switching Options.” Journal of Energy Markets, 4, 1 (2011a), 91120.
Adkins, R., and Paxson, D.. “Renewing Assets with Uncertain Revenues and Operating Costs.” Journal of Financial and Quantitative Analysis, 46 (2011b), 785813.
Adkins, R., and Paxson, D.. “Deterministic Models for Premature and Postponed Replacement.” Omega, 41 (2013a), 10081019.
Adkins, R., and Paxson, D.. “The Effect of Tax Depreciation on the Stochastic Replacement Policy.” European Journal of Operational Research, 229 (2013b), 155164.
Adkins, R., and Paxson, D.. “Stochastic Equipment Capital Budgeting with Technological Progress.” European Financial Management, 20 (2014), 10311049.
Adkins, R., and Paxson, D.. “The Effects of an Uncertain Abandonment Value on the Investment Decision.” European Journal of Finance, 23 (2017a), 10831106.
Adkins, R., and Paxson, D.. “Replacement Decisions with Multiple Stochastic Values and Depreciation.” European Journal of Operational Research, 257 (2017b), 174184.
Adkins, R., and Paxson, D.. “Sequential Investments with Stage-Specific Risks and Drifts.” European Journal of Finance, 23 (2017c), 11501175.
Ahlberg, J., and Nilson, E.. “Convergence Properties of the Spline Fit.” Journal of the Society for Industrial and Applied Mathematics, 11 (1963), 95104.
Andersen, L., and Broadie, M.. “Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options.” Management Science, 50 (2004), 12221234.
Armada, M. J. R.; Pereira, P. J.; and Rodrigues, A.. “Optimal Investment with Two-Factor Uncertainty.” Mathematics and Financial Economics, 7 (2013), 509530.
Bally, V., and Printems, J.. “A Quantization Tree Method for Pricing and Hedging Multidimensional American Options.” Mathematical Finance, 15 (2005), 119168.
Bensoussan, A., and Lions, J.-L.. Applications of Variational Inequalities in Stochastic Control, Vol. 12, Amsterdam, Netherlands: Elsevier (1982).
Biagini, F., and Björk, T.. “On the Timing Option in a Futures Contract.” Mathematical Finance, 17 (2007), 267283.
Birge, J. R., and Linetsky, V.. Handbooks in Operations Research and Management Science: Financial Engineering, Vol. 15, Amsterdam, Netherlands: Elsevier (2007).
Bjerksund, P., and Stensland, G.. “Closed Form Spread Option Valuation.” Quantitative Finance, 14 (2014), 17851794.
Brennan, M. J., and Schwartz, E. S.. “Evaluating Natural Resource Investments.” Journal of Business, 58 (1985), 135157.
Bunch, D. S., and Johnson, H.. “The American Put Option and Its Critical Stock Price.” Journal of Finance, 55 (2000), 23332356.
Carmona, R., and Dayanik, S.. “Optimal Multiple Stopping of Linear Diffusions.” Mathematics of Operations Research, 33 (2008), 446460.
Carmona, R., and Touzi, N.. “Optimal Multiple Stopping and Valuation of Swing Options.” Mathematical Finance, 18 (2008), 239268.
Carr, P.Randomization and the American Put.” The Review of Financial Studies, 11 (1998), 597626.
Compernolle, T.; Huisman, K.; Kort, P.; Lavrutich, M.; Nunes, C.; and Thijssen, J.. “Investment Decisions with Two-Factor Uncertainty.” CentER Discussion Paper Series No. 2018-003 (2018).
Cont, R., and Fournié, D.-A.. “Functional Itô Calculus and Stochastic Integral Representation of Martingales.” Annals of Probability, 41 (2013), 109133.
Cortazar, G.; Gravet, M.; and Urzua, J.. “The Valuation of Multidimensional American Real Options Using the LSM Simulation Method.” Computers & Operations Research, 35 (2008), 113129.
Cox, J. C.; Ross, S. A.; and Rubinstein, M.. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7 (1979), 229263.
d’Halluin, Y.; Forsyth, P. A.; and Labahn, G.. “A Penalty Method for American Options with Jump Diffusion Processes.” Numerische Mathematik, 97 (2004), 321352.
Dixit, A., and Pindyck, R.. Investment under Uncertainty. Princeton, NJ: Princeton University Press (1994).
Dockendorf, J., and Paxson, D.. “Continuous Rainbow Options on Commodity Outputs: What Is the Real Value of Switching Facilities?European Journal of Finance, 19 (2013), 645673.
Dupuis, P., and Wang, H.. “Optimal Stopping with Random Intervention Times.” Advances in Applied Probability, 34 (2002), 141157.
Ekern, S.An Option Pricing Approach to Evaluating Petroleum Projects.” Energy Economics, 10 (1988), 9199.
Farzan, F.; Mahani, K.; Gharieh, K.; and Jafari, M. A.. “Microgrid Investment under Uncertainty: A Real Option Approach Using Closed Form Contingent Analysis.” Annals of Operations Research, 235 (2015), 259276.
Feng, L., and Linetsky, V.. “Pricing Options in Jump-Diffusion Models: An Extrapolation Approach.” Operations Research, 56 (2008), 304325.
Forsyth, P. A., and Vetzal, K. R.. “Quadratic Convergence for Valuing American Options Using a Penalty Method.” SIAM Journal on Scientific Computing, 23 (2002), 20952122.
Glowinski, R. Lectures on Numerical Methods for Non-Linear Variational Problems. Berlin, Germany: Springer Science & Business Media (2008).
Heydari, S.; Ovenden, N.; and Siddiqui, A.. “Real Options Analysis of Investment in Carbon Capture and Sequestration Technology.” Computational Management Science, 9 (2012), 109138.
Lange, R.-J.; Ralph, D.; and van Casteren, J.. “A Vanilla Framework for Solving Multidimensional Options Using Poisson Optional Stopping Times.” Working Paper, Erasmus University Rotterdam (2018).
Lange, R.-J., and Teulings, C.. “The Option Value of Vacant Land and the Optimal Timing of City Extensions.” CEPR Discussion Paper No. DP12847 (2018).
Lempa, J.Optimal Stopping with Information Constraint.” Applied Mathematics & Optimization, 66 (2012), 147173.
Longstaff, F. A., and Schwartz, E. S.. “Valuing American Options by Simulation: A Simple Least-Squares Approach.” Review of Financial Studies, 14 (2001), 113147.
Majd, S., and Pindyck, R. S.. “Time to Build, Option Value, and Investment Decisions.” Journal of Financial Economics, 18 (1987), 727.
McDonald, R., and Siegel, D.. “The Value of Waiting to Invest.” Quarterly Journal of Economics, 101 (1986), 707727.
Peskir, G., and Shiryeav, A.. Optimal Stopping and Free-Boundary Problems. Basel, Switzerland: Birkhäuser (2006).
Pindyck, R. S.“Irreversibility, Uncertainty, and Investment.” Technical Report, National Bureau of Economic Research (1990).
Plemmons, R. J. M-Matrix Characterizations. I—Nonsingular M-Matrices.” Linear Algebra and Its Applications, 18 (1977), 175188.
Revuz, D., and Yor, M.. Continuous Martingales and Brownian Motion, Vol. 293, Berlin, Germany: Springer (1999).
Rogers, L. C.Monte Carlo Valuation of American Options.” Mathematical Finance, 12 (2002), 271286.
Rogers, L. C., and Williams, D.. Diffusions, Markov Processes, and Martingales, Vol. 1, Cambridge, UK: Cambridge University Press (2000).
Rogers, L. C., and Zane, O.. “A Simple Model of Liquidity Effects.” In Advances in Finance and Stochastics, Sandmann, K. and Schönbucher, P. J., eds. Berlin, Germany: Springer (2002), 161176.
Rohlfs, W., and Madlener, R.. “Valuation of CCS-Ready Coal-Fired Power Plants: A Multi-Dimensional Real Options Approach.” Energy Systems, 2 (2011), 243261.
Schwartz, E. S.The Valuation of Warrants: Implementing a New Approach.” Journal of Financial Economics, 4 (1977), 7993.
Stentoft, L.Convergence of the Least Squares Monte Carlo Approach to American Option Valuation.” Management Science, 50 (2004), 11931203.
Støre, K.; Fleten, S.-E.; Hagspiel, V.; and Nunes, C.. “Switching from Oil to Gas Production in a Depleting Field.” European Journal of Operational Research, 271 (2018), 710719.
Stroock, D. W., and Varadhan, S. R. S.. Multidimensional Diffusion Processes. Berlin, Germany: Springer (2007).
Strulovici, B., and Szydlowski, M.. “On the Smoothness of Value Functions and the Existence of Optimal Strategies in Diffusion Models.” Journal of Economic Theory, 159 (2015), 10161055.
Stutzman, S.; Weiland, B.; Preckel, P.; and Wetzstein, M.. “Optimal Replacement Policies for an Uncertain Rejuvenated Asset.” International Journal of Production Economics, 185 (2017), 2133.
Svenstrup, M.On the Suboptimality of Single-Factor Exercise Strategies for Bermudan Swaptions.” Journal of Financial Economics, 78 (2005), 651684.
Trigeorgis, L.A Log-Transformed Binomial Numerical Analysis Method for Valuing Complex Multi-Option Investments.” Journal of Financial and Quantitative Analysis, 26 (1991), 309326.
Trigeorgis, L.The Nature of Option Interactions and the Valuation of Investments with Multiple Real Options.” Journal of Financial and Quantitative Analysis, 28 (1993), 120.
Wang, S.; Yang, X.; and Teo, K.. “Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation.” Journal of Optimization Theory and Applications, 129 (2006), 227254.
Zhang, K.; Wang, S.; Yang, X.; and Teo, K. L.. “A Power Penalty Approach to Numerical Solutions of Two-Asset American Options.” Numerical Mathematics: Theory, Methods and Applications, 2 (2009), 202223.
Zhang, K.; Yang, X.; and Teo, K. L.. “Convergence Analysis of a Monotonic Penalty Method for American Option Pricing.” Journal of Mathematical Analysis and Applications, 348 (2008), 915926.
Zvan, R.; Forsyth, P. A.; and Vetzal, K. R.. “Penalty Methods for American Options with Stochastic Volatility.” Journal of Computational and Applied Mathematics, 91 (1998), 199218.

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Real-Option Valuation in Multiple Dimensions Using Poisson Optional Stopping Times

  • Rutger-Jan Lange, Daniel Ralph and Kristian Støre

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