Skip to main content

2017 | OriginalPaper | Buchkapitel

6. Pricing Multiple Exercise American Options by Linear Programming

verfasst von : Monia Giandomenico, Mustafa Ç. Pınar

Erschienen in: Optimal Financial Decision Making under Uncertainty

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We consider the problem of computing the lower hedging price of American options of the call and put type written on a non-dividend paying stock in a non-recombinant tree model with multiple exercise rights. We prove using a simple argument that an optimal exercise policy for an option with h exercise rights is to delay exercise until the last h periods. The result implies that the mixed-integer programming model for computing the lower hedging price and the optimal exercise and hedging policy has a linear programming relaxation that is exact, i.e., the relaxation admits an optimal solution where all variables required to be integral have integer values.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
1
An earlier version of the paper had quite a long proof for the case h = 2 and restricted to binomial and trinomial trees. It was based on an elaborate primal-dual construction. The present proof was offered by an anonymous reviewer of the earlier version, to whom we are thankful.
 
Literatur
1.
Zurück zum Zitat O. Bardou, S. Bouthemy, G. Pagès, Optimal quantization for the pricing of swing options. Appl. Math. Financ. 16 (2), 187–217 (2009)CrossRef O. Bardou, S. Bouthemy, G. Pagès, Optimal quantization for the pricing of swing options. Appl. Math. Financ. 16 (2), 187–217 (2009)CrossRef
2.
Zurück zum Zitat O. Bardou, S. Bouthemy, G. Pagès, When are swing options bang-bang and how to use it? Int. J. Theor. Appl. Finance 13 (6), 867–899 (2010)CrossRef O. Bardou, S. Bouthemy, G. Pagès, When are swing options bang-bang and how to use it? Int. J. Theor. Appl. Finance 13 (6), 867–899 (2010)CrossRef
3.
Zurück zum Zitat C. Barrera-Esteve, F. Bergeret, C. Dossal, E. Gobet, A. Meziou, R. Munos, D. Reboul-Salze, Numerical methods for the pricing of swing options: a stochastic control approach. Methodol. Comput. Appl. Probab. 8 (4), 517–540 (2006)CrossRef C. Barrera-Esteve, F. Bergeret, C. Dossal, E. Gobet, A. Meziou, R. Munos, D. Reboul-Salze, Numerical methods for the pricing of swing options: a stochastic control approach. Methodol. Comput. Appl. Probab. 8 (4), 517–540 (2006)CrossRef
4.
Zurück zum Zitat C. Bender, Dual pricing of multi-exercise options under volume constraints. Finance Stochast. 15 (1), 1–26 (2010)CrossRef C. Bender, Dual pricing of multi-exercise options under volume constraints. Finance Stochast. 15 (1), 1–26 (2010)CrossRef
5.
Zurück zum Zitat C. Bender, Primal and dual pricing of multiple exercise options in continuous time. SIAM J. Finan. Math. 2, 562–586 (2011)CrossRef C. Bender, Primal and dual pricing of multiple exercise options in continuous time. SIAM J. Finan. Math. 2, 562–586 (2011)CrossRef
6.
Zurück zum Zitat A. Bensoussan, On the theory of option pricing. Acta Appl. Math. 2, 139–158 (1984) A. Bensoussan, On the theory of option pricing. Acta Appl. Math. 2, 139–158 (1984)
7.
Zurück zum Zitat F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)CrossRef F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)CrossRef
8.
Zurück zum Zitat B. Bouchard, E. Temam, On the hedging of American options in discrete time markets with proportional transaction costs. Electron. J. Probab. 10 (22), 746–760 (2005)CrossRef B. Bouchard, E. Temam, On the hedging of American options in discrete time markets with proportional transaction costs. Electron. J. Probab. 10 (22), 746–760 (2005)CrossRef
9.
Zurück zum Zitat R. Buckdahn, Y. Hu, Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23, 177–203 (1998)CrossRef R. Buckdahn, Y. Hu, Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23, 177–203 (1998)CrossRef
10.
Zurück zum Zitat A. Camcı, M.Ç. Pınar, Pricing American contingent claims by stochastic linear programming. Optimization 58 (6), 627–640 (2009)CrossRef A. Camcı, M.Ç. Pınar, Pricing American contingent claims by stochastic linear programming. Optimization 58 (6), 627–640 (2009)CrossRef
11.
Zurück zum Zitat R. Carmona, S. Dayanık, Optimal multiple stopping of linear diffusions. Math. Oper. Res. 33 (2), 446–460 (2008)CrossRef R. Carmona, S. Dayanık, Optimal multiple stopping of linear diffusions. Math. Oper. Res. 33 (2), 446–460 (2008)CrossRef
12.
Zurück zum Zitat R. Carmona, N. Touzi, Optimal multiple stopping and valuation of swing options. Math. Financ. 18 (2), 239–268 (2008)CrossRef R. Carmona, N. Touzi, Optimal multiple stopping and valuation of swing options. Math. Financ. 18 (2), 239–268 (2008)CrossRef
13.
Zurück zum Zitat P. Chalasani, S. Jha, Randomized stopping times and American option pricing with transaction costs. Math. Financ. 11, 33–77 (2001)CrossRef P. Chalasani, S. Jha, Randomized stopping times and American option pricing with transaction costs. Math. Financ. 11, 33–77 (2001)CrossRef
14.
Zurück zum Zitat J.C. Cox, M. Rubinstein, Options Markets (Prentice-Hall, Englewood Cliffs, NJ, 1985) J.C. Cox, M. Rubinstein, Options Markets (Prentice-Hall, Englewood Cliffs, NJ, 1985)
15.
Zurück zum Zitat M. Davis, T. Zariphopoulou, American options and transaction fees, in The IMA Volumes in Mathematics and Its Applications, vol. 65, ed. by M. Davis (Springer, Berlin, 1995), pp. 47–62 M. Davis, T. Zariphopoulou, American options and transaction fees, in The IMA Volumes in Mathematics and Its Applications, vol. 65, ed. by M. Davis (Springer, Berlin, 1995), pp. 47–62
16.
Zurück zum Zitat J. Detemple, American Style Derivatives: Valuation and Computation. Financial Mathematics Series (Chapman & Hall/CRC, Boca Raton, 2005)CrossRef J. Detemple, American Style Derivatives: Valuation and Computation. Financial Mathematics Series (Chapman & Hall/CRC, Boca Raton, 2005)CrossRef
17.
Zurück zum Zitat C. Edirisinghe, V. Naik, R. Uppal, Optimal replication of options with transaction costs and trading restrictions. J. Financ. Quant. Anal. 28, 117–138 (1993)CrossRef C. Edirisinghe, V. Naik, R. Uppal, Optimal replication of options with transaction costs and trading restrictions. J. Financ. Quant. Anal. 28, 117–138 (1993)CrossRef
18.
Zurück zum Zitat M. Figueroa, Pricing Multiple Interruptible-Swing Contracts. Technical report, Birkbeck, School of Economics, Mathematics & Statistics, Birkbeck Working Papers in Economics and Finance: 0606 (2006) M. Figueroa, Pricing Multiple Interruptible-Swing Contracts. Technical report, Birkbeck, School of Economics, Mathematics & Statistics, Birkbeck Working Papers in Economics and Finance: 0606 (2006)
19.
Zurück zum Zitat S.D. Flåm, Option pricing by mathematical programming. Optimization 57 (1), 165–182 (2008)CrossRef S.D. Flåm, Option pricing by mathematical programming. Optimization 57 (1), 165–182 (2008)CrossRef
20.
Zurück zum Zitat H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time. De Gruyter Studies in Mathematics, vol. 27, 2nd edn. (Walter de Gruyter, Berlin, 2004) H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time. De Gruyter Studies in Mathematics, vol. 27, 2nd edn. (Walter de Gruyter, Berlin, 2004)
21.
Zurück zum Zitat G. Haarbrücker, D. Kuhn, Valuation of electricity swing options by multistage stochastic programming. Automatica 45 (4), 889–899 (2009)CrossRef G. Haarbrücker, D. Kuhn, Valuation of electricity swing options by multistage stochastic programming. Automatica 45 (4), 889–899 (2009)CrossRef
22.
Zurück zum Zitat J.M. Harrison, D.M. Kreps, Martingales and arbitrage in multiperiod securities markets. J. Econ. Theor. 20, 381–408 J.M. Harrison, D.M. Kreps, Martingales and arbitrage in multiperiod securities markets. J. Econ. Theor. 20, 381–408
23.
Zurück zum Zitat A. Ibáñez, Valuation by simulation of contingent claims with multiple exercise opportunities. Math. Financ. 14 (2), 223–248 (2004)CrossRef A. Ibáñez, Valuation by simulation of contingent claims with multiple exercise opportunities. Math. Financ. 14 (2), 223–248 (2004)CrossRef
24.
Zurück zum Zitat P. Jaillet, E.I. Ronn, S. Tompaidis, Valuation of commodity-based swing options. Manage. Sci. 50 (7), 909–921 (2004)CrossRef P. Jaillet, E.I. Ronn, S. Tompaidis, Valuation of commodity-based swing options. Manage. Sci. 50 (7), 909–921 (2004)CrossRef
25.
Zurück zum Zitat I. Karatzas, On the pricing of American options. Appl. Math. Optim. 17, 37–60 (1988)CrossRef I. Karatzas, On the pricing of American options. Appl. Math. Optim. 17, 37–60 (1988)CrossRef
26.
Zurück zum Zitat I. Karatzas, S.G. Kou, Hedging American contingent claims with constrained portfolios. Finance Stochast. 2, 215–258 (1998)CrossRef I. Karatzas, S.G. Kou, Hedging American contingent claims with constrained portfolios. Finance Stochast. 2, 215–258 (1998)CrossRef
27.
Zurück zum Zitat J. Keppo, Pricing of electricity swing options. J. Derivatives 11, 26–43 (2004)CrossRef J. Keppo, Pricing of electricity swing options. J. Derivatives 11, 26–43 (2004)CrossRef
28.
Zurück zum Zitat A.J. King, Duality and martingales: a stochastic programming perspective on contingent claims. Math. Programm. Ser. B 91, 543–562 (2002)CrossRef A.J. King, Duality and martingales: a stochastic programming perspective on contingent claims. Math. Programm. Ser. B 91, 543–562 (2002)CrossRef
29.
Zurück zum Zitat F. Longstaff, E. Schwartz, Valuing American options by simulation: a simple least squares approach. Rev. Financ. Stud. 14 (1), 113–147 (2001)CrossRef F. Longstaff, E. Schwartz, Valuing American options by simulation: a simple least squares approach. Rev. Financ. Stud. 14 (1), 113–147 (2001)CrossRef
30.
Zurück zum Zitat D.G. Luenberger, Investment Science (Oxford University Press, Oxford, 1998) D.G. Luenberger, Investment Science (Oxford University Press, Oxford, 1998)
31.
Zurück zum Zitat N. Meinshausen, B.M. Hambly, Monte Carlo methods for the valuation of multiple-exercise options. Math. Financ. 14 (4), 557–583 (2004)CrossRef N. Meinshausen, B.M. Hambly, Monte Carlo methods for the valuation of multiple-exercise options. Math. Financ. 14 (4), 557–583 (2004)CrossRef
32.
Zurück zum Zitat R. Myeni, The pricing of the American option. Ann. Appl. Probab. 2, 1–23 (1992)CrossRef R. Myeni, The pricing of the American option. Ann. Appl. Probab. 2, 1–23 (1992)CrossRef
33.
Zurück zum Zitat T. Pennanen, A. King, Arbitrage Pricing of American Contingent Claims in Incomplete Markets - A Convex Optimization Approach, Working paper, June (2006) T. Pennanen, A. King, Arbitrage Pricing of American Contingent Claims in Incomplete Markets - A Convex Optimization Approach, Working paper, June (2006)
34.
Zurück zum Zitat M.Ç. Pınar, A. Camcı, An integer programming model for pricing American contingent claims under transaction costs. Comput. Econ. 39, 1–12 (2012)CrossRef M.Ç. Pınar, A. Camcı, An integer programming model for pricing American contingent claims under transaction costs. Comput. Econ. 39, 1–12 (2012)CrossRef
35.
Zurück zum Zitat T.R. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970)CrossRef T.R. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970)CrossRef
36.
Zurück zum Zitat J. Schoenmakers, A pure martingale dual for multiple stopping. Finance Stochast. 16, 319–334 (2012)CrossRef J. Schoenmakers, A pure martingale dual for multiple stopping. Finance Stochast. 16, 319–334 (2012)CrossRef
37.
Zurück zum Zitat A. Thompson, Valuation of path-dependent contingent claims with multiple exercise decisions over time: the case of take-or-pay. J. Financ. Quant. Anal. 30 (2), 271–293 (1995)CrossRef A. Thompson, Valuation of path-dependent contingent claims with multiple exercise decisions over time: the case of take-or-pay. J. Financ. Quant. Anal. 30 (2), 271–293 (1995)CrossRef
38.
Zurück zum Zitat K. Tokarz, T. Zastawniak, American contingent claims under small proportional transaction costs. J. Math. Econ. 43, 65–85 (2006)CrossRef K. Tokarz, T. Zastawniak, American contingent claims under small proportional transaction costs. J. Math. Econ. 43, 65–85 (2006)CrossRef
39.
Zurück zum Zitat P. Vayanos, W. Wiesemann, D. Kuhn, Hedging electricity swing options in incomplete markets, in Proceedings of the 18th IFAC World Congress (2011), pp. 846–853 P. Vayanos, W. Wiesemann, D. Kuhn, Hedging electricity swing options in incomplete markets, in Proceedings of the 18th IFAC World Congress (2011), pp. 846–853
40.
Zurück zum Zitat C. Winter, M. Wilhelm, Finite element valuation of swing options. J. Comput. Finance 11 (3), 107–132 (2008)CrossRef C. Winter, M. Wilhelm, Finite element valuation of swing options. J. Comput. Finance 11 (3), 107–132 (2008)CrossRef
Metadaten
Titel
Pricing Multiple Exercise American Options by Linear Programming
verfasst von
Monia Giandomenico
Mustafa Ç. Pınar
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-41613-7_6