Top

Finance and Stochastics

Published in:

03-11-2016

Model uncertainty and the pricing of American options

Authors: David Hobson, Anthony Neuberger

Published in: Finance and Stochastics | Issue 1/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when we do not impose a model, but rather consider the class of all models which are consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error.

previous article Market completion with derivative securities

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

Available only for authorised users

A rare exception is the paper of Cox and Hoeggerl [11]. In that paper, the aim is to find consistency conditions on the possible shapes (as a function of strike) of the family of prices of American put options with fixed maturity, given the values of co-maturing European puts.

In this context, a consistent pricing measure is any measure under which the stock price is a martingale and model-based prices of call options, i.e., their expected values, agree with the quoted prices.

Note that the definition of a consistent model includes the fact that the expected value of the payoff of a vanilla calls equals the quoted price, so that European calls are priced by expectation under the model.

Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26, 233–251 (2016) MathSciNetCrossRefMATH

Andersen, L., Broadie, M.: Primal–dual simulation algorithm for pricing multi-dimensional American options. Manag. Sci. 50, 1222–1234 (2004) CrossRef

Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices—a mass-transport approach. Finance Stoch. 17, 477–501 (2013) MathSciNetCrossRefMATH

Bouchard, P., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823–859 (2015) MathSciNetCrossRefMATH

Breeden, D., Litzenberger, R.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978) CrossRef

Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Finance 11, 285–314 (2001) MathSciNetCrossRefMATH

Carr, P., Lee, R.: Hedging variance options on continuous semimartingales. Finance Stoch. 14, 179–207 (2010) MathSciNetCrossRefMATH

Carr, P., Madan, D.: A note on sufficient conditions for no arbitrage. Finance Res. Lett. 2, 125–130 (2005) CrossRef

Carr, P., Nadtochiy, S.: Static hedging under time-homogeneous diffusions. SIAM J. Financ. Math. 2, 794–838 (2011) MathSciNetCrossRefMATH

10.

Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J. Finance 53, 1165–1190 (1998) CrossRef

11.

Cox, A.M.G., Hoeggerl, C.: Model-independent no-arbitrage conditions on American put options. Math. Finance 26, 431–458 (2016) MathSciNetCrossRefMATH

12.

Cox, A.M.G., Obłoj, J.: Robust pricing and hedging of double no-touch options. Finance Stoch. 15, 573–605 (2011) MathSciNetCrossRefMATH

13.

Cox, A.M.G., Hobson, D., Obłoj, J.: Time-homogeneous diffusions with a given marginal at a random time. ESAIM, Probab. Stat. 15, S11–S24 (2011) MathSciNetCrossRef

14.

Davis, M.H.A., Hobson, D.: The range of traded option prices. Math. Finance 17, 1–14 (2007) MathSciNetCrossRefMATH

15.

Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields 160, 391–427 (2014) MathSciNetCrossRefMATH

16.

Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multi-period securities markets. J. Econ. Theory 20, 381–408 (1979) MathSciNetCrossRefMATH

17.

Haugh, M., Kogan, L.: Pricing American options: a duality approach. Oper. Res. 52, 258–270 (2004) MathSciNetCrossRefMATH

18.

Hobson, D.: Robust hedging of the lookback option. Finance Stoch. 2, 329–347 (1998) CrossRefMATH

19.

Hobson, D.: The Skorokhod embedding problem and model-independent bounds for options prices. In: Carmona, R.A., et al. (eds.) Paris–Princeton Lectures on Mathematical Finance 2010, LNM 2003, pp. 267–318. Springer, Berlin (2011) CrossRef

20.

Hobson, D., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611–649 (2012) MathSciNetCrossRefMATH

21.

Hobson, D., Neuberger, A.: Robust bounds for the forward-start option. Math. Finance 22, 33–56 (2012) MathSciNetCrossRefMATH

22.

Hobson, D., Laurence, P., Wang, T-H.: Static-arbitrage upper bounds for the prices of basket options. Quant. Finance 5, 329–342 (2005) MathSciNetCrossRefMATH

23.

Kahalé, N.: Super-replication of financial derivatives via convex programming. Working paper (2016), available online at http://ssrn.com/abstract=2172315

24.

Neuberger, A.: Bounds on the American option. Working paper (2007), available online at http://ssrn.com/abstract=966333

25.

Rogers, L.C.G.: Monte Carlo valuation of American options. Math. Finance 12, 271–286 (2002) MathSciNetCrossRefMATH

26.

Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965) MathSciNetCrossRefMATH

27.

Vanderbei, R.J.: Linear Programming: Foundations and Extensions, 3rd edn. International Series in Operations Research & Management Science, vol. 114. Springer, Berlin (2008) CrossRefMATH

Title: Model uncertainty and the pricing of American options
Authors: David Hobson
Anthony Neuberger
Publication date: 03-11-2016
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 1/2017
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-016-0314-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 1/2017

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

Continuous-time perpetuities and time reversal of diffusions

Optimal consumption and investment with Epstein–Zin recursive utility

Market completion with derivative securities

Hedging with small uncertainty aversion

Watermark options