Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Finance and Stochastics
Published in: Finance and Stochastics 3/2020

18-05-2020

Conditional Davis pricing

Authors: Kasper Larsen, Halil Mete Soner, Gordan Žitković

Published in: Finance and Stochastics | Issue 3/2020

Log in

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

search-config
loading …

Abstract

We study the set of Davis (marginal utility-based) prices of a financial derivative in the case where the investor has a non-replicable random endowment. We give a new characterisation of the set of all such prices, and provide an example showing that even in the simplest of settings – such as Samuelson’s geometric Brownian motion model –, the interval of Davis prices is often a non-degenerate subinterval of the set of all no-arbitrage prices. This is in stark contrast to the case with a constant or replicable endowment where non-uniqueness of Davis prices is exceptional. We provide formulas for the endpoints of these intervals and illustrate the theory with several examples.
next article Adapted Wasserstein distances and stability in mathematical finance

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
Appendix
Available only for authorised users
Footnotes
1
The theory of Davis pricing is simpler for utility functions, such as the exponential utility, defined on ℝ. For example, in any such model, Davis prices are unique because Bellini and Frittelli [2] have shown that the dual utility minimiser is a countably additive probability measure.
 
2
It has been shown in Larsen and Žitković [29, Lemma 3.12] that under the reasonable elasticity condition, infimisation over the set of countably additive martingale measures – as opposed to its finitely additive enlargement as in Cvitanić et al. [7] – leads to the same value function.
 
3
We thank our AE and a referee for this suggestion.
 
Literature
1.
Becherer, D.: Utility indifference valuation. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1854–1860. Wiley, New York (2010)
2.
3.
Björk, T., Slinko, I.: Towards a general theory of good-deal bounds. Rev. Finance 10, 221–260 (2006) CrossRef
4.
Carmona, R.: Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2009) CrossRef
5.
Shiryaev, A., Cherny, A.: Vector stochastic integrals and the fundamental theorems of asset pricing. Proc. Steklov Math. Inst. 237, 12–56 (2002) MathSciNet
6.
Cochrane, J., Saá-Requejo, J.: Beyond arbitrage: good-deal asset price bounds in incomplete markets. J. Polit. Econ. 108, 79–119 (2000) CrossRef
7.
Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 237–259 (2001) MathSciNetCrossRef
8.
Cvitanić, J., Schachermayer, W., Wang, H.: Erratum to: Utility maximization in incomplete markets with random endowment. Finance Stoch. 21, 867–872 (2017) MathSciNetCrossRef
9.
Davis, M.: Option pricing in incomplete markets. In: Dempster, M.A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities, pp. 216–226. Cambridge University Press, New York (1997)
10.
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetCrossRef
11.
Delbaen, F., Schachermayer, W.: A simple counterexample to several problems in the theory of asset pricing. Math. Finance 8, 1–11 (1998) MathSciNetCrossRef
12.
Duffie, D.: Dynamic Asset Pricing Theory, 3rd edn. Princeton Press (2001) MATH
13.
Dunford, N., Schwartz, J.T.: Linear Operators. Part I. Wiley, New York (1988) MATH
14.
Gu, L., Lin, Y., Yang, J.: On the dual problem of utility maximization in incomplete markets. Stoch. Process. Appl. 126, 1019–1035 (2016) MathSciNetCrossRef
15.
Föllmer, H., Schied, A.: Stochastic Finance, 4th edn. De Gruyter Studies in Mathematics. De Gruyter, Berlin (2016) CrossRef
16.
Föllmer, H., Schweizer, M.: The minimal martingale measure. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1200–1204. Wiley, New York (2010)
17.
Hicks, J.R.: A Revision of Demand Theory. Oxford University Press, London (1956)
18.
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001) CrossRef
19.
Hugonnier, J., Kramkov, D.: Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14, 845–864 (2004) MathSciNetCrossRef
20.
Hugonnier, J., Kramkov, D., Schachermayer, W.: On utility based pricing of contingent claims in incomplete markets. Math. Finance 15, 203–212 (2005) MathSciNetCrossRef
21.
Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003) MathSciNetCrossRef
22.
Kraft, H.: Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant. Finance 5, 303–313 (2005) MathSciNetCrossRef
23.
Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999) MathSciNetCrossRef
24.
Kramkov, D., Sîrbu, M.: Sensitivity analysis of utility based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16, 2140–2194 (2006) MathSciNetCrossRef
25.
Kramkov, D., Weston, K.: Muckenhoupt’s (\(A_{p}\)) condition and the existence of the optimal martingale measure. Stoch. Process. Appl. 126, 2615–2633 (2016) CrossRef
26.
Kim, T.S., Omberg, E.: Dynamic nonmyopic portfolio behavior. Rev. Financ. Stud. 9, 141–161 (1996) CrossRef
27.
28.
29.
Larsen, K., Žitković, G.: Utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23, 665–692 (2013) MathSciNetCrossRef
30.
Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117, 1642–1662 (2007) MathSciNetCrossRef
31.
Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, London (1995) MATH
32.
Munk, C.: The valuation of contingent claims under portfolio constraints: reservation buying and selling prices. Eur. Finance Rev. 3, 347–388 (1999) CrossRef
33.
Owen, M., Žitković, G.: Optimal investment with an unbounded random endowment and utility based pricing. Math. Finance 19, 129–159 (2009) MathSciNetCrossRef
34.
35.
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002) CrossRef
36.
Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1996) MATH
Metadata
Title
Conditional Davis pricing
Authors
Kasper Larsen
Halil Mete Soner
Gordan Žitković
Publication date
18-05-2020
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 3/2020
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-020-00424-5

Other articles of this Issue 3/2020

Finance and Stochastics 3/2020 Go to the issue

OriginalPaper

Realised volatility and parametric estimation of Heston SDEs

OriginalPaper

Time reversal and last passage time of diffusions with applications to credit risk management

OriginalPaper

A splitting strategy for the calibration of jump-diffusion models

OriginalPaper

Fast mean-reversion asymptotics for large portfolios of stochastic volatility models

OriginalPaper

Option valuation and hedging using an asymmetric risk function: asymptotic optimality through fully nonlinear partial differential equations

OriginalPaper

Adapted Wasserstein distances and stability in mathematical finance