Advertisement

Finance and Stochastics

, Volume 22, Issue 4, pp 1007–1036 | Cite as

Weak time-derivatives and no-arbitrage pricing

  • Massimo Marinacci
  • Federico SeverinoEmail author
Article
  • 171 Downloads

Abstract

We prove a risk-neutral pricing formula for a large class of semimartingale processes through a novel notion of weak time-differentiability that permits to differentiate adapted processes. In particular, the weak time-derivative isolates drifts of semimartingales and is null for martingales. Weak time-differentiability enables us to characterize no-arbitrage prices as solutions of differential equations, where interest rates play a key role. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (Econometrica 77:177–234, 2009) by employing weak time-derivatives.

Keywords

No-arbitrage pricing Weak time-derivative Martingale component Special semimartingales Stochastic interest rates 

Mathematics Subject Classification (2010)

60G07 91G80 49J40 

JEL Classification

C02 

Notes

Acknowledgements

We thank Anna Battauz, Francesco Caravenna, Andrea Carioli, Simone Cerreia-Vioglio, Lars Peter Hansen, Ioannis Karatzas, Luigi Montrucchio, Fulvio Ortu, Emanuela Rosazza-Gianin, Martin Schweizer and two anonymous referees for useful comments. We also thank seminar participants at XL AMASES Annual Meeting in Catania (2016) and at XVIII Quantitative Finance Workshop at Università Bicocca, Milan (2017).

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, San Diego (2003) zbMATHGoogle Scholar
  2. 2.
    Aliprantis, C., Border, K.: Infinite Dimensional Analysis. Springer, Berlin (2006) zbMATHGoogle Scholar
  3. 3.
    Alvarez, F., Jermann, U.J.: Using asset prices to measure the persistence of the marginal utility of wealth. Econometrica 73, 1977–2016 (2005) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ansel, J., Stricker, C.: Lois de martingale, densités et décomposition de Föllmer Schweizer. Ann. Inst. Henri Poincaré Probab. Stat. 28, 375–392 (1992) zbMATHGoogle Scholar
  5. 5.
    Bachelier, L.: Théorie de la Spéculation. Gauthier-Villars, Paris (1900) zbMATHGoogle Scholar
  6. 6.
    Björk, T.: Arbitrage Theory in Continuous Time, 3rd edn. Oxford University Press, London (2009) zbMATHGoogle Scholar
  7. 7.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010) CrossRefGoogle Scholar
  9. 9.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002) CrossRefGoogle Scholar
  10. 10.
    Cox, J.C., Ross, S.A.: The valuation of options for alternative stochastic processes. J. Financ. Econ. 3, 145–166 (1976) CrossRefGoogle Scholar
  11. 11.
    Craven, B.D.: Two properties of Bochner integrals. Bull. Aust. Math. Soc. 3, 363–368 (1970) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Diestel, J., Uhl, J.J.: Vector Measures. Am. Math. Soc., Providence (1977) CrossRefGoogle Scholar
  15. 15.
    Duffie, D.: Dynamic Asset Pricing Theory. Princeton University Press, Princeton (2010) zbMATHGoogle Scholar
  16. 16.
    Émery, M.: Compensation de processus VF non localement intégrables. In: Azema, J., Yor, M. (eds.) Séminaire de Probabilités XIV. Lecture Notes in Mathematics, vol. 784, pp. 152–160. Springer, Berlin (1980) Google Scholar
  17. 17.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter, Berlin (2004) CrossRefGoogle Scholar
  18. 18.
    Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis, Stochastics Monographs, vol. 5, pp. 389–414. Gordon and Breach, London/New York (1991) Google Scholar
  19. 19.
    Geman, H., El Karoui, N., Rochet, J.C.: Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab. 32, 443–458 (1995) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hansen, L.P., Heaton, J.C., Li, N.: Consumption strikes back? Measuring long-run risk. J. Polit. Econ. 116, 260–302 (2008) CrossRefGoogle Scholar
  21. 21.
    Hansen, L.P., Richard, S.F.: The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models. Econometrica 55, 587–613 (1987) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hansen, L.P., Scheinkman, J.A.: Long-term risk: an operator approach. Econometrica 77, 177–234 (2009) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991) zbMATHGoogle Scholar
  24. 24.
    Kou, S.G.: A jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101 (2002) CrossRefGoogle Scholar
  25. 25.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971) CrossRefGoogle Scholar
  26. 26.
    Margrabe, W.: The value of an option to exchange one asset for another. J. Finance 33, 177–186 (1978) CrossRefGoogle Scholar
  27. 27.
    Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1976) CrossRefGoogle Scholar
  28. 28.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000) CrossRefGoogle Scholar
  29. 29.
    Modigliani, F., Miller, M.H.: The cost of capital, corporation finance and the theory of investment. Am. Econ. Rev. 48, 261–297 (1958) Google Scholar
  30. 30.
    Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin (2010) CrossRefGoogle Scholar
  31. 31.
    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  32. 32.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) CrossRefGoogle Scholar
  33. 33.
    Samuelson, P.A.: Proof that properly anticipated prices fluctuate randomly. Ind. Manage. Rev. 6(2), 41–49 (1965) Google Scholar
  34. 34.
    Severino, F.: Long-term risk with stochastic interest rates. Working paper (2018). Available online at https://ssrn.com/abstract=3113718
  35. 35.
    Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Decision Sciences & IGIERUniversità BocconiMilanItaly
  2. 2.Department of EconomicsUniversità della Svizzera Italiana (USI)LuganoSwitzerland
  3. 3.Department of FinanceUniversità BocconiMilanItaly

Personalised recommendations