Finance and Stochastics

, Volume 20, Issue 3, pp 589–634

Additive subordination and its applications in finance

Article

Abstract

This paper studies additive subordination, which we show is a useful technique for constructing time-inhomogeneous Markov processes with analytical tractability. This technique is a natural generalization of Bochner’s subordination that has proved to be extremely useful in financial modeling. Probabilistically, Bochner’s subordination corresponds to a stochastic time change with respect to an independent Lévy subordinator, while in additive subordination, the Lévy subordinator is replaced by an additive one. We generalize the classical Phillips theorem for Bochner’s subordination to the additive subordination case, based on which we provide Markov and semimartingale characterizations for a rich class of jump-diffusions and pure jump processes obtained from diffusions through additive subordination, and obtain spectral decomposition for them. To illustrate the usefulness of additive subordination, we develop an analytically tractable cross-commodity model for spread option valuation that is able to calibrate the implied volatility surface of each commodity. Moreover, our model can generate implied correlation patterns that are consistent with market observations and economic intuitions.

Keywords

Bochner’s subordination Additive subordination Time-inhomogeneous Markov processes Spread options 

Mathematics Subject Classification

47D06 47D07 60J35 60J60 60J75 91G20 

JEL Classification

G13 

References

  1. 1.
    Alexander, C., Scourse, A.: Bivariate normal mixture spread option valuation. Quant. Finance 4, 1–12 (2004) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009) CrossRefMATHGoogle Scholar
  3. 3.
    Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68 (1998) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barndorff-Nielsen, O.E., Levendorskiĭ, S.: Feller processes of normal inverse Gaussian type. Quant. Finance 1, 318–331 (2001) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beals, R., Wong, R.: Special Functions: A Graduate Text. Cambridge University Press, Cambridge (2010) CrossRefMATHGoogle Scholar
  6. 6.
    Bjerksund, P., Stensland, G.: Closed form spread option valuation. Quant. Finance 14, 1785–1794 (2011) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bochner, S.: Diffusion equations and stochastic processes. Proc. Natl. Acad. Sci. USA 35, 368–370 (1949) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bochner, S.: Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley (1955) MATHGoogle Scholar
  9. 9.
    Boyarchenko, M., Levendorskiĭ, S.: Valuation of continuously monitored double barrier options and related securities. Math. Finance 22, 419–444 (2012) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boyarchenko, N., Levendorskiĭ, S.: The eigenfunction expansion method in multifactor quadratic term structure models. Math. Finance 17, 503–539 (2007) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Breiman, L.: Probability. SIAM, Philadelphia (1992) CrossRefMATHGoogle Scholar
  12. 12.
    Caldana, R., Fusai, G.: A general closed-form spread option pricing formula. J. Bank. Finance 37, 4893–4906 (2013) CrossRefGoogle Scholar
  13. 13.
    Carmona, R., Durrleman, V.: Pricing and hedging spread options. SIAM Rev. 45, 627–685 (2003) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Carmona, R., Sun, Y.: Implied and local correlations from spread options. Technical report, Princeton University (2012). Available online at https://www.princeton.edu/~rcarmona/download/fe/CS.pdf
  15. 15.
    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
  16. 16.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Finance 13, 345–382 (2003) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: Self-decomposability and option pricing. Math. Finance 17, 31–57 (2007) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Carr, P., Linetsky, V.: A jump to default extended CEV model: an application of Bessel processes. Finance Stoch. 10, 303–330 (2006) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Carr, P., Madan, D.B.: Option pricing and the fast Fourier transform. J. Comput. Finance 2, 61–73 (1999) CrossRefGoogle Scholar
  20. 20.
    Chen, R., Scott, L.: Pricing interest rate options in a two-factor Cox–Ingersoll–Ross model of the term structure. Rev. Financ. Stud. 5, 613–636 (1992) CrossRefGoogle Scholar
  21. 21.
    Cheridito, P., Filipović, D., Yor, M.: Equivalent and absolutely continuous measure change for jump-diffusion processes. Ann. Appl. Probab. 15, 1713–1732 (2005) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Clark, P.: A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155 (1973) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall, Cambridge (2004) MATHGoogle Scholar
  24. 24.
    Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Cox, J.C.: Notes on option pricing, I: constant elasticity of variance diffusions. J. Portf. Manag. 22, 15–17 (1996) CrossRefGoogle Scholar
  26. 26.
    Crosby, J.: Pricing a class of exotic commodity options in a multi-factor jump-diffusion model. Quant. Finance 8, 471–483 (2008) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Davydov, D., Linetsky, V.: Pricing options on scalar diffusions: an eigenfunction expansion approach. Oper. Res. 51, 185–209 (2003) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Dempster, M., Hong, S.: Spread option valuation and the fast Fourier transform. In: Pliska, S.R., Vorst, T. (eds.) Mathematical Finance–Bachelier Congress, Paris, pp. 203–220. Springer, Berlin (2000) Google Scholar
  29. 29.
    Dempster, M., Medova, E., Tang, K.: Long term spread option valuation and hedging. J. Bank. Finance 32, 2530–2540 (2008) CrossRefGoogle Scholar
  30. 30.
    Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, Cambridge (1999) CrossRefMATHGoogle Scholar
  31. 31.
    Eberlein, E., Prause, K.: The generalized hyperbolic model: financial derivatives and risk measures. In: Pliska, S.R., Vorst, T. (eds.) Mathematical Finance–Bachelier Congress, Paris, pp. 245–267. Springer, Berlin (2000) Google Scholar
  32. 32.
    Eydeland, A., Wolyniec, K.: Energy and Power Risk Management. Wiley, Hoboken (2003) Google Scholar
  33. 33.
    Feng, L., Linetsky, V.: Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach. Math. Finance 18, 337–384 (2008) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Feng, L., Linetsky, V.: Computing exponential moments of the discrete maximum of a Lévy process and lookback options. Finance Stoch. 13, 501–529 (2009) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (2010) MATHGoogle Scholar
  36. 36.
    Goldstein, J.A.: Abstract evolution equations. Trans. Am. Math. Soc. 141, 159–185 (1969) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Gulisashvili, A., Van Casteren, J.A.: Non-Autonomous Kato Classes and Feynman–Kac Propagators. World Scientific, Singapore (2006) CrossRefMATHGoogle Scholar
  38. 38.
    Hurd, T.R., Zhou, Z.: A Fourier transform method for spread option pricing. SIAM J. Financ. Math. 1, 142–157 (2010) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) CrossRefMATHGoogle Scholar
  40. 40.
    Kirk, E.: Correlation in the energy markets. In: Managing Energy Price Risk, 1st edn., pp. 71–78. Risk Publications and Enron, London (1995) Google Scholar
  41. 41.
    Knight, F.B.: Essentials of Brownian Motion and Diffusion. Am. Math. Soc., Providence (1981) CrossRefMATHGoogle Scholar
  42. 42.
    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Berlin (2010) CrossRefMATHGoogle Scholar
  43. 43.
    Kudryavtsev, O., Levendorskiĭ, S.: Fast and accurate pricing of barrier options under Lévy processes. Finance Stoch. 13, 531–562 (2009) MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Li, L., Linetsky, V.: Optimal stopping in infinite horizon: an eigenfunction expansion approach. Stat. Probab. Lett. 85, 122–128 (2014) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Li, L., Linetsky, V.: Time-changed Ornstein–Uhlenbeck processes and their applications in commodity derivative models. Math. Finance 24, 289–330 (2014) MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Li, L., Linetsky, V.: Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach. Finance Stoch. 19, 941–977 (2015) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Li, L., Mendoza-Arriaga, R.: Ornstein–Uhlenbeck processes time-changed with additive subordinators and their applications in commodity derivative models. Oper. Res. Lett. 41, 521–525 (2013) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Li, L., Mendoza-Arriaga, R.: Equivalent measure changes for subordinate diffusions. Preprint (2015). Available online at http://ssrn.com/abstract=2633817
  49. 49.
    Li, L., Mendoza-Arriaga, R., Mo, Z., Mitchell, D.: Modeling electricity prices: a time change approach. Quant. Finance (2015). 10.1080/14697688.2015.1125521 Google Scholar
  50. 50.
    Li, L., Qu, X., Zhang, G.: An efficient algorithm based on eigenfunction expansions for some optimal timing problems in finance. J. Comput. Appl. Math. 294, 225–250 (2016) MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Li, L., Linetsky, V.: Optimal stopping and early exercise: an eigenfunction expansion approach. Oper. Res. 61, 625–643 (2013) MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Lim, D., Li, L., Linetsky, V.: Evaluating callable and putable bonds: an eigenfunction expansion approach. J. Econ. Dyn. Control 36, 1888–1908 (2012) MathSciNetCrossRefGoogle Scholar
  53. 53.
    Linetsky, V.: Spectral methods in derivatives pricing. In: Birge, J.R., Linetsky, V. (eds.) Handbook of Financial Engineering. Handbooks in Operations Research and Management Sciences, vol. 6, pp. 213–289. Elsevier, Amsterdam (2008) Google Scholar
  54. 54.
    Longstaff, F., Schwartz, E.S.: Interest rate volatility and the term structure: a two factor general equilibrium model. J. Finance 47, 1259–1282 (1992) CrossRefGoogle Scholar
  55. 55.
    Lorig, M., Lozano-Carbassé, O., Mendoza-Arriaga, R.: Variance swaps on defaultable assets and market implied time-changes. Preprint (2013). Available online at http://ssrn.com/abstract=2141380
  56. 56.
    Madan, D., Carr, P., Chang, E.C.: The variance gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998) CrossRefMATHGoogle Scholar
  57. 57.
    Madan, D., Yor, M.: Representing the CGMY and Meixner Lévy processes as time changed Brownian motions. J. Comput. Finance 12, 27–47 (2008) MathSciNetCrossRefGoogle Scholar
  58. 58.
    Mandelbrot, B., Taylor, H.M.: On the distribution of stock price differences. Oper. Res. 15, 1057–1062 (1967) CrossRefGoogle Scholar
  59. 59.
    McKean, H.P.: Elementary solutions for certain parabolic partial differential equations. Trans. Am. Math. Soc. 82, 519–548 (1956) MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Mendoza-Arriaga, R., Carr, P., Linetsky, V.: Time changed Markov processes in unified credit-equity modeling. Math. Finance 20, 527–569 (2010) MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Mendoza-Arriaga, R., Linetsky, V.: Time-changed CIR default intensities with two-sided mean-reverting jumps. Ann. Appl. Probab. 24, 811–856 (2014) MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Mijatović, A., Pistorius, M.: On additive time-changes of Feller processes. In: Ruzhansky, M., Wirth, J. (eds.) Progress in Analysis and Its Applications, Proceedings of the 7th International ISAAC Congress, 13–18 July 2009, pp. 431–437. Imperial College/World Scientific, London (2010) CrossRefGoogle Scholar
  63. 63.
    Nielsen, L.: Weak convergence and Banach space-valued functions: improving the stability theory of Feynman’s operational calculi. Math. Phys. Anal. Geom. 14, 279–294 (2011) MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhäuser, Basel (1988) CrossRefMATHGoogle Scholar
  65. 65.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) MATHGoogle Scholar
  66. 66.
    Phillips, R.S.: On the generation of semigroups of linear operators. Pac. J. Math. 2, 343–369 (1952) MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. II: Special Functions. Gordon and Breach, New York (1986) MATHGoogle Scholar
  68. 68.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I: Functional Analysis. Academic Press, San Diego (1980) MATHGoogle Scholar
  69. 69.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  70. 70.
    Schilling, R.L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications. de Gruyter, Berlin (2012) CrossRefMATHGoogle Scholar
  71. 71.
    Schoutens, W., Teugels, J.L.: Lévy processes, polynomials and martingales. Commun. Stat., Stoch. Models 14, 335–349 (1998) MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Schwartz, E.S.: The stochastic behavior of commodity prices: implications for valuation and hedging. J. Finance 52, 923–973 (1997) CrossRefGoogle Scholar
  73. 73.
    Van Casteren, J.A.: Markov Processes, Feller Semigroups and Evolution Equations. World Scientific, Singapore (2011) MATHGoogle Scholar
  74. 74.
    Venkatramanan, A., Alexander, C.: Closed form approximations for spread options. Appl. Math. Finance 18, 447–472 (2011) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CITIC SecuritiesBeijingChina
  2. 2.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong KongShatinHong Kong
  3. 3.Department of Information, Risk and Operations Management, McCombs School of BusinessThe University of Texas at AustinAustinUSA

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