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Stochastic Dominance Option Pricing I: The Frictionless Case

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Stochastic Dominance Option Pricing

Abstract

This chapter presents the stochastic dominance (SD) approach to option pricing in frictionless markets, which was developed piecemeal from the outset in a discrete time context in a series of articles published in the mid-1980s. It derives multiperiod bounds for call and put options in a discrete time context, valid for any distribution. The bounds yield the Black-Scholes-Merton option price for both index and equity options when the underlying asset returns converge to any type of univariate diffusion when the time partition tends to 0. They also converge to tight bounds for both index and equity option prices when the underlying distribution includes rare events. It is shown that the index bounds can also explain several failures of the alternative equilibrium models based on a representative investor in the rare events case. Last, the SD frictionless bounds methodology is shown to produce useful results when applied to financial instruments issued by the insurance industry and indexed on catastrophe events.

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Notes

  1. 1.

    See Oancea and Perrakis (2009, 2014) and Perrakis and Boloorforoosh (2013, 2018).

  2. 2.

    For simplicity, dividends are ignored throughout this chapter. All results can be easily extended to the case where the stock has a known and constant dividend yield, as in index options. In the latter case the instantaneous mean in (2.1) and (2.2) is net of the dividend yield.

  3. 3.

    Although this assumption may not be justified in practice, its effect on option values is generally recognized as minor in short- and medium-lived options. It has been adopted without any exception in all equilibrium-based jump-diffusion option valuation models that have appeared in the literature. See the comments in Bates (1991, p. 1039, note 30) and Amin and Ng (1993, p. 891). In order to evaluate the various features of option pricing models, Bakshi, Cao and Chen (1997) applied without deriving it a risk-neutral model featuring stochastic interest rate, stochastic volatility and jumps. They found that stochastic interest rates offer no goodness of fit improvement.

  4. 4.

    The results are unchanged if the traders are assumed to maximize the expected utility of the consumption stream.

  5. 5.

    We relax this assumption in a subsequent section when we examine derivatives for underlying assets indexed on catastrophe events.

  6. 6.

    Bogle (2005) reports that in 2004 index funds accounted for about one-third of equity fund cash inflows since 2000 and represented about one-seventh of equity fund assets.

  7. 7.

    The convexity of the option with respect to the underlying stock price holds in all cases in which the return distribution had independent and identically distributed (iid) time increments, in all univariate state-dependent diffusion processes, and in bivariate (stochastic volatility) diffusions under most assumed conditions; see Merton (1973) and Bergman, Grundy and Wiener (1996).

  8. 8.

    See Ritchken (1985, section III). Actually, the upper bound in that LP is equal to the stock price minus the strike price discounted by the highest possible return; this last term goes to 0 in the multiperiod case.

  9. 9.

    See Ritchken (1985, p. 1227).

  10. 10.

    The discretization (2.26) is sometimes referred to as the Euler discretization.

  11. 11.

    For more on weak convergence for Markov processes, see Ethier and Kurtz (1986) or Stroock and Varadhan (1979).

  12. 12.

    See Cox and Rubinstein (1985, pp. 361–364).

  13. 13.

    In evaluating \( {U}_t,\kern0.5em {L}_t \) the terms \( \overline{\nu}\left({\widehat{z}}_{t+\Delta t}\right) \) and \( \overline{\nu}\left({z}_{\min, t+\Delta t}\right) \) should replace \( {\widehat{z}}_{t+\Delta t} \) and \( {z}_{\min, t+\Delta t} \).

  14. 14.

    See, for instance, Froot (2001), Froot and O’Connell (2008), Ibragimov et al. (2009) and Lane and Mahul (2008).

  15. 15.

    See, for instance, Bates (1991), Amin (1993) and Amin and Ng (1993).

  16. 16.

    Duffie et al. (2000) have presented option prices under general \( Q \)-distributions containing both stochastic volatility and jumps but without any linkage to the underlying market’s dynamics.

  17. 17.

    See, for instance, Rosenberg and Engle (2002) and Bliss and Panigirtzoglou (2004), who find coefficients ranging from 2 to 12 and 1.97 to 7.91 respectively.

  18. 18.

    For simplicity, dividends are ignored throughout this paper. All results can be easily extended to the case where the stock has a known and constant dividend yield, as in index options. In the latter case the instantaneous mean in (2.40)–(2.42) is net of the dividend yield.

  19. 19.

    See Perrakis and Boloorforoosh (2013, Lemma 2).

  20. 20.

    Note that we do not assume here that the jump risk is diversifiable.

  21. 21.

    More details can be found in Oancea and Perrakis (2014) and Merton (1982).

  22. 22.

    See Fama and French (2002), Constantinides (2002) and Dimson, Marsh and Staunton (2008).

  23. 23.

    See, for instance, Bates (1991, 2006), Liu et al. (2005) and Zhang et al. (2012).

  24. 24.

    The equilibrium model does not allow stochastic volatility and jumps in linking the P- and Q-distributions . Although Duffie et al. (2000) have presented option prices under general Q-distributions containing both stochastic volatility and jumps, to our knowledge the only stochastic volatility pricing kernel was derived by Christoffersen et al. (2013) in the context of the Heston (1993) model. For stochastic volatility in the SD context, see the conclusions section.

  25. 25.

    See Eraker et al. (2003, p. 1294), Broadie et al. (2007, 2009) and Ross (2015).

  26. 26.

    See also the survey article by Kocherlakota (1996).

  27. 27.

    See Campanale et al. (2010).

  28. 28.

    See GOP (2018).

  29. 29.

    See the comments in Martin (2013, Section 2).

  30. 30.

    See GOP (2018, Appendix D) for the formulation of the truncated case that implies \( {j}_{I\min }>-1. \)

  31. 31.

    See MMC securities (2005) and Guy Carpenter (2016).

  32. 32.

    See, for instance, Dassios and Jang (2003), Duan and Yu (2005), Lee and Yu (2007) and Chang et al. (2010).

  33. 33.

    See Froot (2001), Froot and O’Connell (2008) and Lane and Mahul (2008).

  34. 34.

    See, respectively, Froot and O’Connell (2008) and Ibragimov et al. (2009).

  35. 35.

    See Perrakis and Boloorforoosh (2013, 2018).

  36. 36.

    These assumptions mirror the conditions prevailing in the traded hurricane futures in the Chicago Mercantile Exchange (CME).

  37. 37.

    See the article “The insurance industry has been turned upside down by catastrophe bonds”, Wall Street Journal, August 8, 2016, as well as the survey article by Edesses (2015).

  38. 38.

    See Edesses (2015, p. 9).

  39. 39.

    See Corollary 2 in CHJ (p. 1970, 2013); the kernel is monotone if the parameter \( \xi \) is equal to 0.

  40. 40.

    See Heston (1993).

  41. 41.

    We also use the proof of the convergence of the diffusion process discussed in Oancea and Perrakis (2014). In the extension of the proof to stochastic volatility, the only difference is related to the vector \( {\phi}_t \) in applying the Lindeberg condition, which is now a two-dimensional \( \left({S}_t,{\sigma}_t^2\right) \) vector.

  42. 42.

    The pricing kernel monotonicity holds if the kernel does not include a separate variance preference parameter; see Christoffersen et al. (2013, pp. 1966–1967). For the convexity, see the results of Bergman et al. (1996) for a bivariate diffusion.

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  1. Financial Derivatives, John Molson School of Business, Concordia University, Montreal, QC, Canada

    Stylianos Perrakis

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Perrakis, S. (2019). Stochastic Dominance Option Pricing I: The Frictionless Case. In: Stochastic Dominance Option Pricing. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-11590-6_2

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