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From innovation to obfuscation: continuous time finance fifty years later

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Abstract

This paper surveys several of the most important applications of the continuous time finance paradigm in portfolio selection and derivatives pricing. While it recognizes the powerful insights that the paradigm offered to researchers and practitioners, it finds that several methodological approaches that it introduced have themselves hardened into paradigms and become dysfunctional. They have downgraded and neglected significant real-world problems because of their inability to model them, or adopted simplifications that had little relevance to the problems that they were supposed to solve. The paper then offers in all cases an alternative methodology that can reach the desired solution via rigorous economic and mathematical reasoning, by replacing mathematical elegance with numerical estimations and approximations.

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Notes

  1. See the Foreword by Samuelson and the Preface in Merton (1992), as well as Bachelier’s mathematical antecedents in Veisdal (2019).

  2. See The Black Swan by Nassim Taleb (2010, pp. 277–285).

  3. See their Eq. (12) in p. 4266, which holds only for a frictionless market.

  4. The values for both \(b_{T}\) and \(v_{T}\) are averages over the 4 days preceding the start of the maturity period.

  5. See, for instance, Bates (1991, 2006), Pan (2002), Liu et al (2005).

  6. An example of such an empirical SD application is Constantinides et al. (2020), that uses the observable bias-adjusted VIX index to estimate the P distribution at every time point.

  7. A colorful way of stating this indeterminacy of the “true” option price is in one of the earliest empirical option market studies by Rubinstein (1985, p. 465).

  8. See in p. 1968 of Christoffersen et al (2013), where the risk neutral parameter \(\vartheta ^{*}\) in our notation should exceed \(\vartheta\), while the estimated values in Tables 4 and 5 show \(\vartheta > \vartheta ^{*}\).

  9. In real-life such a replication would not be feasible in the presence of market frictions, but this cannot be invoked to explain the frictionless market results.

  10. This also applies to the Beare and Schmidt (2016) study, who derived a kernel that violated monotonicity in several dates in their sample, but relied for their derivation on a technical paper by Monteiro et al. (2008), that implicitly assumed option market efficiency.

  11. The material here borrows heavily from Perrakis (2017), Perrakis (2019, Chapter 3).

  12. A different type of friction was introduced by Leippold and Su (2015), who considered margin costs and different borrowing and lending rates in otherwise frictionless trading of both underlying and options, which preserves the replication property but creates bounds that do not admit put-call parity. Since their proofs are based on replication, they cannot be extended to conventional trading frictions. In principle SD can be extended to admit margins and differential borrowing and lending rates, but this has not been attempted yet and may present challenges.

  13. This assumption can be easily relaxed when dealing with American options, as analyzed in detail in Chapter 4 of Perrakis (2019). For the empirically important cases where the dividends accrue to the bond account and for normal parameter values it can be shown that including the dividends in the risky asset yields a very close approximation to the optimal policies; see Czerwonko and Perrakis (2016b).

  14. The results extend routinely to the case that consumption occurs at each trading date and utility is defined over consumption at each of the trading dates and over the net worth at the terminal date.

  15. If utility is defined only for non-negative net worth, then the decision variable is constrained to be a member of a convex set that ensures the non-negativity of the net worth.

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Acknowledgements

I thank George Constantinides, Michal Czerwonko and Thierry Post for their valuable comments and support in writing this essay. I also thank the editor Markus Schmid and two anonymous referees for helpful advice and comment.

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  1. Department of Finance, Concordia University, Montreal, Canada

    Stylianos Perrakis

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RBC Distinguished Professor of Derivatives, Concordia University. I wish to thank George Constantinides, Michal Czerwonko and Thierry Post for their valuable comments and support in writing this essay. I also thank the editor and two anonymous referees for helpful advice and comment.

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Perrakis, S. From innovation to obfuscation: continuous time finance fifty years later. Financ Mark Portf Manag 36, 369–401 (2022). https://doi.org/10.1007/s11408-021-00399-z

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