nach oben

Finance and Stochastics

Erschienen in:

15.09.2016

Hedging with small uncertainty aversion

verfasst von: Sebastian Herrmann, Johannes Muhle-Karbe, Frank Thomas Seifried

Erschienen in: Finance and Stochastics | Ausgabe 1/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.

Nächster Artikel Continuous-time perpetuities and time reversal of diffusions

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Jetzt informieren

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!

Jetzt informieren

Nur mit Berechtigung zugänglich

A related approach is Mykland’s “conservative delta hedging” [52, 53].

This terminology stems from the literature on robust control [33] and also from the robust representation of convex risk measures (see e.g. [25, Sect. 5.2]). Note that the penalty is not imposed on the agent, but on her fictitious adversary who minimises over \(P\). Hence, the penalty in (1.1) is added and not subtracted.

In view of the well-known deficiencies of local volatility models in capturing the dynamics of option prices (cf. e.g. [21, 30]) an extension to more general reference models is an important direction for future research; see Remark 2.7 for some further discussion.

A similar penalty has been used by [5] in the context of local volatility calibration with prior beliefs.

The term \(U'(Y_{t})\) in (1.2) renders the preferences invariant under affine transformations of the utility function; see also Remark 2.6.

Asymptotic analyses of option pricing and hedging problems with the worst-case approach have been carried out by [48, 2, 3, 26].

In Sect. 1, we only display the formulas for the special case of the penalty functional (1.2). Our analysis also applies to more general penalty functionals; cf. Sect. 2.3.

Our results can be formally linked to the UVM with a random, time-dependent volatility band depending on the option’s cash gamma and the agent’s uncertainty aversion; cf. Remark 3.7. Note that the cash gamma also plays a crucial role in the asymptotic analysis of other frictions such as discrete rebalancing [9], transaction costs [64], price impact [51], or jumps [13].

Specifically, the payoff is the Black–Scholes value of a standard put option with strike 100 and maturity 1 day.

As the option payoff is convex, this spread is simply the difference between the Black–Scholes values of the “smooth put” corresponding to the two endpoints of the volatility band.

Here and in the following, subscripts on functions denote the corresponding partial derivatives.

The strategy adjustment may become dependent on risk aversion for other penalty functionals; see the discussion following Theorem 3.4.

A rigorous verification of these results would proceed along the same lines as for the simpler benchmark case discussed here. In order not to drown the ideas in further technicalities resulting from even more state variables, regularity conditions, etc., we do not pursue this here.

Worst-case hedges for some exotics have been derived by [36, 12, 16, 15, 41, 40, 39, 62], for example.

This is the analogue of the Lagrangian uncertain volatility model [7]; also compare [54].

The same formula is obtained—mutatis mutandis—for exotics of Asian, lookback or barrier type, after adding the appropriate state variables to the reference cash gamma.

This error incurred from hedging with a misspecified volatility is well known in the literature and in practice. To the best of our knowledge, it appeared first in Lyons [48, Eq. (27)] (see also [47, Eq. (6.7)]).

Under each \(P\), the stochastic integrals of sufficiently integrable, progressively measurable integrands against Itô process integrators are constructed as \(\mathbb{F}\)-progressively measurable stochastic processes with \(P\)-a.s. continuous paths; see the discussion before Lemma 4.3.3 in Stroock and Varadhan [63].

Since \(\mathfrak{P}(\theta ,\sigma )\) may contain more than one measure, we also have to allow “nature” to choose a specific measure.

The penalty functional in the sense of the criterion (1.1) is \(\alpha (P) = E^{P} [ \mathfrak{a}(\sigma^{P}) ]\), where \(\sigma^{P}\) is the volatility of returns of \(S\) under \(P\).

Recall from footnote 2 that \(\mathfrak{a}\) penalises the fictitious adversary (“nature”) and not the agent.

Notably, [5] show that penalty functionals of this form can arise as the continuous-time limit of the relative entropy in a discrete-time approximation.

Using \(U'(y_{0})\) instead of \(U'(Y_{t})\) would yield the same expansion for \(v(\psi )\) as in Theorem 3.4 and, formally and at the leading order, the same almost optimal strategies and volatilities. This is because in the asymptotic limit for small uncertainty aversion, the P&L process converges to a constant.

In the context of robust portfolio choice, Maenhout [50] also observes that some modification of the standard (non wealth-dependent) entropic penalty is reasonable to avoid that the agent’s uncertainty aversion wears off as her wealth rises, and tackles this effect by directly modifying the HJBI equation.

Here, we assume that all relevant partial derivatives exist; precise conditions are given in Assumption 3.2.

Hölder-continuity uniformly on \((0,T)\times (K ^{-1},K)\) suffices for this step.

Strictly speaking, \(\theta^{\psi }\) and \(\sigma^{\psi }\) have to be defined for every \(\omega \in \varOmega \), even those for which \(S\) or \(Y\) exceeds the bounds (3.1). Outside these bounds, however, the functions \(\bar{V}_{s}\), \(\widetilde{\theta }\), \(\bar{\sigma }\), \(\widetilde{\sigma }\) are not defined. As we only consider measures \(P\) such that (3.1) holds, we do not make explicit the corresponding straightforward adjustments, which would only hamper readability.

Mutatis mutandis, the threshold 1 in the definition of the stopping time \(\tau \) can be replaced by any other constant. The same modification also appears in the asymptotic analysis of models with transaction costs [44].

A sufficient condition for \(\widetilde{w}_{sy} = 0\) is that \(f''(t,s,y;\varsigma )\) does not depend on \(y\).

A second-order expansion for the ask price can also be obtained, but does not offer much additional insight.

This symmetry generally breaks down for the second-order term \(\widehat{w}\); cf. the corresponding source term (3.5). Hence, for a second-order expansion of the indifference bid price, we have to use the \(\widehat{w}\) corresponding to the negative of the option.

This specification is a particular case of the general “random \(G\)-expectation” [56].

[48] imposes bounds on the instantaneous variance of prices instead of the volatility of returns. Hence, the PDE for \(\widetilde{V}\) there looks slightly different. The PDE presented here is a slight generalisation of the one derived in [26].

For example, if \(N\) is integrable under \(P^{\theta ,\sigma }\) for each \((\theta ,\sigma ) \in \mathcal{A} \times \mathcal{V}\), then \(v\) is a well-defined number in the extended real line \([-\infty ,+\infty ]\).

We obtain the same results if we interchange the order of the infimum and the supremum. In the language of two-player, zero-sum stochastic differential games, this indicates that the game “has a value”.

For the heuristic derivation in this section, we tacitly assume that for each \((\theta ,\sigma )\), \(P^{\theta ,\sigma }\) attains the infimum in (2.13), so that the additional infimum over measures in (2.13) disappears.

This covers most of the specific choices that are dealt with in the following subsections, except for additional state variables needed for some exotic options in Sect. 4.2. To explain the general procedure, we first focus here on the easiest case with just two state variables, the stock price \(S\) and the P&L process \(Y\).

Note that this hedge \(\bar{\varDelta }\) reflects the option’s sensitivity to price moves in the underlying both directly through \(S\) and indirectly through the additional state variable \(A\).

Portfolios including exotic options can be treated along the same lines; we do not pursue this here to ease notation.

That is, the local volatility model is calibrated to the observed market prices of the liquid options at time 0.

Portfolios of barrier options as in [4] or other exotics can be treated along the same lines, but require a more extensive notation.

For instance, if \(\theta \) is of finite variation, then the stochastic integral can be defined pathwise via the integration by parts formula.

Unlike [14], we disregard bid-ask spreads for the liquidly traded options.

A Mathematica file containing these calculations is available from the authors upon request.

Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26, 233–251 (2016) MathSciNetCrossRefMATH

Ahn, H., Muni, A., Swindle, G.: Misspecified asset price models and robust hedging strategies. Appl. Math. Finance 4, 21–36 (1997) CrossRefMATH

Ahn, H., Muni, A., Swindle, G.: Optimal hedging strategies for misspecified asset price models. Appl. Math. Finance 6, 197–208 (1999) CrossRefMATH

Avellaneda, M., Buff, R.: Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options. Appl. Math. Finance 6, 1–18 (1999) CrossRefMATH

Avellaneda, M., Friedman, C., Holmes, R., Samperi, D.: Calibrating volatility surfaces via relative-entropy minimization. Appl. Math. Finance 4, 37–64 (1997) CrossRefMATH

Avellaneda, M., Levy, A., Parás, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2, 73–88 (1995) CrossRef

Avellaneda, M., Parás, A.: Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model. Appl. Math. Finance 3, 21–52 (1996) CrossRefMATH

Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices—a mass transport approach. Finance Stoch. 17, 477–501 (2013) MathSciNetCrossRefMATH

Bertsimas, D., Kogan, L., Lo, A.: When is time continuous? J. Financ. Econ. 55, 173–204 (2000) CrossRefMATH

10.

Biagini, S., Bouchard, B., Kardaras, C., Nutz, M.: Robust fundamental theorem for continuous processes. Math. Finance (2016). doi:10.1111/mafi.12110/full

11.

Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823–859 (2015) MathSciNetCrossRefMATH

12.

Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Finance 11, 285–314 (2001) MathSciNetCrossRefMATH

13.

Černý, A., Denkl, S., Kallsen, J.: Hedging in Lévy models and the time step equivalent of jumps. Preprint (2013). Available online at http://arxiv.org/abs/1309.7833

14.

Cont, R.: Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance 16, 519–547 (2006) MathSciNetCrossRefMATH

15.

Cox, A., Obłój, J.: Robust hedging of double touch barrier options. SIAM J. Financ. Math. 2, 141–182 (2011) MathSciNetCrossRefMATH

16.

Cox, A., Obłój, J.: Robust pricing and hedging of double no-touch options. Finance Stoch. 15, 573–605 (2011) MathSciNetCrossRefMATH

17.

Davis, M.: Martingale methods in stochastic control. In: Kohlmann, M., Vogel, W. (eds.) Stochastic Control and Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol. 16, pp. 85–117. Springer, Berlin (1979) CrossRef

18.

Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978) MATH

19.

Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006) MathSciNetCrossRefMATH

20.

Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields 160, 391–427 (2014) MathSciNetCrossRefMATH

21.

Dumas, B., Fleming, J., Whaley, R.: Implied volatility functions: empirical tests. J. Finance 53, 2059–2106 (1998) CrossRef

22.

Dupire, B.: Pricing with a smile. Risk 7(1), 18–20 (1994)

23.

Fleming, W., Hernández-Hernández, D.: On the value of stochastic differential games. Commun. Stoch. Anal. 5, 341–351 (2011) MathSciNetMATH

24.

Fleming, W., Souganidis, P.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38, 293–314 (1989) MathSciNetCrossRefMATH

25.

Föllmer, H., Schied, A.: Probabilistic aspects of finance. Bernoulli 19, 1306–1326 (2013) MathSciNetCrossRefMATH

26.

Fouque, J.-P., Ren, B.: Approximation for option prices under uncertain volatility. SIAM J. Financ. Math. 5, 260–383 (2014) MathSciNetCrossRefMATH

27.

Frey, R.: Superreplication in stochastic volatility models and optimal stopping. Finance Stoch. 4, 161–187 (2000) MathSciNetCrossRefMATH

28.

Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) MATH

29.

Galichon, A., Henry-Labordère, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24, 312–336 (2014) MathSciNetCrossRefMATH

30.

Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley, Hoboken (2006)

31.

Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989) MathSciNetCrossRefMATH

32.

Hansen, L., Sargent, T.: Robust control and model uncertainty. Am. Econ. Rev. 91(2), 60–66 (2001) CrossRef

33.

Hansen, L., Sargent, T.: Robustness. Princeton Univ. Press, Princeton (2007) MATH

34.

Hayashi, T., Mykland, P.: Evaluating hedging errors: an asymptotic approach. Math. Finance 15, 309–343 (2005) MathSciNetCrossRefMATH

35.

Herrmann, S., Muhle-Karbe, J.: Model uncertainty, dynamic recalibration, and the emergence of delta-vega hedging. Preprint (2015). Available online at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2694718

36.

Hobson, D.: Robust hedging of the lookback option. Finance Stoch. 2, 329–347 (1998) CrossRefMATH

37.

Hobson, D.: Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8, 193–205 (1998) MathSciNetCrossRefMATH

38.

Hobson, D.: The Skorokhod embedding problem and model-independent bounds for option prices. In: Carmona, R., et al. (eds.) Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Mathematics, vol. 2003, pp. 267–318. Springer, Berlin (2011) CrossRef

39.

Hobson, D., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611–649 (2012) MathSciNetCrossRefMATH

40.

Hobson, D., Klimmek, M.: Robust price bounds for the forward starting straddle. Finance Stoch. 19, 189–214 (2015) MathSciNetCrossRefMATH

41.

Hobson, D., Neuberger, A.: Robust bounds for forward start options. Math. Finance 22, 31–56 (2012) MathSciNetCrossRefMATH

42.

Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Preprint (2015). Available online at http://arxiv.org/abs/1503.02822

43.

Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2002) CrossRefMATH

44.

Kallsen, J., Li, S.: Portfolio optimization under small transaction costs: a convex duality approach. Preprint (2013). Available online at http://arxiv.org/abs/1309.3479

45.

Kallsen, J., Muhle-Karbe, J.: Option pricing and hedging with small transaction costs. Math. Finance 25, 702–723 (2015) MathSciNetCrossRefMATH

46.

Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1998) CrossRefMATH

47.

El Karoui, N., Jeanblanc-Picqué, M., Shreve, S.: Robustness of the Black and Scholes formula. Math. Finance 8, 93–126 (1998) MathSciNetCrossRefMATH

48.

Lyons, T.: Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133 (1995) CrossRef

49.

Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498 (2006) MathSciNetCrossRefMATH

50.

Maenhout, P.: Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17, 951–983 (2004) CrossRef

51.

Moreau, L., Muhle-Karbe, J., Soner, H.M.: Trading with small price impact. Math. Finance (2016). doi:10.1111/mafi.12098/full

52.

Mykland, P.: Conservative delta hedging. Ann. Appl. Probab. 10, 664–683 (2000) MathSciNetCrossRefMATH

53.

Mykland, P.: Financial options and statistical prediction intervals. Ann. Stat. 31, 1413–1438 (2003) MathSciNetCrossRefMATH

54.

Mykland, P.: The interpolation of options. Finance Stoch. 7, 417–432 (2003) MathSciNetCrossRefMATH

55.

Neufeld, A., Nutz, M.: Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18, 1–14 (2013) MathSciNetCrossRefMATH

56.

Nutz, M.: Random \(G\)-expectations. Ann. Appl. Probab. 23, 1755–1777 (2013) MathSciNetCrossRefMATH

57.

Nutz, M.: Superreplication under model uncertainty in discrete time. Finance Stoch. 18, 791–803 (2014) MathSciNetCrossRefMATH

58.

Pham, T., Zhang, J.: Two person zero-sum game in weak formulation and path dependent Bellman–Isaacs equation. SIAM J. Control Optim. 52, 2090–2121 (2014) MathSciNetCrossRefMATH

59.

Possamaï, D., Royer, G., Touzi, N.: On the robust superhedging of measurable claims. Electron. Commun. Probab. 18(95), 1–13 (2013) MathSciNetMATH

60.

Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 2: Itô Calculus, 2nd edn. Cambridge Univ. Press, Cambridge (2000) CrossRefMATH

61.

Seifried, F.: Optimal investment for worst-case crash scenarios: a martingale approach. Math. Oper. Res. 35, 559–579 (2010) MathSciNetCrossRefMATH

62.

Stebegg, F.: Model-independent pricing of Asian options via optimal martingale transport. Preprint (2014). Available online at http://arxiv.org/abs/1412.1429

63.

Stroock, D., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979) MATH

64.

Whalley, A., Wilmott, P.: An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7, 307–324 (1997) MathSciNetCrossRefMATH

Titel: Hedging with small uncertainty aversion
verfasst von: Sebastian Herrmann
Johannes Muhle-Karbe
Frank Thomas Seifried
Publikationsdatum: 15.09.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 1/2017
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-016-0309-z

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 1/2017

Arbitrage-free pricing of multi-person game claims in discrete time

Model uncertainty and the pricing of American options

Watermark options

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

Continuous-time perpetuities and time reversal of diffusions

Optimal consumption and investment with Epstein–Zin recursive utility