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Model-independent pricing with insider information: a skorokhod embedding approach

Part of: Stochastic processes Mathematical finance

Published online by Cambridge University Press:  17 March 2021

Beatrice Acciaio ,
Alexander M. G. Cox Open the ORCID record for Alexander M. G. Cox [Opens in a new window]  and
Martin Huesmann
Beatrice Acciaio*
Affiliation:
ETH Zurich
Alexander M. G. Cox*
Affiliation:
University of Bath
Martin Huesmann*
Affiliation:
Universität Münster
*
*Postal address: Department of Mathematics, ETH Zurich, Switzerland.
**Postal address: Department of Mathematical Sciences, University of Bath, UK. Email address: [email protected]
***Postal address: Institute of Mathematical Stochastics, Universität Münster, Germany.
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Abstract

In this paper we consider the pricing and hedging of financial derivatives in a model-independent setting, for a trader with additional information, or beliefs, on the evolution of asset prices. In particular, we suppose that the trader wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the payoff of the derivative, and the insider’s information or beliefs, which take the form of a set of impossible paths, are time-invariant. In this way we accommodate drawdown constraints, as well as information/beliefs on quadratic variation or on the levels hit by asset prices. Our setup allows us to adapt recent work of [12] to prove duality results and a monotonicity principle. This enables us to determine geometric properties of the optimal models. Moreover, for specific types of information, we provide simple conditions for the existence of consistent models for the informed agent. Finally, we provide an example where our framework allows us to compute the impact of the information on the agent’s pricing bounds.

Keywords

Inside informationmodel-independent pricing and hedgingSkorokhod embedding problemmonotonicity principle

MSC classification

Primary: 91G20: Derivative securities
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 30 - 56
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Acciaio, B., Beiglböck, M., Penkner, F. and Schachermayer, W. (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26, 233–251.CrossRefGoogle Scholar
Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W. and Temme, J. (2013). A trajectorial interpretation of Doob’s martingale inequalities. Ann. Appl. Prob. 23, 1494–1505.CrossRefGoogle Scholar
Acciaio, B. and Larsson, M. (2017). Semi-Static completeness and robust pricing by informed investors. Ann. Appl. Prob. 27, 2270–2304.CrossRefGoogle Scholar
Aksamit, A., Hou, Z. and Obłój, J. (2016). Robust framework for quantifying the value of information in pricing and hedging. Preprint. Available at https://arxiv.org/abs/1605.02539.Google Scholar
Amendinger, J., Imkeller, P. and Schweizer, M. (1998). Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263–286.CrossRefGoogle Scholar
Ankirchner, S., Hobson, D. and Strack, P. (2015). Finite, integrable and bounded time embeddings for diffusions. Bernoulli 21, 1067–1088.CrossRefGoogle Scholar
Ankirchner, S. and Strack, P. (2011). Skorokhod embeddings in bounded time. Stoch. Dynamics 11, 215–226y.CrossRefGoogle Scholar
Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités XIII, Springer, Berlin, Heidelberg, pp. 90–115.10.1007/BFb0070852CrossRefGoogle Scholar
Bartl, D., Kupper, M. and Neufeld, A. (2020). Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–248.CrossRefGoogle Scholar
Bayraktar, E., Zhang, X. and Zhou, Z. (2018). Transport plans with domain constraints. To appear in Appl. Math. Optimization.Google Scholar
Bayraktar, E. and Zhou, Z. (2017). On arbitrage and duality under model uncertainty and portfolio constraints. Math. Finance 27, 988–1012.CrossRefGoogle Scholar
Beiglböck, M., Cox, A. M. and Huesmann, M. (2017). Optimal transport and Skorokhod embedding. Invent. Math. 208, 327–400.CrossRefGoogle Scholar
Beiglböck, M., Cox, A. M., Huesmann, M., Perkowski, N. and Prömel, D. J. (2017). Pathwise superreplication via Vovk’s outer measure. Finance Stoch. 21, 1141–1166.CrossRefGoogle Scholar
Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-Independent bounds for option prices—a mass transport approach. Finance Stoch. 17, 477–501.CrossRefGoogle Scholar
Beiglböck, M., Nutz, M. and Touzi, N. (2017). Complete duality for martingale optimal transport on the line. Ann. Prob. 45, 3038–3074.CrossRefGoogle Scholar
Biagini, F. and Øksendal, B. (2005). A general stochastic calculus approach to insider trading. Appl. Math. Optimization 52, 167–181.CrossRefGoogle Scholar
Biagini, S., Bouchard, B., Kardaras, C. and Nutz, M. (2017). Robust fundamental theorem for continuous processes. Math. Finance 27, 963–987.10.1111/mafi.12110CrossRefGoogle Scholar
Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Prob. 25, 823–859.CrossRefGoogle Scholar
Breeden, D. and Litzenberger, R. (1978). Prices of state-contingent claims implicit in option prices. J. Business 51, 621–651.10.1086/296025CrossRefGoogle Scholar
Campi, L. (2005). Some results on quadratic hedging with insider trading. Stochastics 77, 327–348.CrossRefGoogle Scholar
Carr, P. and Lee, R. (2010). Hedging variance options on continuous semimartingales. Finance Stoch. 14, 179–207.CrossRefGoogle Scholar
Cox, A. M., Obłój, J. and Touzi, N. (2019). The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach. Prob. Theory Relat. Fields 173, 211–259.CrossRefGoogle ScholarPubMed
Cox, A. M. G., Hou, Z. and Obłój, J. (2016). Robust pricing and hedging under trading restrictions and the emergence of local martingale models. Finance Stoch. 20, 669–704.CrossRefGoogle Scholar
Cox, A. M. G. and Wang, J. (2013). Optimal robust bounds for variance options. Preprint. Available at https://arxiv.org/abs/1308.4363.Google Scholar
Cox, A. M. G. and Wang, J. (2013). Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Prob. 23, 859–894.CrossRefGoogle Scholar
Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland, Amsterdam.Google Scholar
Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Prob. Theory Relat. Fields 160, 391–427.CrossRefGoogle Scholar
Dolinsky, Y. and Soner, H. M. (2015). Martingale optimal transport in the Skorokhod space. Stoch. Process. Appl. 125, 3893–3931.CrossRefGoogle Scholar
Fahim, A. and Huang, Y.-J. (2016). Model-Independent superhedging under portfolio constraints. Finance Stoch. 20, 51–81.CrossRefGoogle Scholar
Galichon, A., Henry-Labordère, P. and Touzi, N. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Prob. 24, 312–336.CrossRefGoogle Scholar
Gassiat, P., Oberhauser, H. and dos Reis, G. (2015). Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs. Stoch. Process. Appl. 125, 4601–4631.CrossRefGoogle Scholar
Grorud, A. and Pontier, M. (1998). Insider trading in a continuous time market model. Internat. J. Theoret. Appl. Finance 01, 331–347.CrossRefGoogle Scholar
Guo, G., Tan, X. and Touzi, N. (2016). Optimal Skorokhod embedding under finitely many marginal constraints. SIAM J. Control Optimization 54, 2174–2201.CrossRefGoogle Scholar
Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris-Princeton Lectures on Mathematical Finance 2010, Springer, Berlin, pp. 267–318.CrossRefGoogle Scholar
Hobson, D. G. (1998). Robust hedging of the lookback option. Finance Stoch. 2, 329–347.CrossRefGoogle Scholar
Hou, Z. and Obłój, J. (2018). Robust pricing–hedging dualities in continuous time. Finance Stoch. 22, 511–567.CrossRefGoogle Scholar
Karandikar, R. L. (1995). On pathwise stochastic integration. Stoch. Process. Appl. 57, 11–18.CrossRefGoogle Scholar
Lee, R. (2010). Realized volatility options. In Encyclopedia of Quantitative Finance, John Wiley, New York. Available at https://doi.org/10.1002/9780470061602.eqf07040.Google Scholar
Meilijson, I. (1982). There exists no ultimate solution to Skorokhod’s problem. In Séminaire de Probabilités XVI 1980/81, eds. Azéma, J. and Yor, M., Springer, Berlin, Heidelberg, pp. 392–399.CrossRefGoogle Scholar
Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43, 1293–1311.10.1214/aoms/1177692480CrossRefGoogle Scholar
Nutz, M. (2014). Superreplication under model uncertainty in discrete time. Finance Stoch. 18, 791–803.CrossRefGoogle Scholar
Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Prob. Surveys 1, 321–390.CrossRefGoogle Scholar
Pikovsky, I. and Karatzas, I. (1996). Anticipative portfolio optimization. Adv. Appl. Prob. 28, 1095–1122.CrossRefGoogle Scholar
Root, D. H. (1969). The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40, 715–718.CrossRefGoogle Scholar
Spoida, P. (2014). Robust pricing and hedging with beliefs about realized variance. Doctoral Thesis, University of Oxford.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.CrossRefGoogle Scholar
Van der Vecht, D. P. (1986). Ultimateness and the Azéma–Yor stopping time. In Séminaire de probabilités XX 1984/85, eds. Azéma, J. and Yor, M., Springer, Berlin, Heidelberg, pp. 375–378.CrossRefGoogle Scholar
Vovk, V. (2012). Continuous-Time trading and the emergence of probability. Finance Stoch. 16, 561–609.CrossRefGoogle Scholar
Vovk, V. (2015). Itô calculus without probability in idealized financial markets. Lithuanian Math. J. 55, 270–290.10.1007/s10986-015-9280-1CrossRefGoogle Scholar