The Annals of Applied Probability

Robust hedging of options on a leveraged exchange traded fund

Alexander M. G. Cox and Sam M. Kinsley

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A leveraged exchange traded fund (LETF) is an exchange traded fund that uses financial derivatives to amplify the price changes of a basket of goods. In this paper, we consider the robust hedging of European options on a LETF, finding model-free bounds on the price of these options.

To obtain an upper bound, we establish a new optimal solution to the Skorokhod embedding problem (SEP) using methods introduced in Beiglböck–Cox–Huesmann. This stopping time can be represented as the hitting time of some region by a Brownian motion, but unlike other solutions of, for example, Root, this region is not unique. Much of this paper is dedicated to characterising the choice of the embedding region that gives the required optimality property. Notably, this appears to be the first solution to the SEP where the solution is not uniquely characterised by its geometric structure, and an additional condition is needed on the stopping region to guarantee that it is the optimiser. An important part of determining the optimal region is identifying the correct form of the dual solution, which has a financial interpretation as a model-independent superhedging strategy.

Article information

Ann. Appl. Probab., Volume 29, Number 1 (2019), 531-576.

Received: February 2017
Revised: February 2018
First available in Project Euclid: 5 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91G20: Derivative securities
Secondary: 60G44: Martingales with continuous parameter 60J60: Diffusion processes [See also 58J65]

Leveraged exchange traded fund optimal Skorokhod embedding problem monotonicity principle robust pricing and hedging


Cox, Alexander M. G.; Kinsley, Sam M. Robust hedging of options on a leveraged exchange traded fund. Ann. Appl. Probab. 29 (2019), no. 1, 531--576. doi:10.1214/18-AAP1427.

Export citation


  • [1] Ahn, A., Haugh, M. and Jain, A. (2015). Consistent pricing of options on leveraged ETFs. SIAM J. Financial Math. 6 559–593.
  • [2] Avellaneda, M. and Zhang, S. (2010). Path-dependence of leveraged ETF returns. SIAM J. Financial Math. 1 586–603.
  • [3] Beiglböck, M., Cox, A. M. G. and Huesmann, M. (2017). Optimal transport and Skorokhod embedding. Invent. Math. 208 327–400.
  • [4] Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. J. Bus. 51 621–651.
  • [5] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). Robust hedging of barrier options. Math. Finance 11 285–314.
  • [6] Chacon, R. M. (1985). Barrier Stopping Times and the Filling Scheme. Ph.D. thesis, Univ. Washington.
  • [7] Cheng, M. and Madhavan, A. (2009). The dynamics of leveraged and inverse exchange-traded funds. J. Invest. Manag. 43–62.
  • [8] Cox, A. M. G., Hobson, D. and Obłój, J. (2008). Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. 18 1870–1896.
  • [9] 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.
  • [10] Cox, A. M. G. and Kinsley, S. M. (2018). Discretisation and duality of optimal Skorokhod embedding problems. Stochastic Process. Appl. To appear.
  • [11] Cox, A. M. G. and Obłój, J. (2011). Robust hedging of double touch barrier options. SIAM J. Financial Math. 2 141–182.
  • [12] Cox, A. M. G., Obłój, J. and Touzi, N. (2018). The Root solution to the multi-marginal embedding problem: An optimal stopping and time-reversal approach. Probab. Theory Related Fields. DOI:10.1007/s00440-018-0833-1.
  • [13] Cox, A. M. G. and Peskir, G. (2015). Embedding laws in diffusions by functions of time. Ann. Probab. 43 2481–2510.
  • [14] Cox, A. M. G. and Wang, J. (2013). Optimal robust bounds for variance options. Available at
  • [15] Cox, A. M. G. and Wang, J. (2013). Root’s barrier: Construction, optimality and applications to variance options. Ann. Appl. Probab. 23 859–894.
  • [16] De Angelis, T. (2018). From optimal stopping boundaries to Rost’s reversed barriers and the Skorokhod embedding. Ann. Inst. Henri Poincaré Probab. Stat. 54 1098–1133.
  • [17] Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
  • [18] Dolinsky, Y. and Soner, H. M. (2014). Robust hedging with proportional transaction costs. Finance Stoch. 18 327–347.
  • [19] Dolinsky, Y. and Soner, H. M. (2015). Martingale optimal transport in the Skorokhod space. Stochastic Process. Appl. 125 3893–3931.
  • [20] Henry-Labordère, P. (2016). Robust hedging of options on spot and variance: Exotic options made almost simple. Personal communication.
  • [21] Henry-Labordère, P., Obłój, J., Spoida, P. and Touzi, N. (2016). The maximum maximum of a martingale with given $n$ marginals. Ann. Appl. Probab. 26 1–44.
  • [22] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Springer, Berlin.
  • [23] Hobson, D. and Klimmek, M. (2012). Model-independent hedging strategies for variance swaps. Finance Stoch. 16 611–649.
  • [24] Hobson, D. G. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
  • [25] Hobson, D. G. and Pedersen, J. L. (2002). The minimum maximum of a continuous martingale with given initial and terminal laws. Ann. Probab. 30 978–999.
  • [26] Hou, Z. and Obłój, J. (2018). Robust pricing-hedging dualities in continuous time. Finance Stoch. 22 511–567.
  • [27] Loynes, R. M. (1970). Stopping times on Brownian motion: Some properties of Root’s construction. Z. Wahrsch. Verw. Gebiete 16 211–218.
  • [28] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [29] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [30] Root, D. H. (1969). The existence of certain stopping times on Brownian motion. Ann. Math. Stat. 40 715–718.
  • [31] Rost, H. (1971). The stopping distributions of a Markov Process. Invent. Math. 14 1–16.
  • [32] Rost, H. (1976). Skorokhod stopping times of minimal variance. In Séminaire de Probabilités X Université de Strasbourg. Lecture Notes in Math. 511 194–208. Springer, Berlin.
  • [33] Zhang, J. (2010). Path-Dependence Properties of Leveraged Exchange-Traded Funds: Compounding, Volatility and Option Pricing. Ph.D. thesis, New York Univ.