Robust price bounds for the forward starting straddle. (English) Zbl 1396.91735
The paper deals with the problem of constructing no-arbitrage lower price bounds and semi-static superreplicating hedging strategies for the forward starting straddle. The authors provide a robust, model-independent, lower bound on the price of a forward starting straddle with payoff \(|F_{T_1} - F_{T_0}|\), with \(T_0 <T_1\) two future times. It is assumed that the call prices for the maturities \(T_0\) and \(T_1\) are given. Using a Lagrangian approach and under a certain assumption on the distributions, the authors solve the primal and dual problems and demonstrate that there is no duality gap. The primal problem is to minimise \(\mathbb{E}[|F_{T_1} - F_{T_0}|]\) over all martingale models for \(F\) with given marginal laws at times \(T_0\) and \(T_1\). The dual problem is to find the cheapest semi-static subhedge for the strategy \(|F_{T_1} - F_{T_0}|\).
By considering a certain assumption on the supports of the marginal laws, the authors derive explicit expressions for the coupling which minimises the price of the option, and the form of the semi-static subhedge. A geometric representation for the optimal coupling is found, which can be applied to arbitrary distributions. Several examples for the marginal laws are discussed.
By considering a certain assumption on the supports of the marginal laws, the authors derive explicit expressions for the coupling which minimises the price of the option, and the form of the semi-static subhedge. A geometric representation for the optimal coupling is found, which can be applied to arbitrary distributions. Several examples for the marginal laws are discussed.
Reviewer: Iulian Stoleriu (Iaşi)
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 60G42 | Martingales with discrete parameter |
Keywords:
martingale coupling; martingale optimal transport; model-independent bounds; static hedging; forward starting straddleReferences:
| [1] | 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 (2014, to appear). doi:10.1111/mafi.12060 · Zbl 1378.91129 |
| [2] | Ambrosio, L., Lecture notes on optimal transport problems, No. 1812, 1-52 (2010), Berlin · Zbl 1047.35001 · doi:10.1007/978-3-540-39189-0_1 |
| [3] | Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices: a mass transport approach. Finance Stoch. 17(3), 477-501 (2013) · Zbl 1277.91162 · doi:10.1007/s00780-013-0205-8 |
| [4] | Beiglboeck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. (2014, to appear) · Zbl 1134.91425 |
| [5] | Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621-651 (1978) · doi:10.1086/296025 |
| [6] | Brown, H., Hobson, D.G., Rogers, L.C.G.: The maximum of a martingale constrained by an intermediate law. Probab. Theory Relat. Fields 119, 558-578 (2001) · Zbl 0980.60048 · doi:10.1007/PL00008771 |
| [7] | Brown, H., Hobson, D.G., Rogers, L.C.G.: Robust hedging of barrier options. Math. Finance 11, 285-314 (2001) · Zbl 1047.91024 · doi:10.1111/1467-9965.00116 |
| [8] | Carr, P., Lee, R.: Hedging variance options on semi-martingales. Finance Stoch. 14, 179-208 (2010) · Zbl 1224.91149 · doi:10.1007/s00780-009-0110-3 |
| [9] | Cousot, L.: Conditions on options prices for absence of arbitrage and exact calibration. J. Bank. Finance 31(11), 3377-3397 (2007) · doi:10.1016/j.jbankfin.2007.04.006 |
| [10] | Cox, A., Wang, J.: Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23, 859-894 (2013) · Zbl 1266.91101 · doi:10.1214/12-AAP857 |
| [11] | d’Aspremont, A., El Ghaoui, L.: Static arbitrage bounds on basket option prices. Math. Program., Ser. A 106(3), 467-489 (2006) · Zbl 1099.91067 · doi:10.1007/s10107-005-0642-z |
| [12] | Davis, M.H.A., Hobson, D.G.: The range of traded option prices. Math. Finance 17, 1-14 (2007) · Zbl 1278.91158 · doi:10.1111/j.1467-9965.2007.00291.x |
| [13] | Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.: The concept of co-monotonicity in actuarial science and finance: applications. Insur. Math. Econ. 31, 133-161 (2002) · Zbl 1037.62107 · doi:10.1016/S0167-6687(02)00135-X |
| [14] | Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields 160, 391-427 (2014) · Zbl 1305.91215 · doi:10.1007/s00440-013-0531-y |
| [15] | Galichon, A., Henry-Labordere, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds with given marginals, with an application to lookback options. Ann. Appl. Probab. 24(1), 312-336 (2014) · Zbl 1285.49012 · doi:10.1214/13-AAP925 |
| [16] | Henry-Labordere, P., Touzi, N.: An explicit martingale version of Brenier’s theorem (2013). Preprint arXiv:1302.4854 · Zbl 1346.60058 |
| [17] | Hobson, D.G.: Robust hedging of the lookback option. Finance Stoch. 2, 329-347 (1998) · Zbl 0907.90023 · doi:10.1007/s007800050044 |
| [18] | Hobson, D. G., The Skorokhod embedding problem and model independent bounds for option prices (2010), Berlin |
| [19] | Hobson, D.G., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611-649 (2012) · Zbl 1262.91134 · doi:10.1007/s00780-012-0190-3 |
| [20] | Hobson, D.G., Laurence, P., Wang, T.-H.: Static-arbitrage upper bounds for the prices of basket options. Quant. Finance 5, 329-342 (2005) · Zbl 1134.91425 · doi:10.1080/14697680500151392 |
| [21] | Hobson, D.G., Neuberger, A.: Robust bounds for forward-start options. Math. Finance 22, 31-56 (2008) · Zbl 1278.91162 · doi:10.1111/j.1467-9965.2010.00473.x |
| [22] | Kahale, N.: Model-independent lower bound on variance swaps. Available at SSRN 1493722 (2011) · Zbl 1360.91142 |
| [23] | Kantorovich, L.V.: Lecture to the memory of Alfred Nobel: mathematics in economics: achievements, difficulties, perspectives (1975). www.nobelprize.org/nobel_prizes/economics/laureates/1975/kantorovich-lecture.html · Zbl 0346.90001 |
| [24] | Rüschendorf, L.: Monge-Kantorovich transportation problem and optimal couplings. Jahresber. Dtsch. Math.-Ver. 109, 113-137 (2007) · Zbl 1132.60019 |
| [25] | Strassen, V., Almost sure behavior of sums of independent random variables and martingales, 315-343 (1967), Berkeley, CA · Zbl 0201.49903 |
| [26] | Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2009) · Zbl 1156.53003 · doi:10.1007/978-3-540-71050-9 |
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