Continuous-time duality for superreplication with transient price impact. (English) Zbl 1444.91194

The authors consider a financial model with large investor who can invest in a riskless savings account bearing zero interest and whose trades into and out of a risky asset move bid- and ask prices that, in addition, are also driven by some exogenous noise, which is specified by a continuous, adapted process on a filtered probability space \((\Omega,(\mathcal{F}_{t})_{t\ge0},\mathrm{P})\). It is assumed that all \((\mathcal{F}_{t})_{t\ge0}\)-martingales have a continuous version. The large investor’s trading strategy is described by his given initial holdings \(x_0\in\mathrm{R}\) and a right-continuous, predictable process \(X=({X}_{t})_{t\ge0}\) of bounded variation specifying the number of risky assets held at any time. Let \(X^{\uparrow}\) and \(X^{\downarrow}\) be the right-continuous predictable increasing and decreasing part resulting from the Hahn-decomposition of \(X_{t}=x_0+X^{\uparrow}_{t}-X^{\downarrow}_{t}\), \(t\geq 0\), \(X_{0-}=x_0\), \(X^{\uparrow}_{0-}=X^{\downarrow}_{0-}=0\). It is assumed that the half-spread follows the dynamics \(d\zeta_{t}^{X}=(dX^{\uparrow}_{t}+dX^{\downarrow}_{t})/\delta_{t}-r_{t}\zeta_{t}^{X}dt\), \(\zeta_{0-}^{X}=\zeta_0\geq 0\) for a given market depth process \(\delta\) and resilience rate \(r\). The authors assume that the market depth is continuous and adapted, the resilience rate is predictable and such that \(\delta\) and \(\rho\) are bounded away from zero and infinity, where \(\rho_{t}=\exp\left(\int_{0}^{t}r_{s}ds\right)\), \(t\ge0\), and \(\kappa_{t}=\delta_{t}/\rho^2_{t}\) is strictly decreasing in \(t\geq 0\). By time \(T\in (0,\infty)\), the induced investor’s cash position will have evolved from its given initial value \(\xi_0\in\mathrm{R}\) to the terminal cash position \(\xi_{T}^{X}=\xi_0-\int_{[0,T]}P_{t}^{X}\circ dX_{t}-\int_{[0,T]}\zeta_{t}^{X}\circ d(X_{t}^{\uparrow}+X_{t}^{\downarrow})\), where \(P^{X}\) is the mid-price. It is considered the classical superreplication problem for a cash-settled European contingent claim with \(\mathcal{F}_{T}\)-measurable payoff \(H\geq 0\) at time \(T\geq 0\). An exogenous payoff’s superreplication cost is defined by \(\pi(H)=\inf\{\xi_0\in\mathrm{R}:\ \xi_{T}^{X}\geq H\) for some \(X\in\mathcal{X}\) with \(X_{T}=0\}\). The main result of this paper is following. Under previous assumptions the superreplication costs of contingent claim \(H\geq 0\) have the dual description \(\pi(H)=\sup_{(Q,M,\alpha)}\left\{\mathrm{E}_{Q}[H]-\frac{1}{2}\Vert\alpha-\zeta_0\Vert_{L^2(Q\otimes\mu)}-M_0 x_0-\frac{1}{2}\iota x^2_0\right\}>-\infty\), where the supremum is taken over all triples \((Q,M,\alpha)\) of probability measures \(Q<<\mathrm{P}\) on \(\mathcal{F}_{T}\), martingales \(M\in\mathcal{M}^2(Q)\) and all optimal \(\alpha\in L^2(Q\otimes\mu)\) which control the fluctuations of \(P\) in the sense that \(\vert P_{t}-M_{t}\vert\leq\frac{\rho_{t}}{\delta_{t}}\mathrm{E}_{Q}\left[\int_{[0,T]}\alpha_{u}\mu(du)\vert\mathcal{F}_{t}\right]\), \(0\leq t\leq T\). Here \(\mu(dt)=\mathrm{1}_{(0,T)}(t)\vert d\kappa_{t}\vert+\kappa_{T}\mathrm{Dirac}_{T}(dt)\), \(\iota\) is some impact parameter. As an application, the authors show that in proposed transient price impact model the best way to superreplicate a call option is, under natural conditions, to buy and hold the asset until maturity, and it is proved a verification theorem for utility maximizing investment strategies.


91G10 Portfolio theory
91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI arXiv Euclid


[1] Alfonsi, A., Fruth, A. and Schied, A. (2010). Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10 143-157. · Zbl 1185.91199
[2] Almgren, R. and Chriss, N. (2001). Optimal execution of portfolio transactions. J. Risk 3 5-39.
[3] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1-18. · Zbl 1119.91040
[4] Bank, P. and El Karoui, N. (2004). A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Probab. 32 1030-1067. · Zbl 1058.60022
[5] Bank, P. and Fruth, A. (2014). Optimal order scheduling for deterministic liquidity patterns. SIAM J. Financial Math. 5 137-152. · Zbl 1308.91198
[6] Bank, P. and Kauppila, H. (2017). Convex duality for stochastic singular control problems. Ann. Appl. Probab. 27 485-516. · Zbl 1360.93765
[7] Bank, P. and Riedel, F. (2001). Optimal consumption choice with intertemporal substitution. Ann. Appl. Probab. 11 750-788. · Zbl 1022.90045
[8] Bank, P. and Voß, M. (2018). Optimal investment with transient price impact. Available at arXiv:1804.07392. · Zbl 1429.91302
[9] Bayraktar, E. and Yu, X. (2019). Optimal investment with random endowments and transaction costs: Duality theory and shadow prices. Math. Financ. Econ. 13 253-286. · Zbl 1410.91409
[10] Becherer, D. and Bilarev, T. (2018). Hedging with transient price impact for non-covered and covered options. Available at arXiv:1807.05917. · Zbl 1403.35330
[11] Becherer, D., Bilarev, T. and Frentrup, P. (2019). Stability for gains from large investors’ strategies in \(M_1/J_1\) topologies. Bernoulli 25 1105-1140. · Zbl 1459.60121
[12] Bouchard, B., Loeper, G. and Zou, Y. (2017). Hedging of covered options with linear market impact and gamma constraint. SIAM J. Control Optim. 55 3319-3348. · Zbl 1415.91278
[13] Bouchard, B. and Touzi, N. (2000). Explicit solution to the multivariate super-replication problem under transaction costs. Ann. Appl. Probab. 10 685-708. · Zbl 1083.91510
[14] Campi, L. and Schachermayer, W. (2006). A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch. 10 579-596. · Zbl 1126.91024
[15] Chiarolla, M. B. and Ferrari, G. (2014). Identifying the free boundary of a stochastic, irreversible investment problem via the Bank-El Karoui representation theorem. SIAM J. Control Optim. 52 1048-1070. · Zbl 1298.91118
[16] Cont, R., Kukanov, A. and Stoikov, S. (2014). The price impact of order book events. J. Financ. Econom. 12 47-88.
[17] Cvitanić, J. and Karatzas, I. (1996). Hedging and portfolio optimization under transaction costs: A martingale approach. Math. Finance 6 133-165. · Zbl 0919.90007
[18] Czichowsky, C. and Schachermayer, W. (2016). Duality theory for portfolio optimisation under transaction costs. Ann. Appl. Probab. 26 1888-1941. · Zbl 1415.91258
[19] Czichowsky, C. and Schachermayer, W. (2017). Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion. Ann. Appl. Probab. 27 1414-1451. · Zbl 1414.91336
[20] Davis, M. H. A. and Norman, A. R. (1990). Portfolio selection with transaction costs. Math. Oper. Res. 15 676-713. · Zbl 0717.90007
[21] Duffie, D. and Protter, P. (1992). From discrete- to continuous-time finance: Weak convergence of the financial gain process. Math. Finance 2 1-15. · Zbl 0900.90046
[22] Ferrari, G. (2015). On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Probab. 25 150-176. · Zbl 1307.93455
[23] Gatheral, J., Schied, A. and Slynko, A. (2012). Transient linear price impact and Fredholm integral equations. Math. Finance 22 445-474. · Zbl 1278.91061
[24] Gerhold, S., Muhle-Karbe, J. and Schachermayer, W. (2013). The dual optimizer for the growth-optimal portfolio under transaction costs. Finance Stoch. 17 325-354. · Zbl 1319.91142
[25] Guasoni, P. (2002). Optimal investment with transaction costs and without semimartingales. Ann. Appl. Probab. 12 1227-1246. · Zbl 1016.60065
[26] Guasoni, P. and Rásonyi, M. (2015). Hedging, arbitrage and optimality with superlinear frictions. Ann. Appl. Probab. 25 2066-2095. · Zbl 1403.91311
[27] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18 491-520. · Zbl 1133.91422
[28] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6 157-191. · Zbl 1239.91190
[29] Huberman, G. and Stanzl, W. (2004). Price manipulation and quasi-arbitrage. Econometrica 72 1247-1275. · Zbl 1141.91450
[30] Jakubėnas, P., Levental, S. and Ryznar, M. (2003). The super-replication problem via probabilistic methods. Ann. Appl. Probab. 13 742-773. · Zbl 1029.60052
[31] Jarrow, R. (1994). Derivative securities markets, market manipulation and option pricing theory. Journal of Financial and Quantitative Analysis 29 241-261.
[32] Jouini, E. and Kallal, H. (1995). Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory 66 178-197. · Zbl 0830.90020
[33] Kabanov, Y. M. and Stricker, C. (2002). Hedging of contingent claims under transaction costs. In Advances in Finance and Stochastics 125-136. Springer, Berlin. · Zbl 1016.91043
[34] Kabanov, Y. M. (1999). Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3 237-248. · Zbl 0926.60036
[35] Kallsen, J. and Muhle-Karbe, J. (2010). On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20 1341-1358. · Zbl 1194.91175
[36] Komiya, H. (1988). Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11 5-7. · Zbl 0646.49004
[37] Kusuoka, S. (1995). Limit theorem on option replication cost with transaction costs. Ann. Appl. Probab. 5 198-221. · Zbl 0834.90049
[38] Levental, S. and Skorohod, A. V. (1997). On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7 410-443. · Zbl 0883.90018
[39] Obizhaeva, A. A. and Wang, J. (2013). Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets 16 1-32.
[40] Predoiu, S., Shaikhet, G. and Shreve, S. (2011). Optimal execution in a general one-sided limit-order book. SIAM J. Financial Math. 2 183-212. · Zbl 1222.91062
[41] Schachermayer, W. (2014). The super-replication theorem under proportional transaction costs revisited. Math. Financ. Econ. 8 383-398. · Zbl 1309.91136
[42] Schachermayer, W. (2017). Asymptotic Theory of Transaction Costs. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich. · Zbl 1422.91008
[43] Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 609-692. · Zbl 0813.60051
[44] Soner, H. M., Shreve, S. E. and Cvitanić, J. (1995). There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 327-355. · Zbl 0837.90012
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