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October 2018 Option pricing with linear market impact and nonlinear Black–Scholes equations
Gregoire Loeper
Ann. Appl. Probab. 28(5): 2664-2726 (October 2018). DOI: 10.1214/17-AAP1367


We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a nonlinear Black–Scholes equation that provides an exact replication strategy.

This equation is fully nonlinear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms.

The close connections with the problem of hedging with gamma constraints [SIAM J. Control Optim. 39 (2000) 73–96, Math. Finance 17 (2007) 59–80, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 633–666], with the problem of hedging under liquidity costs [Finance Stoch. 14 (2010) 317–341] are discussed. The optimal strategy and associated diffusion are related with the second-order target problems of [Ann. Appl. Probab. 23 (2013) 308–347], and with the solutions of optimal transport problems by diffusions of [Ann. Probab. 41 (2013) 3201–3240].

We also derive a modified Black–Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.


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Gregoire Loeper. "Option pricing with linear market impact and nonlinear Black–Scholes equations." Ann. Appl. Probab. 28 (5) 2664 - 2726, October 2018.


Received: 1 August 2016; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974762
MathSciNet: MR3847970
Digital Object Identifier: 10.1214/17-AAP1367

Primary: 35K55 , 49L20 , 91G20 , 93E20

Keywords: fully nonlinear parabolic equations , hedging , price impact

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 5 • October 2018
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