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Option pricing with linear market impact and nonlinear Black-Scholes equations. (English) Zbl 1417.91511
Summary: 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 [H. M. Soner and N. Touzi, SIAM J. Control Optim. 39, No. 1, 73–96 (2000; Zbl 0960.91036); Math. Finance 17, No. 1, 59–79 (2007; Zbl 1278.91151); with P. Cheridito, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 5, 633–666 (2005; Zbl 1078.91010)], with the problem of hedging under liquidity costs [U. Çetin et al., Finance Stoch. 14, No. 3, 317–341 (2010; Zbl 1226.91072)] are discussed. The optimal strategy and associated diffusion are related with the second-order target problems of [H. M. Soner et al., Ann. Appl. Probab. 23, No. 1, 308–347 (2013; Zbl 1293.60063)], and with the solutions of optimal transport problems by diffusions of [X. Tan and N. Touzi, Ann. Probab. 41, No. 5, 3201–3240 (2013; Zbl 1283.60097)].
We also derive a modified Black-Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
35K55 Nonlinear parabolic equations
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