OPTION PRICING WITH LINEAR MARKET IMPACT AND NONLINEAR BLACK-SCHOLES EQUATIONS

• 成果类型: Article
• 署名作者: Loeper, Gregoire
• 署名单位: Monash University; Monash University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1367
• 发表日期: 2018
• 页码: 2664-2726
• 关键词: gamma; RISK

摘要

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. Poincare Anal. Non Lineaire 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.