Option pricing with an illiquid underlying asset market. (English) Zbl 1198.91210

Liu, Hong; Yong, Jiongmin

J. Econ. Dyn. Control 29, No. 12, 2125-2156 (2005).

Summary: We examine how price impact in the underlying asset market affects the replication of a European contingent claim. We obtain a generalized Black-Scholes pricing PDE and establish the existence and uniqueness of a classical solution to this PDE. Unlike the case with transaction costs, we prove that replication with price impact is always cheaper than superreplication. Compared to the Black-Scholes case, a trader generally buys more stock and borrows more (shorts and lends more) to replicate a call (put). Furthermore, price impact implies endogenous stochastic volatility and an out-of-money option has lower implied volatility than an in-the-money option. This finding has important implications for empirical analysis on volatility smile.

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
91G80	Financial applications of other theories

Keywords:

price impact; option pricing; illiquidity; volatility smile

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