Hedging with Transient Price Impact for Non-Covered and Covered Options

Abstract

We solve the superhedging problem for European options in a market with finite liquidity where trading has transient impact on prices, and possibly a permanent one in addition. Impact is multiplicative to ensure positive asset prices. Hedges and option prices depend on the physical and cash delivery specifications of the option settlement. For non-covered options, where impact at the inception and maturity dates matters, we characterize the superhedging price as a viscosity solution of a degenerate semilinear pde that can have gradient constraints. The non-linearity of the pde is governed by the transient nature of impact through a resilience function. For covered options, the pricing pde involves gamma constraints but is not affected by transience of impact. We use stochastic target techniques and geometric dynamic programming in reduced coordinates.

Continuous-time duality for superreplication with transient price impact

“…But this would require a rather product-specific analysis and is thus beyond the scope of this duality paper. See, however, the PDE approaches in, e.g., [13,11] as well as [31,3] for some results in this direction. Note also that for a vanilla option, whose payoff only depends on the terminal mid-price at time T , the liquidation constraint X T = 0 ensures that P X T = P T and thus prevents any manipulation possibilities, making our duality result below applicable to these products.…”