Hedging with transient price impact for non-covered and covered options

Continuous-time duality for superreplication with transient price impact

“…This together with (11) and the ε/3-bound on the fluctuations of P from the definition of A N,σ ±,n gives that g + P − M N,σ satisfies the required bound (10) Q N,σ -a.s. for N > N 0 (σ).…”

“…. , N , Q N,σ -a.s., as follows readily by induction using the definition of A N,σ ±,n and the Lipschitz continuity of g. Theorem 4.4 in [21] in conjunction with (11) yields that, as N ↑ ∞, the distribution of Z N,σ…”

“…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.…”