Hedge impact shifts a call's strike by λN

A hedge that moves the settlement print can change the option payoff it is meant to cover. Itkin and Sánchez-Betancourt price that feedback. For a short call book of N contracts, the effective strike becomes K − λN, with λ denoting permanent price impact per share. The strike shift is simple. The continuous-time existence result behind it applies only when execution costs sit exactly at the midpoint.

The hedge sets the payoff

The observed price equals the fundamental price plus λ times the hedger's inventory. At maturity T, that impacted price fixes the payoff; the hedge is unwound before cash settlement at T+. The authors point to 0DTE options, which accounted for 59% of S&P 500 index option volume in 2025, averaging 2.3 million contracts a day. Cboe's end-of-month S&P 500 options supply their example of the timing split: closing prices fix the payoff, and cash arrives the next business day. A replicating strategy must have a liquidation value equal to the payoff at the price its own trades produced. Its target moves as it hedges.

The construction begins in a one-period tree, where replication becomes a scalar fixed-point equation. Any Lipschitz payoff has a solution, and every solution holds at most L_V shares; L_V is the payoff's Lipschitz constant. For monotone payoffs, the solution is unique when L_V times the impact's Lipschitz constant falls below the up-down price spread. The threshold is exact. At equality, with K at the up-state price, every position in [0, N] replicates. The multiperiod construction works backward through the recombining fundamental tree, even when the impacted-price tree does not recombine. In continuous time, pricing becomes a nonlinear partial differential equation (PDE) with a terminal condition given by an implicit ODE. Solving that ODE yields Ṽ, the payoff adjusted for the hedge's own impact. The PDE then takes Ṽ as its terminal value.

The terminal date is the departure from the closest work the paper names, Bouchard, Loeper and Zou (2016), which characterises permanent-impact superhedging through a different nonlinear PDE. Itkin and Sánchez-Betancourt separate fundamental price dynamics from impact and fix the payoff before liquidating the terminal hedge. That timing creates the implicit terminal ODE. It matters to a desk hedging into the print.

Where does the proof hold?

For monotone, convex, Lipschitz payoffs, including calls and puts, the paper proves PDE well-posedness and exact replication in the midpoint regime 2φ = λ. Execution there occurs halfway between the fundamental and impacted prices. The cost lies between the frictionless price of Ṽ and that price plus ½λL_V²(1 − e^(−rT)). For calls and puts, L_V = |N|, making the upper bound grow with N². Holdings never exceed one share per contract. The conditions r ≥ 0 and 2φ ≥ λ rule out price manipulation; both are also necessary in continuous time.

A digital call exposes the limit even in the one-period tree. Under linear impact, strikes in an interval of width Nλ/(S_T(u) − S_T(d)) above the down-state price cannot be replicated. The interval is capped at the up-state price, for N > 0. As the paper says, "the replicating attempt itself causes the hedger to miss the mark."

Strike shift and size

For a call, Ṽ shifts the strike to K̃ = K − λN. At midpoint costs, delta jumps from 0 to N at K̃. When 2φ > λ, holdings instead rise linearly across (K − 2φN, K − λN), so inventory starts building before the shifted strike. With the Figure 6 inputs K = 100, λ = 0.1 and φ = 0.06, the formula places that ramp at 99.88 to 99.90 for one contract and 99.76 to 99.80 for two.

Size changes the result sharply. At midpoint costs, per-contract price and hedge depend on N and λ through Nλ alone. In the paper's quadratic example, N = 1, λ = 0.1 matches N = 10, λ = 0.01 exactly. In the call runs, φ stays at 0.06. There, increasing N tenfold moves per-contract values more than increasing λ tenfold. Our reading is that N raises both effective parameters the paper identifies, Nλ and N(2φ − λ). Raising λ from 0.01 to 0.1 while holding φ at 0.06 raises Nλ but reduces N(2φ − λ) from 0.11N to 0.02N. Every case shown has a hedge above the Black-Scholes delta.

Those call curves need a qualification. At φ = 0.06, either λ = 0.1 or λ = 0.01 gives 2φ > λ, outside the regime covered by Theorem 4.7. The conclusion confines well-posedness to 2φ = λ. For 2φ > λ, the authors give a verification theorem conditional on a smooth solution, solve the terminal ODE and provide a closed-form quadratic example. They give no existence proof for calls in that regime. The call curves are numerical PDE solutions where existence remains unproved.

Nine paths driven across the strike

In one illustration, 9 of 1,000 simulated paths ended with S_T < K < P_T. The hedger's buying alone made the option pay. Nine in a thousand is 0.9%, with a binomial standard error of about 0.3 points. This is a simulation result at the paper's chosen λ and σ.

The model places no randomness between T and T+, leaving no price risk when the terminal hedge is unwound. Only the hedger has impact, and nobody trades against predictable settlement flow. Figure 2 explains its options-book shape with a one-period tree: S0 = 100, up 105, down 95, λ ∈ {0.1, 0.2, 0.3}, execution cost φ√|x| and φ = 0.5λ. The option price is concave in N because the underlying execution cost is specified as square-root. We did not find a comparison with observed option quotes.

We could not backtest this. Estimating λ and φ near the close requires order-book depth and trade prints; our data contain minute bars and end-of-day option prices. History never shows the settlement price that would have prevailed without your own flow, so it cannot recover the counterfactual behind those nine paths. A real test would require closing-auction depth to estimate λ and executable option quotes by size for comparison with the Figure 2 shape.

For now, I would use it as a stress calculation. Take a desk estimate of λ, move the strike to K − λN, and compare the midpoint-regime bound ½λN²(1 − e^(−rT)) above the frictionless price of Ṽ with the trade's edge. If closing-auction calibration put λN well inside a tick at typical size, I would stop running it.