Perfect Hedging with Endogenous Permanent Market Impact

Summary

The document develops a market model in which quoted prices follow a nonlinear curve derived from the utility indifference of a representative liquidity supplier. The supplier’s utility is represented through a g-expectation. Because trades change the supplier’s inventory, each trade can permanently affect prices, so traders cannot assume they are price takers. Strategy profit and loss is consequently represented by a nonlinear stochastic integral.

Within this framework, the authors give a completeness condition that permits any derivative to be replicated exactly by dynamic trading. In a Markovian special case, the associated pricing and hedging problem can be solved through a semilinear partial differential equation. The excerpt states the theoretical structure and replication result but does not specify the condition, utility assumptions, or examples. It therefore offers no empirical evidence about how well the model describes actual markets or how practical perfect replication is under real trading constraints.

Key ideas

• Prices are modeled through the utility indifference curve of a liquidity supplier.
• Trades have permanent price impact because they alter the supplier’s inventory.
• Trading profit and loss is expressed using a nonlinear stochastic integral.
• A stated completeness condition allows derivatives to be replicated by dynamic strategies.
• In a Markovian setting, pricing and hedging reduce to a semilinear PDE.

Tags

# Perfect hedging under endogenous permanent market impacts


# Perfect hedging under endogenous permanent market impacts









We model a nonlinear price curve quoted in a market as the utility indifference curve of a representative liquidity supplier. As the utility function we adopt a g-expectation. In contrast to the standard framework of financial engineering, a trader is no more price taker as any trade has a permanent market impact via an effect to the supplier's inventory. The P&L of a trading strategy is written as a nonlinear stochastic integral. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. In the special case of a Markovian setting the corresponding pricing and hedging can be done by solving a semi-linear PDE.

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