Hedging Strategies Under Temporary Market Impact - Brackmann, H (Oxford University;Mathematical Institute, 2013)

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This thesis is concerned with the problem of hedging derivatives under temporary market impact. We are going to examine the problem of hedging derivatives in an optimal control setting similar to [14] and [10]. By considering specific combinations of volatility and liquidity specifications, we are able to obtain analytic solutions. Under the optimal control, the holdings in the stock exhibit a mean reverting behavior. While the reversion-speed is similar to the results in the literature, we are able to find a new characterization of the target portfolio as weighted average of future Δ-hedge portfolios. If the stock-price process is given by a Brownian motion, this characterization collapses to the Δ-hedge. In that case, the final hedging error is proportional to the variance of the Δ-hedge. Further, we extend the analysis by including drift using asymptotic expansions. We find that under the optimal control, the wealth process is mean-reverting towards the expectation of the derivative. The hedging portfolio adjusts more to the inclusion of drift if future hedging is less aggressive. The comparison with numerical solutions suggests that the approximations perform reasonably well.