Self-financing strategy expression in general shape limit order book with market impacts in continuous time

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In this paper, we extend the self-financing strategy expression in the linear supply curve model with market impacts by Roch (2011) and Cetin et al. (2004) to a non-linear case. Option hedging with liquidity costs and market impacts has been a key issue since the financial crises. We generalize the continuous time expression of a self-financing strategy in the linear supply curve model with market impacts proposed by Roch (2011) and Cetin et al. (2004), which is a useful result when one considers an option hedging strategy under an illiquid market, to a non-linear case. After showing an expression of the maximum price to when a hedger buys a certain amount under the non-linear supply curve, we define a non-linear market impact and show the self-financing expression under the non-linear supply curve model with market impacts. We also show examples of the strategy in non-linear supply curves observed in practice.