OPTIMAL HEDGING OF DERIVATIVES WITH TRANSACTION COSTS
We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a HamiltonâJacobiâBellman equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("BlackâScholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a BlackâScholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a BlackâScholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.
Growth optimal criteria, transaction costs, Black and Scholes
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