Optimal hedging of Derivatives with transaction costs

Details Optimal hedging of Derivatives with transaction costs Optimal hedging of Derivatives with transaction costs

2005-09-16

arxiv.org/abs/physics/0509150v2

IJTAF Vol 9, No 7 (2006), 1051-1070

Physics

Physics and Society

Revised version, expanded introduction and references 17 pages, submitted to International Journal of Theoretical and Applied

Scientific paper

10.1142/S0219024906003901

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 Hamilton-Bellman-Jacobi 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 ("Black-Scholes") 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 Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.

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