In this chapter, we have shown some practical problems that arise in the hedging of derivatives. Although the Black-Scholes-Merton model assumes continuous time trading, resulting in continuous rebalancing of the hedging portfolio without transaction costs, in reality, trading occurs in discrete time, and it does have costs. Consequently, the cost of hedging depends on the future path of the spot price of the underlying asset; thus, it is not a single value presented by the analytical formula any more, but it is a stochastic variable that can be described by its probability distribution. In this chapter, we simulated different paths, calculated the cost of hedging, and presented the probability distribution assuming different rebalancing frequencies. We received that in the absence of transaction costs the volatility reduces with the shortening of the rebalancing period. On the other hand, transaction costs can boost not only the expected value of the cost of the hedge but also its variance. We presented several optimization algorithms to find the optimal hedging strategy.

We created several user-defined functions in R to simulate price movements and to generate the cost distribution. Finally, we applied numerical optimization according to the given optimization model.