Delta-Hedging of a European Call Options in Black-Scholes Under the Replicating Portfolio Strategy

Investing money involves different levels of risks depending on the choice of the investment. In finance, an investor is faced with the problems of where and when to invest; ability to regularly and dynamically build his portfolio of investments; ability to find investment strategies that give profits with zero initial expenditures; proper risk management; option pricing; and many more. Delta-hedging, which involves trading financial instruments strategically, helps investors to eliminate or reduce risk associated with option trading. This can be achieved by continuous re-balancing the portfolio of the stock and option to always have after re-balancing, a total delta of zero. Practically, hedging is being done periodically. This work deals with the Delta-hedging of a European Call Options in Black-Scholes under the replicating portfolio strategy. This replicating portfolio contains stocks and money market accounts. We obtain the initial value required to build a trading strategy that produces exact payoff and has similar cash flow as that of the Call Option at any time which is the replicating portfolio. From there, we derive the delta of a European Call Option. The condition of the self-financing trading strategy is satisfied by the replicating strategy. Generally, the payoff from delta-hedging a European option depends on the stock path.