Delta Hedging:

A single option or a portfolio of options can be hedged by taking positions on the spot market of the underlying asset. The delta of the option portfolio is the amount of the spot position that the portfolio generates.

For example if there is a portfolio of currency options on EUR/USD and delta of the portfolio is EUR 500,00. This means that the investor holding the option portfolio has a long position of EUR 500,000 on EUR/USD currency. If the investor wants to hedge the position, he should sell EUR 500,000 at the spot market. This action makes the portfolio delta neutral against the small changes on the spot price of EUR/USD.

On the other hand for big changes in the spot market this hedge will not be enough. Because portfolio’s delta is also affected by the changes in the spot price. This changes are measured with the gamma greek.

In the previous example if the gamma of the portfolio is EUR 100,000 then when spot increases by % 1, delta of the portfolio becomes EUR 600,000. The investor should sell EUR 100,000 from the new spot level to rebalence the portfolio.

In theoraticaly if investor buys an option and hedges dynamicaly through the life of the option and the realized volatility becomes the same as the implied volatility which is paid while buying the option then, p&l generated by delta hedging will be equal to the options value at the expiry date minus the premium paid to the option.
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