Monday, January 28, 2013

Calculating your portfolio's delta

Because deltas are nothing but (partial) differentiation operators, and differentiation operators add in the natural way, deltas also add in the natural way. Thus, the delta of a portfolio is the sum of the deltas of the pieces of the portfolio:

\[ \delta_{P}=\sum_{i}w_{i}\delta_{i} \]

Here, $\delta_{p}$ is the delta of the portfolio, $w_{i}$ are the weights of the components of the portfolio, and $\delta_{i}$ are the deltas of the components of the portfolio.

For example, suppose you have a portfolio made up of a stock and call and put options on the stock, as follows:

  • 25 percent in the underlying stock.
  • 40 percent in call options on the underlying stock, each with a delta of 0.6.
  • 35 percent in put options on the underlying stock, each with a delta of -0.3.

Because the delta of a stock is 1, the delta of your portfolio is:

0.25 * 1 + 0.4 * 0.6 + 0.35 * (-0.3)



In other words, using the implications of the definition of delta, as noted here, your portfolio moves 38.5 percent of the move in the underlying stock.


Additional material: To understand the essence of delta hedging, read this.

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