Delta and the Replicating Portfolio

[colby2152]

Here's some more connection of Greeks to the replicating portfolio. Am I correct to think of these concepts by saying the following? Specifically, using the Black-Scholes formula, Delta is the expected shares of stock to be purchased in the replicating portfolio, so a probability factor must be taken into account.

Also, I have a question concerning portfolio of Delta. The Delta, or any Greek for that matter, of a portfolio is some number. Say the result of Delta of a three option portfolio is -0.42. What can we say about the portfolio? What does this number mean?

[JohnLocke]

Originally Posted by colby2152

Also, I have a question concerning portfolio of Delta. The Delta, or any Greek for that matter, of a portfolio is some number. Say the result of Delta of a three option portfolio is -0.42. What can we say about the portfolio? What does this number mean?

Wouldn't that mean that if the stock went up by one, the portfolio would go down in value by .42?

[colby2152]

Originally Posted by JohnLocke

Wouldn't that mean that if the stock went up by one, the portfolio would go down in value by .42?

Wow, I had a brain fart with that one. Thanks JL!

[DrVanNostrin]

Delta is just the first derivative of option price with respect to stock price. When you graph the option price vs. stock price the result will not be a straight line.

To find the change in option price, given a change in stock price you could use a Taylor series expansion. (The answer JL posted is the result of using only the first term in a Taylor series expansion.) For small changes in stock price this is fine. Using gamma (the second derivative of option price with respect to stock price) and a second term in the Taylor series expansion will give you a better answer.