5.17.2005

Q&A: How to create a delta-neutral porfolio with...

...call options and stocks given in this Q&A?

CppQuant Answer

  • S0 = $100, X = $100, r = 0.06, T = 1, and stderr = 0.1. The call price is $7.46 and the call delta N(d1) = 0.7422.
  • Consider a portfolio, P, of a short position of one call on the stock, combined with a long position of 0.7422 units of the stock
  • The portfolio would have the value: P = -c + N(d1) S0 = -7.46 + 0.7422 x 100 = $66.76.
    • If the stock price were up by $1, the portfolio's value would be -8.21 + 0.7422 x 101 = $66.7522.
    • if the stock price were up by $10, the portfolio's value would be -16.09 + 0.7422 x 110 = $65.552.
    • If the change in the stock price were larger, the change of the portfolio would be larger, but it would still be quite small relative to the change in the stock price.
    • If the change in the stock price were infinitesimal, the price of the portfolio would not change at all.

Bonus Points

  • the delta of this portfolio is zero: the value of the portfolio is insensitive to the value of the stock.
  • Investors often use delta to construct hedges to offset the risk they have assumed by buying and selling options. For example, if a dealer sells 1,000 call options discussed above, he would buy 742 shares of the stock to construct a delta-neutral portfolio.
  • An option is delta-hedged when a position has been taken in the underlying which matches its delta. Such a hedge is only effective instantaneously, because the option's delta is itself altered by changes in the price of the underlying, interest rates, the option's volatility and time to expiry.
  • A delta-hedge must, therefore, be rebalanced continuously to be effective. aka dynamic hedging.

Category: C++ Quant > Portfolio

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