5.15.2005

Q&A: What's the Delta of a call option on...

...a stock trading at $100. The continuously compounding risk-free interest rate is 6%. The option has an exercise price of $100 and expires in one year. The standard deviation of the stock's returns is 0.1.

CppQuant Answer

Delta = change in option price / change in underlying price, which can be approximated from the Black-Scholes-Merton formula as the value of N(d1) for calls and N(d1) - 1 for puts.

  • d1 = ( ln(S0/X) + (r+stderr^2/2)*T) / (stderr*T^1/2) = ( ln(100/100) + (0.06 + 0.12/2)*1) / (0.1*1^(1/2)) = 0.65
  • N(d1): In Excel, Normsdist(0.65) = 0.74
  • For a $1 up(+) in the price of the stock, we expect the new call option price to be up(+) by 0.74.

Bonus Points

  • The delta of a call option is always positive as the value of a call increases with an underlying price increase.
  • The delta of a put option is always negative as the value of a put decreases with an underlying price increase.
    • The put delta = N(d1) - 1 = 0.74 - 1 = -.026.
    • For a $1 up (+) in the price of the stock, we expect the new put option price to be down(-) by 0.26.
  • In general, the larger the move in the underlying price, the worse the approximation.

Category: C++ Quant > Derivatives > Options

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