dstefan
07-01-2008, 05:35 PM
Chapter 4, Problem 9
(i) Consider an at-the-money call on a non--dividend paying asset; assume the Black-Scholes framework. Show that the Delta of the option is always greater than 0.5.
(ii) If the underlying asset pays dividends at the continuous rate q, when is the Delta of an at-the-money call less than 0.5?
Solution:
(i) Recall that the Delta of a call option is given by
\Delta(C) ~=~ e^{-q T} N(d_1) ~=~ e^{-q T} N \left( \frac{\ln \left( \frac{S}{K} \right) + \left(r-q+\frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}} \right)
For an at-the-money call on a non--dividend paying asset, i.e., for K=S and q=0, we find that
\Delta(C) ~=~ N(d_1) ~=~ N\left( \frac{\left(r+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right) ~\geq~ N(0) ~=~ 0.5
(ii) If the underlying asset pays dividends at the continuous rate q, the Delta of an ATM call is
\Delta(C) ~=~ e^{-q T} N(d_1) ~=~ e^{-q T} N\left( \frac{\left(r-q+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right)
For a fixed risk-free rate r and fixed maturity T, we conclude that \Delta(C) < 0.5 if and only if the dividend yield q and the volatility \sigma of the underlying asset satisfy the condition N\left( \frac{\left(r-q+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right) ~<~ 0.5~e^{q T}
This happens, for example, if r = q and T is large enough, since
\lim_{T \to \infty} N\left( \frac{\sigma \sqrt{T}}{2} \right) ~=~ 1 and \lim_{T \to \infty} 0.5~e^{q T} ~=~ \infty
(i) Consider an at-the-money call on a non--dividend paying asset; assume the Black-Scholes framework. Show that the Delta of the option is always greater than 0.5.
(ii) If the underlying asset pays dividends at the continuous rate q, when is the Delta of an at-the-money call less than 0.5?
Solution:
(i) Recall that the Delta of a call option is given by
\Delta(C) ~=~ e^{-q T} N(d_1) ~=~ e^{-q T} N \left( \frac{\ln \left( \frac{S}{K} \right) + \left(r-q+\frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}} \right)
For an at-the-money call on a non--dividend paying asset, i.e., for K=S and q=0, we find that
\Delta(C) ~=~ N(d_1) ~=~ N\left( \frac{\left(r+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right) ~\geq~ N(0) ~=~ 0.5
(ii) If the underlying asset pays dividends at the continuous rate q, the Delta of an ATM call is
\Delta(C) ~=~ e^{-q T} N(d_1) ~=~ e^{-q T} N\left( \frac{\left(r-q+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right)
For a fixed risk-free rate r and fixed maturity T, we conclude that \Delta(C) < 0.5 if and only if the dividend yield q and the volatility \sigma of the underlying asset satisfy the condition N\left( \frac{\left(r-q+\frac{\sigma^2}{2}\right) \sqrt{T}}{\sigma} \right) ~<~ 0.5~e^{q T}
This happens, for example, if r = q and T is large enough, since
\lim_{T \to \infty} N\left( \frac{\sigma \sqrt{T}}{2} \right) ~=~ 1 and \lim_{T \to \infty} 0.5~e^{q T} ~=~ \infty