Asset or Nothing Options' Greeks under Black Scholes

We derive the formulae for the Greeks/derivatives of a digital or binary option that pays one unit of asset or nothing.

Vega

We now differentiate the Asset or Nothing digital option price formula with respect to \( \sigma \), the volatility of the underlying, in order to derive a formula for the Vega:

$$\frac {\partial Price} {\partial \sigma} =\frac {\partial } {\partial \sigma} \left(S e^{- r_{f} \tau} N{\left( \phi d_{1} \right )}\right) $$ $$=S e^{- r_{f} \tau} \frac{\partial}{\partial \sigma} N{\left( \phi d_{1} \right )}$$ $$=\phi S e^{- r_{f} \tau} n{\left (\phi d_{1} \right )} \frac{\partial}{\partial \sigma} d_{1}$$

Now, by definition:

$$ \frac{\partial}{\partial \sigma} \left( d_{1} \right)=\frac{\partial}{\partial \sigma} \left( \frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right) \right)$$ $$=\frac{\partial}{\partial \sigma} \left( \frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} \right) \tau}\right) + \frac{1}{\sigma \sqrt{\tau}} \left( \frac{\sigma^{2}}{2}\tau \right) \right)$$ $$=\frac{\partial}{\partial \sigma} \left( \frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} \right) \tau}\right) + \frac{\sigma \sqrt{\tau}}{2} \right)$$ $$=-\frac{1}{ {\sigma}^2 \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} \right) \tau}\right) + \frac{\sqrt{\tau}}{2} $$ $$=-\frac{1}{ {\sigma}^2 \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f}+\frac{\sigma^{2}}{2} \right) \tau}\right) +\frac{1}{ {\sigma}^2 \sqrt{\tau}} \frac{\sigma^{2} \tau}{2}+ \frac{\sqrt{\tau}}{2} $$ $$=-\frac{d_{1}}{\sigma} + \frac{\sqrt{\tau}}{2} + \frac{\sqrt{\tau}}{2} = -\frac{d_{1}}{\sigma} +\sqrt{\tau} $$ $$=-\frac{d_{1}-\sigma \sqrt{\tau}}{\sigma} =-\frac{d_{2}}{\sigma} $$

Then

$$\frac {\partial Price} {\partial \sigma}= \phi S e^{- r_{f} \tau} n{\left (\phi d_{1} \right )} \frac{\partial}{\partial \sigma} d_{1}$$ $$=-\phi S e^{- r_{f} \tau} n{\left (\phi d_{1} \right )}\frac{d_{2}}{\sigma}$$ $$=- \frac{\phi S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )} $$