## 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

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