Greeks for binary option?
How to derive an analytic formula of greeks for binary option?
We know a vanilla option can be constructed by an asset-or-nothing call and a cash-or-nothing call, does that help us?
Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.
Does that mean the delta of a binary call is also the gamma of a vanilla call? Can we use the analytical formula for gamma of vanilla call for binary option?
3 Answers 3.
For a digital option with payoff $1_ $, note that, for $\varepsilon > 0$ sufficiently small, \begin 1_ &\approx \frac .\tag \end That is, The value of the digital option \begin D(S_0, T, K, \sigma) &= -\frac , \end where $C(S_0, T, K, \sigma)$ is the call option price with payoff $(S_T-K)^+$. Here, we use $d$ rather than $\partial$ to emphasize the full derivative.
If we ignore the skew or smile, that is, the volatility $\sigma$ does not depend on the strike $K$, then \begin D(S_0, T, K, \sigma) &= -\frac \\ &= N(d_2)\\ &= N\big(d_1-\sigma \sqrt \big). \tag \end That is, the digital option price has the same shape as the corresponding call option delta $N(d_1)$. Similarly, the digital option delta $\frac )> $ has the same shape as the call option gamma $\frac $. Here, we note that they have the same shape, but they are not the same .
However, if we take the volatility skew into consideration, Binary options the above conclusion does not hold. Specifically, \begin D(S_0, T, K, \sigma) &= -\frac \\ &= -\frac - \frac \frac \\ &= N(d_2) - \frac \frac ,\tag \end which may not have the same shape as $N(d_2)=N(d_1-\sigma \sqrt )$. In this case, we prefer to value the digital option using the call-spread approximation given by (1) above instead of the analytical formula (2) or (3).