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, binary options arbitrage we note that they have the same shape, but they are not the same.
However, if we take the volatility skew into consideration, the above conclusion does not hold. Specifically, \begin D(S_0, binary options brokers usa reviews 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).

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