While recently interviewing at a prestigious bank, I received a question that I stumbled on. Went something like this: If one knows a stock price, a strike price and the premium, estimate the delta of the option.
Can I estimate other greeks given these information ?
One way to price a call option is to use the Black Scholes formula. $$ \begin{align} C(S_t, t) &= N(d_1)S_t - N(d_2) Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= d_1 - \sigma\sqrt{T - t} \\ \end{align} $$ You know the price $C$, so you solve for $\sigma$, which would be called the implied volatility.
The greeks can obtained by differentiating the Black Scholes formula.
In the Black Scholes model, the delta has a closed form: $$ \Delta = \frac{\partial C}{\partial S} = N(d_1) $$