# what is the vol in the BS formula?

I need to compute the delta of an option for which I know a) the time to maturity, b) the price of the option, c) the price of the underlying asset.

1. what is the formula to get this delta
2. It seems that the volatility is a parameter of this formula (yes, I have some clue of the answer of the first question ;). What is this vol? Where can I get it?
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Recall that the delta of an option is the sensitivity of its price to changes in the underlying's stock price:

$$\Delta = \frac{\partial V}{\partial S}$$

Now, if you assume the BS framework, you find that:

$$V(t,T,K,\sigma,r) = S_t \Phi(d_1) - e^{-r(T-t)} K \Phi(d_2)$$

Clearly, $\Delta = \frac{\partial V}{\partial S}= \Phi(d_1)$.

Note that $d_1$ is a function of $S,K,r,\sigma,t$ and $T$.

You have

a) time to maturity

This is $\tau=T-t$.

b) the price of the option

This is $V$.

c) the price of the underlying asset

This is $S_t$.

With only this information I do no think you can solve the problem, but I assume you should have somewhere the strike price $K$ and the interest rate $r$.

If you do, then you can find the implied volatility $\hat{\sigma}$ by solving computationally:

$$\hat{\sigma}=\underset{\sigma}{\arg \min} \left[ \left(S_t \Phi(d_1) - e^{-r(T-t)} K \Phi(d_2)\right) - V \right]^2$$

You can then compute the delta...

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