Implied Volatility and Black–Scholes Option Delta

Summary

The document explains how to calculate a European call option’s delta under the Black–Scholes framework. Delta is the sensitivity of the option value to the underlying price and, in this model, equals the cumulative normal probability evaluated at d1. That quantity depends on inputs including the strike, interest rate, time to maturity, and volatility.

When volatility is not known, the answer proposes finding implied volatility by numerically choosing the value that makes the model price match the observed option price. The stated information—time to maturity, option price, and underlying price—is insufficient on its own: strike and interest rate are also needed. This is a model-based method, so the resulting volatility and delta depend on the Black–Scholes assumptions and on having the additional inputs; the document does not discuss numerical solver choices or model limitations in detail.

Key ideas

• Black–Scholes call delta is given by the cumulative normal function evaluated at d1.
• The d1 input depends on volatility, strike, interest rate, underlying price, and time to maturity.
• Implied volatility can be estimated by matching the model price to the observed option price.
• The listed option and underlying prices and maturity alone do not determine delta; strike and interest rate are also required.

Tags

# what is the vol in the BS formula?


# 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.

- what is the formula to get this delta

- 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?

## Answer by SRKX (score 2)

https://quant.stackexchange.com/a/10799

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.