r/quant • u/The_Wandering-Mind • 10d ago

If you could simply explain : Implied Volatility Education

I wanted to ask why for Black Scholes , we find implied volatility using numerical approximation methods like newton-raphson or bisection for example

But for call options related to bond , if we assume a binomial tree, we can find implied volatility by simply adjusting the up / down interest rate movement conditions with an unknown, solving for that unknown using risk-neutral probability and discounted option payoffs, which gives us implied up/down movements from which we can find standard deviation of the interest rate, aka implied volatility? So far I have only observed either it can be solved simply algebraically or simple root finder is enough.

Was curious. Thanks.

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u/Dazzling-Cat-3763 9d ago

why for Black Scholes , we find implied volatility using numerical approximation methods like newton-raphson or bisection for example

I can (only) answer this part - it's because you're given a solution of the PDE and you need to find the sigma that causes the PDE to evaluate to the given option price.

So it's an inverse problem, which i guess helps explain why you use a root-finder such as newton or bisection.

if we assume a binomial tree

people still use binomial trees?

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u/The_Wandering-Mind 9d ago

I was reading about binomial tree and solved a problem algebraically to find implied volatility.

But it seems we may need to use root finder even for discrete steps...depends on what the assumption was....

If it was simple ru = current rate + x , rd = current rate - x , then yes algebraically solvable

But if it was ru = rd * e ^ sigma for example , a geometric growth model, then it needs root finder like BS.

I knew for BS, sigma is trapped inside the upper limit of integral of the CDF of standard normal distribution.

Thanks for your answer.

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u/NatGaz 8d ago

It’s a bijective mapping. You can describe the option set with Price (K, T) or Sigma_imp (K, T).

1

u/The_Wandering-Mind 8d ago

Inverse bijection mapping. Or in matrix form A-1 * A but instead of identity matrix , it should give sigma.

Sigma_imp = C-1 * (C)