Finding Implied Volatility from an Option’s Market Price
Summary
Implied volatility is presented as the volatility input that makes a Black–Scholes–Merton option price equal the observed market price. The market quote is interpreted through the model: with the other pricing inputs fixed, implied volatility is the unknown parameter that reconciles theoretical value with the quote.
The described procedure is to gather the inputs associated with the quote, including the underlying price, time to maturity, interest rate, and dividend yield; hold them fixed in the pricing equation; then solve for volatility numerically. Root-finding methods such as Newton’s method or the secant method are suggested because the equation has no closed-form solution for implied volatility. The answer is a concise explanation of the standard calculation, not a discussion of model risk, quote quality, numerical convergence, or how implied volatility should be interpreted under market conditions.
Key ideas
- Implied volatility is the volatility input that makes a model price match the observed option price.
- The underlying price, expiry, interest rate, and dividend yield are among the inputs to fix first.
- With other inputs held constant, solve for volatility by finding the root of the price difference.
- Numerical methods such as Newton’s method or the secant method can find the implied volatility.
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Full text
# How to find IV from market prices accodring to Bergomi
# How to find IV from market prices accodring to Bergomi
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the PDE as an accouting tool, which makes sense to me given the difficulty to model stocks. Anyway, I am just on the 1st chapter, he might have something to say about modelling later.
To the question!
My interpretation of his introduction is that IV, i.e $\hat{\sigma}$, is a number related to breakeven in the PnL which in turn is related to a family of parameterised(in IV) black-scholes equatations, which the price of the option must statisfy(as usual).
He then elaobeates on how to find this IV when there is a deep and liquid market for the option and that, from what I can tell, is done by finding the right BS PDE(within the family) that matches up to market price.
Am I thinking about this the right way?
Here is the book
https://www.lorenzobergomi.com/_files/ugd/c4ff5c_ba17141422d44ba99daf19ee2b931544.pdf
The relevant parts are on pages 2-5. In particular that $(1.4)$ is a family of PDEs in $\sigma$ and that finding the IV correspond the finding which of these that match up with market price as mentioned on page $5$ in the 3rd paragraph
## Answer by LvM_ (score 2)
https://quant.stackexchange.com/a/73741
In short, the implied volatility is the value of $\sigma$ that you input in the Black-Scholes-Merton equation such that the BSM price = Market price.
To find it, you need to use some Numerical procedure (there is no close form solution).
- First, extract all the parameter available (corresponding to your market price quote), e.g. stock price, stock, time to maturity, risk free rate, dividend yield, etc...
- Then you input and fix these parameter into the BSM equation so you express the equation as a function of $\sigma$ only i.e BSM($\sigma$)
- Finally you implement a numerical procedure(e.g. Newton's method, Secant method etc...) to find the root of the equation: Market Price$ - BSM(\sigma) = 0$
ps: My answer is based on the title of the question only. There was no question in the body.
useful links:
https://www.wallstreetmojo.com/implied-volatility-formula/
https://www.investopedia.com/ask/answers/032515/what-options-implied-volatility-and-how-it-calculated.asp
https://en.wikipedia.org/wiki/Root-finding_algorithmsShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.