I have read several articles about local volatility and implied volatility, but I am still confused with the difference between the two.

Please correct me if I am wrong: Implied volatility is the constant volatility back-solved from the BS model with the current option price and market parameters (e.g. interest rate, spot price, strike, etc). Meanwhile, local volatility is also the volatility calculated via BS model, but additionally it assumed volatility is a function the underlying price and time when we calculate the local volatility.

So my first question: is my interpretation above correct? (I am looking for an intuitive explanation for two concepts.)

And second: if local volatility is calculated from BS model, how can it be calculated at the same time considering it is a function of the underlying price and time?

I do not have a quantitative background, so I am looking for an easy-to-understand explanation for above questions.

Thank you.

  • 1
    $\begingroup$ Hi Katrinachanlh. This question has been ask several times on this site, see e.g. here quant.stackexchange.com/questions/37524/…. I am therefore voting to close this question. $\endgroup$
    – Quantuple
    Jun 6, 2019 at 6:20
  • $\begingroup$ @Quantuple the above post is more on a mathematical explanation to the two concepts instead of an intuitive explanation, I still can't find a good answer to my above questions $\endgroup$ Jun 6, 2019 at 7:32
  • $\begingroup$ Intuitvely, consider you want to price a European vanilla with strike $K$ and expiry $T$. It helps to think of it in terms of Monte Carlo simulation, in particular what is the volatility you use to move from the underlying price at time $t$ to that at time $t+\delta t$. Under BS, you'll always use the same volatility number $\sigma$. Under LV, the number you'll use depends on the time and asset price $\sigma(t,S_t)$. As such, to price a single option you need: 1 vol figure under BS, a whole volatility surface under LV. $\endgroup$
    – Quantuple
    Jun 6, 2019 at 7:41
  • $\begingroup$ @Quantuple so for example t=0, we have the price of a 1 year ATMS call option, do you mean implied volatility is the single vol calculated from BS at t=0 assuming constant vol; while local volatility can be the vol at t=3months/6months/9months given we have the simulated underlying price, interest rate, option price at t=3months/6months/9months, from that we able to create a volatility surface, called local volatility surface? $\endgroup$ Jun 6, 2019 at 9:51