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I have some options prices I found using the Heston Model. How do I calculate the implied volatility? In Matlab there exist a blsimpv function, but is this the right tool for me since I'm working with the Heston Model and not the Black-Scholes model?

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Yes. You should use that function to calculate the implied volatility - market convention is to always quote implied volatility using the Black-Scholes model. Traders may execute a trade simply by agreeing a level of implied volatility combined with the use of the corresponding Bloomberg option pricing page.

Someone once said, "it is the wrong number in the wrong model that gives the right price". And the price is the only number that actually matters since that is what you have to pay. So, as I just said, if you give someone the BS implied volatility then they can type it into their Bloomberg calculator to see the price they have to pay.

However it was a bit unfair to simply call it the "wrong number". It is a meaningful number in the sense that it is the volatility of a Geometric Brownian Motion (GBM) process that will reprice the option in the risk-neutral measure (a world in which it is assumed that the stock price grows at the risk-free rate minus the dividend yield).

GBM is not a perfect model for stock prices, but it is not a very bad model either. Even experts can have trouble telling a simulated GBM process from an actual stock market price process. We can also calculate the realised historical GBM volatility from stock prices which gives us a sense for the typical value it should have and allows us to see if the market's "prediction" was accurate or not.

It also makes comparisons between options easier. If I want to compare the value of two otherwise equal options on two different stocks, implied volatility can tell me the difference in terms of volatility and this makes it easier for you to understand and assign a view as to whether such a difference is correct.

You can also do the same for options of different maturities on the same stock. And when you look at options with different strike prices, their differing implied volatilities can tell quickly about the market's view of the likelihood of deviations from the lognormal dynamics of GBM.

So while not perfect as a model, it is meaningful enough to become the quoting convention for options.

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When you here implied volatility in finance, it usually means Black volatility or Bachelier volatility. In your case, since you have the prices from Heston, you can use Black-scholes to get the implied volatility. In that scenario, don't think about Black-scholes as a model, but as a translator to better understand the option price. For option traders, seeing $40\%$ as implied vol makes more sense than seeing $500$ dollars.

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  • $\begingroup$ What if I used a more complex model - e.g jump-diffusion model. Still use BS Implied Volatility? $\endgroup$ – LocalMartingale Dec 5 '19 at 19:51
  • $\begingroup$ What are you asking for ? Implied volatilities, in other words, what volatility should be used in the Black model to match your price. The latter can come from a more complex model, the market... $\endgroup$ – Canardini Dec 6 '19 at 15:14

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