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How can I reproduce the implied volatility curve (plotted against the strike price) in the Heston model (i.e. the blue line in the graph below)?

What's the equation that I have to set up and solve?

I have a function that evaluates the price of a Heston call:

heston_call$(S_0, v_0, \theta, \kappa, \eta, r, \rho, \tau, K)$ where $K$ is the strike price and the other variables are as explained below.

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    $\begingroup$ The definition of implied volatility $I(K)$ is $BS(S_t,K,I(K)) = C(S_t,K)$, where $BS$ is the Black-Scholes call (or put) formula, and $C$ is the market or model price (in your case Heston). So you need to find the value of $I(K)$ for each strike $K$ such that the BS price matches the market/model price. You can use the bisection method for example to find $I(K)$. $\endgroup$
    – user34971
    Commented Feb 3, 2022 at 19:06
  • $\begingroup$ The values for initial and long-variance look much like volatilities. But for those values (0.2 and 0.3), I find it surprising that implied vol ATM should be so high (it is below 0.3 for almost no strike). $\endgroup$
    – Enrico Schumann
    Commented Feb 16, 2022 at 10:00

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The concept of implied volatility is in fact inseparable from the Black-Scholes-Model. Thus you will have to solve for which implied volatility must be used to result in the desired option price. You can find a nice explanation how it can be done here: A simple formula for calculating implied volatility?

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