# Implied volatility plotted against the strike price in Heston model

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.

• 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 $I(K)$. Feb 3 at 19:06
• 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). Feb 16 at 10:00