Calculating Heston Implied Volatility by Inverting Black-Scholes

Summary

The document explains how to create an implied-volatility curve from option prices generated by a Heston model. For each strike, first evaluate the Heston call price using the chosen model parameters and maturity. Then find the Black-Scholes implied volatility that reproduces that same price, treating volatility as the unknown in the Black-Scholes pricing equation.

Repeating this inversion across strikes produces implied volatility as a function of strike. The answer emphasizes that implied volatility is defined relative to the Black-Scholes pricing framework, even when the price being matched comes from another model. The discussion is brief: it points to a general explanation of implied-volatility calculation but gives no numerical solver, initialization, convergence guidance, or treatment of boundary cases. It therefore describes the core pricing inversion rather than a complete implementation.

Key ideas

• Heston prices can be converted into Black-Scholes implied volatilities by matching option prices.
• For each strike, solve for the Black-Scholes volatility that reproduces the Heston call price.
• Repeating the inversion across strikes yields the implied-volatility curve.
• The short explanation does not specify a numerical method or discuss solver edge cases.

Tags

# Implied volatility plotted against the strike price in Heston model


# 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.

## Answer by Sebastian (score 1, accepted)

https://quant.stackexchange.com/a/69762

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.