Computing Heston Implied Volatility from Model Prices

Summary

The note explains how to obtain Black–Scholes implied volatility for an option priced under the Heston stochastic volatility model. Heston parameters determine the model price; they are not themselves implied volatility inputs to that pricing formula. The proposed workflow is to price a European option under Heston, then find the Black–Scholes volatility that reproduces that price.

The Heston price may be computed with Fourier methods or Monte Carlo, after which a numerical root-finding method such as Newton’s method or bisection can solve for implied volatility. The note gives no worked example, convergence guidance, or comparison of the numerical methods. It also frames the procedure around European options and does not discuss practical issues such as selecting a root-finding bracket or handling prices outside the Black–Scholes attainable range.

Key ideas

• Heston model parameters determine the option price but do not directly give Black–Scholes implied volatility.
• First compute a European option price under the Heston volatility process.
• Fourier methods and Monte Carlo are cited as ways to obtain the Heston price.
• Apply Newton’s method or bisection to find the Black–Scholes volatility matching that price.

Tags

# Implied Volatility in Heston Model


# Implied Volatility in Heston Model












recently I started reading the interesting book about option pricing in the stochastic volatility world from Lewis. He gives very interesting and detailed insights about this topic in general. However the book does not cover the implied volatility topic. That's why I am interested of how I get implied volatility out of the Heston Model. Do I have to calculate an european call price with Heston's formula and then reverse it by the help of Black Scholes to get the implied volatility?

Thanks in advance

## Answer by Gordon (score 8)

https://quant.stackexchange.com/a/22250

The option price with a Heston volatility model depends on the Heston parameters only. That is, the implied volatility parameters does not enter into the Heston option price formula. Unless there is an analytical formula to compute the implied volatility for a given option price, it is impossible to compute the implied volatility directly from the Heston parameters. What people usually do is to first compute the European option, from the Heston volatility process, using either Fourier transformation or Monte Carlo, and then compute the implied volatility using the Newton's method or the bi-section method.

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