Calibrating Heston Models to Implied Volatility or Option Prices
Summary
This document considers how to calibrate a Heston stochastic volatility model to a panel of European S&P 500 option observations. It distinguishes quoted implied volatility, which is defined through the Black–Scholes framework, from prices generated by the Heston model. Consequently, quoted implied volatility is not a model-independent measure that can be matched directly without specifying how model outputs are converted.
Two calibration approaches are described. One converts observed implied volatilities into Black–Scholes option prices and minimizes price errors against Heston prices. The other prices options under Heston, converts those prices back to Black–Scholes implied volatilities, and minimizes volatility errors. The questioner’s proposed risk-free rate and input values are not confirmed in the answer, and the document provides no comparison of weighting schemes, numerical optimization, or calibration performance. The choice between price and volatility errors therefore remains a modeling decision that may affect which strikes and maturities dominate the fit.
Key ideas
- Implied volatility is defined relative to a pricing model such as Black–Scholes.
- One approach converts market implied volatilities into Black–Scholes prices and fits Heston prices to them.
- Another approach converts Heston prices into Black–Scholes implied volatilities and fits those values.
- The two objective functions measure different errors and can produce different calibration emphasis.
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# Calibrating Heston model using implied volatilities
# Calibrating Heston model using implied volatilities
I'm trying to understand how the authers of a paper calibrated their model.
We got data on European type options on the S&P500-index period from early 2005 to mid-2009. We have daily data on option prices; 182 implied volatilities for each day in our data-set with moneyness from -30% to +30% (of current underlying) and between one month and three years to expiry (See the image attached below as an example for 31-jul-2009)
I want to calibrate a Heston Model to this data by minimizing
\begin{align} \sum_{t,k}(IV_{t,k}-IV_{t,k}^{\Theta})^2 \end{align} where $IV_{t,k}$ and $IV_{t,k}^\Theta$ are the quoted and model implied volatilities, respectively. (t,k) are the maturity-strike combinations.
My question is:
> How do I compute $IV_{t,k}^{\Theta}$ using the data given in the screenshot?
EDIT:
I want to use the first method and calculate the BS prices.
So for example consider the call from 31-Jul-2009 with strike 1283.72 and tenor = 3M.
I use the following function:
blackscholes(call, S0, K, r, time till expiration in years, volatility, dividend_yield)
blackscholes(call, 987.48, 1283.72, 0.47%, 0.25, 16.8%, 2.36%)
Are these the correct values for this example? (I took the ZeroRate as risk-free interest rate)
And then do the same for every combination of strikes and tenors.
## Answer by Sebastian (score 3, accepted)
https://quant.stackexchange.com/a/69682
The implied volatility based moneyness has no meaning in Heston model. There are two possible solutions:
- Use the (quoted) implied volatility and compute the quoted option price $C_{t,k}$ with the Black-Scholes-Model. Calibrate the Heston model by minimizing $$\sqrt{\sum_{t,k}\left( C_{t,k} - C_{t,k}^{\Theta} \right)^2}$$
- Compute the option price $C_{t,k}^{\Theta}$ bases on your Heston Model. Use the Black-Scholes-Model to get the corresponding implied volatility $IV(C_{t,k}^{\Theta})$. For the details see A simple formula for calculating implied volatility?
The concept of implied volatility is in fact inseparable from the Black-Scholes-Model.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.