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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?

enter image description here

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.

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The implied volatility based moneyness has no meaning in Heston model. There are two possible solutions:

  1. 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 $$\sqrt{\sum_{t,k}\left( C_{t,k} - C_{t,k}^{\Theta} \right)^2}$$

  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.

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  • $\begingroup$ Thanks a lot! I edited my question and added an example. I want to use the 1. method. Can you tell me if these are the correct values to put in the Black Scholes pricing formula? $\endgroup$
    – user826130
    Commented Jan 30, 2022 at 9:22
  • $\begingroup$ Yes that seems to be correct. I would however advise you to check the strike prices. The displayed strikes aren't even which means they are probably not traded but are calculated to match some delta (1%, 10%, 20%, 25%, 30%, ...). If that is the case the different tenors will have different strike prices associated with the delta levels. $\endgroup$
    – Sebastian
    Commented Jan 30, 2022 at 20:24

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