# 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.

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?