# Values for Heston Model Parameters

Under the Heston model, the stock price and volatility follow the processes \begin{align*} dS & = \mu S dt + \sqrt{V} S dW^1, \\ dV & = \kappa (\theta - V)dt + \sigma \sqrt{V} dW^2, \\ dW^1 dW^2 & = \rho dt. \end{align*}

The parameters to be calibrated are $\kappa$, $\theta$, $\sigma$, $\rho$, and $V_0$, which appears in the pricing formula. Below, I've used MATLAB's lsqnonlin function to minimize the objective function $$\sum_{i=1}^N \frac{1}{ask_i - bid_i}(C_i - \hat{C}_i)^2,$$ where $C_i$ is the $i^{th}$ European call's market price and $\hat{C}_i$ is the corresponding Heston value. After running this calibration to 170 separate days of prices, I've obtained the following empirical distributions for $\kappa$, $\theta$ and $\sigma$, and was curious if the orders of magnitude seem about right?

I should mention I placed the following restrictions on the admissible parameter values during calibration, following the paper by Moodley: $$\kappa > 0 \\ \theta \in [0, 1] \\ \sigma \in [0, 5] \\ \rho \in [-1, 0] \\ V_0 \in [0, 1]$$

• I think that you should change your weight parameter. $\frac{1}{ask_i-bid_i}$ is not so good. – user16891 Jul 16 '15 at 18:20
• @Farahvartish Do you have any suggestions? The one used was proposed in math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdf so I went with that. Actually, is a weight parameter really necessary? – bcf Jul 16 '15 at 18:23
• Another remedy is to use the loss function described in Christoffersen et al. (2009).It uses the reciprocal of the squared Black-Scholes vega as the weight .The parameter estimates from their method are, therefore, based on the loss function $\frac{1}{N}\sum_{t,k}^{N}\frac{(C_{t,k}-C_{t,k}^{\Theta})}{Vega_{t,k}^2}$ – user16891 Jul 16 '15 at 18:35
• How can you justify calibrating vol of vol to a bunch of vanilla options, which have no vol of vol exposure? It would make a lot more sense to me to calibrate to vol/var swaps. – will Feb 26 '16 at 18:09
• @will Good idea, but are those publicly available? – bcf Feb 27 '16 at 13:38