# Heston Model Calibration

I've calibrated the Heston Model using options data and I was wondering if the parameters I've obtained are stable enough. Also, is Feller condition imposed, when calibrating the Heston Model, in the industry ?

Cheers

Edit: The squared returns which is supposed to be an estimator for daily variance has a large peak during around Jan-09. Is it a good explanation for the peak in theta and v0 ?

• What strikes me here are the spikes in $\theta$ (and $v_0$): $\theta=0.2$ ($v_0=0.6$) means a stationary (initial) variance of 44% (77%) which looks huge! Since $(v_0,\theta,\kappa)$ are related through variance swaps' prices they here do seem to move hand in hand but it really seems like a numerical instability to me. Maybe a move from one local minimum to another? What optim algo are you using? Also do you compute smart initial guesses or use the guess of the previous day or? Note that Heston exhibits somehow 'redundant' parameters, it may be worth it to fix some of these (or constraints?). – Quantuple Aug 17 '17 at 14:04
• I used a genetic algorithm (60 generations with 900 individuals) to generate 5 sets of initial parameters. Then, I used Matlab's fmincon (with an interior-point algorithm) for local optimization using each set of initial parameters. I keep the set which gives me the lowest MSE. My constraints are : lb = [e e e e -.999]; ub = [20 2 2 3 .999] for kappa , theta, sigma, v0, rho. I initially tried Gauthier and Rivaille's smart initial guess for rho and sigma and I used ATM IV for v0. That produced similar spikes, that's why I implemented global optimization – Andrei Aug 17 '17 at 14:41
• Seems like a good approach to me. Spikes are around where Lehman collapsed so this might explain that. Especially since your parameters seem stable elsewhere. I just had a look here google.be/url?sa=t&source=web&rct=j&url=http://… and you seem to have parameters in the same ball park. Try cross-checking your results since many authors have done the same exercise. – Quantuple Aug 17 '17 at 16:29
• @Quantuple I have tried to calibrate the Heston model with Jumps using the same approach but the parameters seem to be really unstable ? What would you recommend for calibrating this model in regards to starting parameters and parameter constrains ? Cheers – Andrei Aug 19 '17 at 11:07
• Hmmm never used Bates, I usually prefer adding a local volatility component to fit the vanilla market while preserving a 'nice' forward dynamics. But I guess you could start with jump parameters estimated from history (it is also what this paper seems to mention and it has maybe more clues inside I did not check: papers.ssrn.com/sol3/papers.cfm?abstract_id=2362968). That being said yeah EOY 2008 and start of 2009 were indeed very volatile to say the last hence the spikes. – Quantuple Aug 28 '17 at 18:08