# Calibration of Heston model

I would like to calibrate the Heston model and I am wondering which are the most common approaches used in the literature. Any suggestions (references from the main stream literature, notes or presentations) is greatly appreciated.

• Calibrate on a static vol surface, time series of returns, jointly on both, or something else? – Kiwiakos Apr 11 '16 at 19:20
• mainly on time series. – AlmostSureUser Apr 11 '16 at 19:33

If you want to calibrate on time series, then you have a 'non linear filtering' problem, since volatility is latent. There have been papers from late 90s/ early 00s that do that: Google for Heston together with Ghysels, Gallant, Renault, Chernov, Tauchen, Pan, Bates, Shephard, MCMC, unscented Kalman filter/ particle filter.

Given the significant complexity though, you should understand your motivation and requirements. Ask yourself why calibrate Heston on time series? Why a more straightforward Garch variant is not sufficient?

• Thank you very much. The motivation is mainly for teaching. I taught to my students all the main ideas behind the theory of option pricing for stochastic volatility model and now I would like to do a couple of seminars on how to apply the theory in practice. So which strategy do you suggest for this case? – AlmostSureUser Apr 12 '16 at 7:33
• In practice you would not calibrate Heston to time series, but to a (static) volatility surface instead. Also, since Heston will not fit the surface perfectly, you would augment it with a 'local' component (vol or similar) to ensure that it prices vanillas correctly. If you want to remain within the hedgeable diffusion realm. Given the interplay between speed of mean reversion and vol-of-vol parameters you might use liquid exotics to finetune. But again you have to ask yourself what you want to achieve. Do you want to price/ hedge some exotic that depends heavily on the vol dynamics? – Kiwiakos Apr 12 '16 at 9:51
• The main purpose is to clarify to student how the unknown component of the market price of risk and other parameters can be estimated on data. The model in the $Q$-measure reads as $$\left\{\begin{array}{lll} dS_t & = & r\,S_t\,dt+\sqrt{\xi_t}\,S_t\,dW^Q_{1,t}\\ d\xi_t & = & (k\,(\xi_0-\xi_t)-\eta\,\sqrt{\xi_t}\,\nu_{2,t})\,dt+\eta\,\sqrt{\xi_t}\,\,dW^Q_{2,t}\\ \end{array}\right.,$$ with $dW^Q_{1,t}\,dW^Q_{2,t}=\rho\,dt$. So by parametrizing $\nu_{2,t}=a\,\sqrt{\xi_t}$ one can derive the close form solution for the call option. How can the set of parameter $a,k,\xi_0,\eta$ be estimated? – AlmostSureUser Apr 13 '16 at 12:51