# Calibrating Heston paremeters based on market data for Implied Vol for Call options

Several questions have been asked in here regarding calibration in Heston yet I have not found what I have been looking for, so I will ask:

I am looking at a Heston model: $$dS_t=\lambda \sqrt{v_t}S_tdW_t^S$$ $$dv_t=-k(v_t-1)dt+\theta\sqrt{v_t}dW_t^v$$ $$d\langle W^S, W^v \rangle_t = \rho dt$$ where the two Wiener process have correlation $\rho$.

Say I have some market data on a call option that expires at $T=1$ and $spot=100$. The market data is implied volatilities for 3 different strikes $K=90,100,120$. If we set $k=0.15$ or any other arbitrary value then we end up with 3 parameters to be estimated and 3 points?

Can somone provide a guide for a how to estimate $\lambda$, $\theta$ and $\rho$ from market data in this setup

• What is the $\lambda$ parameter in your model? Also you seem to specify the long term mean reversion level to 100%, why? Indeed if you fix the mean-reversion speed $\kappa$ you are left with the estimation of the volatility of inst.variance and the spot/inst. variance correlation (+ your $\lambda$ but this is not the traditional Heston). Usually this is done by solving an optimisation problem. – Quantuple Apr 16 '18 at 14:01