# Interpretation and intuition behind the Put-Call symmetry under the Heston Model

I am currently working on a report regarding the put-call symmetry relations under the Heston model. I did all the math and managed to prove the relations using PDE approach. However, I wish to have a more intuitive interpretation of the derived relations.

Specifically, suppose a call option (European or American) with strike price $K$ and spot price $S_0$ is priced under the Heston dynamics with initial variance $V_0$:

$$dS_t = (r-q)S_tdt+ \sqrt{v_t}S_tdW_t^1,$$

$$dv_t = \kappa(\theta - v_t)dt + \sigma\sqrt{v_t}dW_t^2,$$

$$\rho dt = dW_t^1dW_t^2,$$ its value will equal to the put option with strike price $S_0$ and spot price $K$ priced under the Heston dynamics with the following parameters:

$$r_p = q,$$ $$q_p = r,$$ $$\kappa_p = \kappa-\rho\sigma,$$ $$\theta_p = \frac{\kappa\theta}{\kappa-\rho\sigma},$$ $$V_{0,p} = V_0,$$ $$\sigma_p = \sigma,$$ $$\rho_p = -\rho.$$

My main question is: what is the interpretation or intuition of $$\kappa_p = \kappa-\rho\sigma,$$ $$\theta_p = \frac{\kappa\theta}{\kappa-\rho\sigma},$$ and $$\rho_p = -\rho.$$ Does anyone have an explanation for the changes in theses three parameters? What are the physical and financial implications? Thanks!

This is a consequence of transforming a Put on $S_T$ with strike $K$ into a Call on $(K S_0)/S_T$ with strike $S_0$ under the stock measure. The new set of parameters $r_p$, $q_p$, $\kappa_p$, ... etc . are those that correspond to the Heston dynamics for the process $((K S_0)/S_t, v_t)$ under the stock measure.
In the Heston case you are looking at, start from the Put price as discounted expectation under the risk neutral measure $P$: $$p = e^{-rT} E^P\left[(K - S_T)^+ \right]$$ Next rewrite it as $$p = e^{-qT} E^P\left[\frac{e^{(q-r)T} S_T}{S_0} \left(\frac{K S_0}{S_T} - S_0\right)^+ \right]=e^{-qT} E^Q\left[\left(\frac{K S_0}{S_T} - S_0\right)^+ \right]$$ where $Q$ is the stock measure defined by the Radon Nikodym derivative $$\frac{dQ}{dP} =\frac{e^{(q-r)T} S_T}{S_0}$$ Now apply Ito's Lemma to $X_t=\frac{K S_0}{S_t}$: $$\frac{dX_t}{X_t}=-(r-q)dt - \sqrt{v_t} dW^1_t+ v_t dt$$ and finally apply Girsanov theorem to obtain the dynamics of $X_t$ and $v_t$ under $Q$: $$\frac{dX_t}{X_t}=-(r-q)dt - \sqrt{v_t} dW'^1_t+v_t dt - v_t dt=-(r-q)dt + \sqrt{v_t} (-dW'^1_t) \\ d v_t = \kappa (\theta - v_t)dt + \sigma \sqrt{v_t} dW'^2_t+ \rho \sigma v_t dt = (\kappa-\rho \sigma ) \left(\frac{\kappa \theta}{\kappa-\rho \sigma } - v_t \right)dt + \sigma \sqrt{v_t} dW'^2_t$$ with $W'^1$ and $W'^2$ standard Brownian motions under $Q$ with correlation $\rho$, so that you are now pricing a call under new Heston parameters $r_p$, $q_p$, $\kappa_p$, ... etc. defined as in your post.
• Yes. The intuitive explanation is the 1st paragraph. If it helps, consider the case where $S_T$ represents the EURUSD FX rate instead of a stock. Then one clearly sees that a Put on EURUSD with strike $K$ is same as a Call on USDEUR with strike $1/K$, with notional $K S_0$. Now if you posit an Heston model $r$, $q$, $\kappa$, etc. for $S_t=$ EURUSD under the USD risk neutral measure, you deduce a new Heston model for USDEUR = 1/EURUSD = $1/S_T$ under the EUR risk neutral measure, with parameters $r_p$, $q_p$, $\kappa_p$, etc. as in your post, after applying Ito and Girsanov. – Antoine Conze Apr 6 '18 at 20:11
• From the Girsanov theorem the drift of any process $Y_t$ is adjusted by its quadratic covariation $d<Y_t,e^{(q-r)t} S_t/S_0>$ with the change of measure. – Antoine Conze Apr 6 '18 at 20:29