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!