What is the rationale behind using SV models with 2 distinct volatility processes?
In the Double Heston model, there are 2 distinct volatility processes. The SDEs read
$$ \begin{align} & d{{S}{t}}=r{{S}{t}}dt+\sqrt{{{v}{1}}(t)}{{S}{t}}d{{W}{1}}(t)+\sqrt{{{v}{2}}(t)}{{S}{t}}d{{W}{2}}(t) \ & d{{v}{1}}(t)={{\kappa }{1}},({{\theta }{1}}-{{v}{1}}),dt+,,{{\sigma }{1}}\sqrt{{{v}{1}}(t)},d{{B}{1}}(t) \ & d{{v}{2}}(t)={{\kappa }{2}}({{\theta }{2}}-{{v}{2}})dt+{{\sigma }{2}}\sqrt{{{v}{2}}(t)},d{{B}{2}}(t) \ & E[d{{W}{1}}d{{B}{1}}]={{\rho }{1}}dt \ & E[d{{W}{2}}d{{B}{2}}]={{\rho }{2}}dt \ & E[d{{W}{1}}d{{B}{2}}]=E[d{{W}{2}}d{{B}{1}}]=E[d{{W}{1}}d{{W}{2}}]=E[d{{B}{1}}d{{B}{2}}]=0 \ \end{align} $$ Could someone point out what could be the advantages of using such a model? Thanks.
I think,the additional volatility factor, $ v_2(t) $ , provides more flexibility in modeling the volatility surface.We know $ \rho $ controls the slope of the implied volatility.In the single-factor Heston model, $ \rho $ is constant over maturities,In deed
$$ Corr[{dS}/{S,,,dv]};=\rho , $$ which means that model has trouble providing an adequate fit to market implied volatilities when the slope of the smile varies substantially across maturities, although it does a good job when the slopes are all relatively flat or all relatively steep. Incorporating a second volatility factor allows for two different correlations and, hence, for two different regimes of volatility, because In the Double Heston model, the correlation between the returns and their variance is stochastic: $$ Corr[{dS}/{S,,,dv]};=\frac{{{\sigma }{1}}{{\rho }{1}}{{v}{1}},+{{\sigma }{2}}{{\rho }{2}}{{v}{2}}}{\sqrt{{{\sigma }{1}}{{}^{2}}{{v}{1}},+{{\sigma }{2}}^{2}{{v}{2}}},\sqrt{{{v}{1}},+{{v}{2}}}}, $$ Edit
Here, I show the correlation between the returns and variance processe is stochastic: for $ j=1,2 $ we have
$$ Cov,[{dS}/{S,,,};d{{v}{j}}]={{\sigma }{j}}{{\rho }{j}}{{v}{j}},dt $$ let $ v=v_1+v_2 $ , as a result $$ Cov,[{dS}/{S,,,};dv]=({{\sigma }{1}}{{\rho }{1}}{{v}{1}},+{{\sigma }{2}}{{\rho }{2}}{{v}{2}},)dt $$ on the other hand $$ \begin{align} & Var,[{dS}/{S};]=({{v}{1}},+{{v}{2}})dt=vdt \ & Var,[dv]=({{\sigma }{1}}{{}^{2}}{{v}{1}},+{{\sigma }{2}}^{2}{{v}{2}},)dt \ \end{align} $$ then $$ Corr[{dS}/{S,,,dv]};=\frac{Cov,[{dS}/{S,,,};dv]}{\sqrt{ Var,[{dS}/{S};]Var,[dv]}}=\frac{{{\sigma }{1}}{{\rho }{1}}{{v}{1}},+{{\sigma }{2}}{{\rho }{2}}{{v}{2}}}{\sqrt{{{\sigma }{1}}{{}^{2}}{{v}{1}},+{{\sigma }{2}}^{2}{{v}{2}}},\sqrt{{{v}{1}},+{{v}{2}}}}, $$