# 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 $$Corr[{dS}/{S\,,\,dv]}\;=\frac{Cov\,[{dS}/{S\,,\,}\;dv]}{\sqrt{ Var\,[{dS}/{S}\;]Var\,[dv]}}=\frac{{{\sigma }_{1}}{{\rho }_{1}}{{v}_{1}}\,+{{\sigma }_{2}}{{\rho }_{2}}{{v}_{2}}}{\sqrt{{{\sigma }_{1}}{{}^{2}}{{v}_{1}}\,+{{\sigma }_{2}}^{2}{{v}_{2}}}\,\sqrt{{{v}_{1}}\,+{{v}_{2}}}}\,$$

• Thanks Behrouz Maleki. How can I demonstrate the last equation? – math May 30 '16 at 16:24
• @Behrouz Maleki +1, although you should remove the $dt$ no? – Quantuple May 30 '16 at 19:15
• @ Quantuple yes you are Right. Thanks+1 – user16651 May 30 '16 at 20:13
• Cheers thank you ;) – Quantuple May 30 '16 at 21:46

Two volatility processes yield a higher flexibility of the model. This is of greater importance if one tries to price derivatives with different maturities in one single model. A additional volatility component helps to capture the term structure of volatility, which can depend greatly on time to maturity. See for example the VIX term structure from CBOE:

http://vixcentral.com/

With maturity average volatility increases/decreases (depends on economic condition) and the increases/decreases are the largest in magnitude for short maturities. The longer the maturity the smaller the change in average volatility per time. To model this properly models have to incorporate a "short-term" volatility component and a "long-term" component. A detailed discussion can be found in

"The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well, Peter Christoffersen, Steven Heston, Kris Jacobs, 2009" -- http://fic.wharton.upenn.edu/fic/papers/09/0905.pdf

The model can be found on P.8 EQ: 3-5 and the parameter estimates on P.40 Table 3. What we see is a a long-term volatiltiy ($V_1$) of high persitence and low volatility-of-volatility $\sigma_1$ and a erratic short-term volatility ($V_2$) of low persitence (high mean reversion speed) and high volatility-of-volatility $\sigma_2$. Therefore, those parameters show impressivly the two-fold dynamic of volatility.

Loosely speaking, it can be seen as inserting an additional degree of freedom in the underlying's dynamics.

This can be useful from a static perspective: with an additional lever to play on, one can hope to better capture the short term implied volatility smile, which "naive" stochastic volatility models (single volatility factor, no jumps) are known to be bad at (the same goes for the ATM implied volatility term structure).

This additional degree of freedom also allows for richer forward dynamics (notably stochastic skew).