In pricing financial derivatives, we often first assume that the volatility of the stock price is constant.

$$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sigma S(t) \mathrm{d}W(t)\text{.}$$

The volatility itself, $\sigma$, may be modelled as a random process, however, like in the Heston Model:

$$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sqrt{\upsilon(t)} S(t) \mathrm{d}W_1(t)$$

$$\mathrm{d}\upsilon(t) = \kappa (\theta(t)-\upsilon(t)) \mathrm{d}t + \xi \sqrt{\upsilon(t)} \mathrm{d}W_2(t)\text{.}$$

Other models that include stochastic volatility can be found here.

We could keep going, however, and treat $\xi$ in the above as a random process. This might be thought of as "stochastic volatility of the volatility."

$$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sqrt{\upsilon(t)} S(t) \mathrm{d}W_1(t)$$

$$\mathrm{d}\upsilon(t) = \kappa (\theta(t)-\upsilon(t)) \mathrm{d}t + \sqrt{\xi(t)} \sqrt{\upsilon(t)} \mathrm{d}W_2(t)\text{.}$$

$$\mathrm{d}\xi(t) = \ldots$$

This process could be repeated infinitely, but I am mostly concerned with whether or not this third step is even a good idea. Are there (reasonable) models that allow the volatility of the volatility to itself be a random process? Has anyone investigated a model like this before?

  • $\begingroup$ For vanilla or even exotic payoffs, the 1st and 2nd order sensitivities (delta, gamma, theta, vega, vanna, volga, cross gamma etc) explains most of the p&l. The third/higher order p&l like $\frac{\partial^3 V}{\partial \sigma^3}d\sigma^3$ have much smaller impact than the 1st and 2nd order sensitivities, and a model with stochastic vol-of-vol cannot be calibrated unless you observe liquid options on vol-of-vol. If you concern very exotic payoffs e.g. options on vol-of-vol, then it maybe suitable to write these models. $\endgroup$
    – ryc
    May 22 at 7:40
  • $\begingroup$ Can you think of a product where the aforementioned model would produce substantially different prices and greeks than those produced by stoch vol models? $\endgroup$
    – Arshdeep
    May 23 at 7:18

In addition to the comments, I think $\xi$ has an important use case in Stochastic Local Vol (SLV) models. Once calibrated to the vanilla market, Local Vol (LV) and Stochastic Vol (SV) offer no extra flexibility in matching the dynamics of implied volatility. That will not change much with making $\xi$ itself random. For example, prices for barriers and touches tend to be undervalued by LV but overvalued by SV.

In SLV, mostly ($\xi$) vol of vol and correlation ($\rho$) control the mixing of LV and SV. Hence, appropriate calibration of the mixing parameters will allow you to closely match market quotes. LV and stochastic SV are simply degenerate cases where the mixing fraction is such that only one or the other is used (e.g. if $\xi = 1$, SLV becomes purely SV).

There is one issue here though; you need (reliable) prices for exotic options to calibrate to. E.g. barrier options as mentioned above.


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