Stochastic Volatility of Volatility and Its Role in SLV Models
Summary
The document asks whether the volatility-of-volatility parameter in a stochastic volatility model should itself vary randomly. It frames this as an extension of Heston-style dynamics, where asset variance is random and its fluctuations are governed by a volatility-of-volatility term. The discussion raises the possibility of extending the hierarchy further, but does not specify a complete model for the new process.
The answer connects volatility of volatility to stochastic local volatility (SLV). After calibration to vanilla options, local volatility and stochastic volatility alone may not reproduce implied-volatility dynamics, and SLV uses volatility-of-volatility and correlation to control the blend between them. The response describes the local-volatility and stochastic-volatility cases as limiting forms of this mixture. It also notes that calibrating the mixing parameters requires dependable exotic-option prices, such as barrier prices. The document offers a qualitative explanation rather than model equations, calibration results, or evidence that adding another random process improves pricing.
Key ideas
- Stochastic volatility models make asset variance random, and a further extension could make volatility of volatility random as well.
- In SLV models, volatility of volatility and correlation help control the balance between local-volatility and stochastic-volatility behavior.
- Vanilla-option calibration alone may leave the models with limited flexibility to match implied-volatility dynamics.
- Calibrating SLV mixing parameters requires reliable exotic-option prices, including barrier-option prices.
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# Non-constant Volatility of the Volatility in Stochastic Volatility Models
# Non-constant Volatility of the Volatility in Stochastic Volatility Models
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?
## Answer by AKdemy (score 0, accepted)
https://quant.stackexchange.com/a/64192
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.