Interpreting Drift in Rough Volatility Option Models
Summary
The document explains why some rough-volatility papers write a stock-like process without an explicit drift. The answer says these models are focused on volatility and are often specified for a forward price under a terminal measure, where the forward has zero drift. This is a modeling convention tied to the pricing measure, rather than simply an omitted term to be restored without regard to the numeraire.
For a European option, the response describes taking the expected payoff under the expiry-date measure and discounting it by the corresponding discount factor. Since the forward price represents the conditional expected future stock price under that measure, its dynamics are the object to model. For non-European products, volatility swaps, and related claims, it states that deterministic interest rates make the terminal and risk-neutral measures coincide, allowing the volatility dynamics to transfer between the forward and stock formulations. The explanation is concise and assumes familiarity with measure changes; it does not cover stochastic rates or provide a numerical example.
Key ideas
- Rough-volatility papers may model a forward price under a terminal measure, giving it zero drift.
- The missing drift reflects the chosen measure and modeled quantity rather than an arbitrary simplification.
- European option values are obtained from discounted expected payoffs under the expiry-date measure.
- With deterministic interest rates, the response treats the terminal and risk-neutral measures as equivalent for the discussed volatility dynamics.
- The explanation does not address the complications introduced by stochastic interest rates.
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# Drift term in rough volatility models
# Drift term in rough volatility models
I'm studying rough volatility papers and was wondering, why the drift term is always missing.
See for example the paper Pricing under rough volatility by Bayer, Friz, Gatheral. On page 2, the fractional stochastic volatility model is introduced and the stock price process is defined by $$\frac{d S_t}{S_t} = \sigma_t d Z_t$$ Why is a drift term missing? First, I thought that it is just a simplification and it has to be incorporated in a practical application of the model. But after reading Turbocharging Monte Carlo pricing for the rough Bergomi model by McCrickerd and Pakkanen I think it is a kind of change of numeraire since they state on page 2 that the stock price needs to have the following property $\mathbb{E}(S_t)=1, \forall t \geq 0$.
If the numeraire is changed in a way, that the stock price is equal to one in expectation, it probably has to be something like $S_t e^{-r t}$.
So, if I'm right, how do you price an option with this model? Do I have to discount the strike price as well? And what about dividends, should one discount with the risk free rate plus a continuous dividend rate?
It would be great, if someone could shed some light on this problem. If you could provide a paper, explaining this stuff, it would be great, too.
## Answer by Antoine Conze (score 5, accepted)
https://quant.stackexchange.com/a/43796
These papers are interested in modelling the stochastic volatility, so implicitely they model the dynamics of the forward price which has zero drift under the corresponding terminal measure. This makes the exposition much simpler.
It is then easy to switch back to the stock price:
- european options: an option with expiry $T$ on the stock price $S_t$ is computed as $PV_t = D_t(T) E^{Q_T}_t[\text{payoff}(S_T)] = D_t(T) E^{Q_T}_t[\text{payoff}(F_T)]$ where $F_t=E^{Q_T}_t[S_T]$, so the only thing that needs modelling is the dynamics for $F_t$ under the terminal measure $Q_T$
- non european option, volatility swaps, etc. : under the assumption that interest rates are deterministic the terminal measure and the risk neutral measure are the same, so the stochastic volatility for $F_t$ under the terminal measure is the same as the stochastic volatility for $S_t$ under the risk neutral measure.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.