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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.

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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.

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