# How to get Risk-Neutral Drift for Trading Volume from Time Series

I am trying to price an option with Monte-Carlo simulation, where the payoff depends on some constants and a time-series (trading volume) which I model to follow a GBM. Now if I understood it correctly, since my goal is to price an option, I have to work with risk-neutral valuation. However it is not clear to me how I can get the risk-neutral drift based on the historical time series of the trading volume which I can use for the simulation? Since the time series is not a stock, I can't just use the risk-free rate $$r$$.

Would I be operating in the real-world measure, I would probably just calculate the mean of the daily returns and use that as drift, which would probably not be a good estimate.

• Can you specify what your payoff looks like? Is it VWAP based or something dependent only on volume and not on price? I would also recommend you to change the title of this post to highlight that you want risk neutral drift for volume time series.
– emot
Aug 11, 2021 at 8:40
• The volume is the only uncertainty which is why I use a stochastic process for it, however the payoff also depends on a constant. I.e. the payoff is \$10 for every 100000 volume. E.g. on a day with 300000 volume, the payoff for that day is \$30. Aug 12, 2021 at 7:05
• Merwin, risk neutral valuation assumes that you can hedge your risk factor, but your risk factors is not observable directly, therefore to hedge such contract you need another contract. If you have real contracts quoted in the market with this kind of payoff you can derive risk neutral drift from such contracts. Are there any real contracts traded with this kind of payoff (or similar) to which you could calibrate your model?
– emot
Aug 12, 2021 at 14:13
• For the payoff, I don't think there exists traded contracts with this payoff. However, the risky part of my payoff is the trading volume. Is it enough if the trading volume has some sort of correlation with a traded stock? I.e. I found a correlation coefficient of 0.8, so this is an assumption that I can make. Is it possible to find the risk-neutral drift for trading volume keeping that correlation in mind? Aug 13, 2021 at 7:11
• If you have such high correlation, you can then model volume as a function of a stock. You should fit linear regression to the datta and this will become something like $volume_t=\alpha+\beta S_t$, then your payoff $H$ becomes $H(volume)=H(\alpha+\beta S_t)$ and you have one source of risk $S_t$ with risk neutral drift $r$ i.e $dS=rSdt+\sigma S dW$. But this approach is not perfect, because it assumes that error $\epsilon$ between predicted and realized value is 0, which is not the case here. Maybe someone else will have better ideas how to approach this.
– emot
Aug 13, 2021 at 8:02