Risk-Neutral Drift for Trading Volume in Option Simulation

Summary

The document asks how to choose a drift for simulating trading volume when pricing an option whose payoff depends on volume. The proposed model treats volume as geometric Brownian motion, and the author recognizes that option pricing calls for risk-neutral valuation. The central issue is that trading volume is not a tradable stock, so its risk-neutral drift cannot simply be set equal to the risk-free rate. The author contrasts this with estimating a real-world drift from average historical daily returns, while noting that such an estimate may be unreliable.

No solution, derivation, or empirical evidence is included: the text is an open question rather than a worked method. It does surface an important modeling limitation: a historical time series alone does not specify a risk-neutral measure for a non-traded quantity. A pricing approach would need additional assumptions or a market calibration framework, and the document does not provide either.

Key ideas

• The question concerns risk-neutral simulation of trading volume in a derivative payoff.
• Trading volume is not a tradable asset whose drift can be replaced directly by the risk-free rate.
• A historical average of daily returns would estimate a real-world drift, not necessarily a pricing drift.
• The document raises the modeling problem but gives no proposed solution or supporting evidence.

Tags

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


# 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.

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