Fluctuations of the time-weighted average price

Motivated by an interview question posed to a friend of mine recently, today I’d like to talk about the time-weighted average price (TWAP) of a financial asset (e.g. a stock).

Let’s consider the price of our stock a short time scale (e.g. intraday) corresponding to the interval . Its price will start at and evolve through to ; the time-weighted average price is

defined in this way is interesting, since this is the effective price obtained when executing a large order in the market by splitting it into smaller chunks and executing them throughout the day at a constant rate. This TWAP algorithm is one of the basic algorithmic execution tools used e.g. by asset managers to minimize market impact costs. More sophisticated ones include volume-weighted average price (VWAP) where one would adjust the volume executed at each time during the day proportionally to the typical trading volume at that time.