So I used matlab and simulated stock prices with the Merton diffusion model. Now I want to take the integral of the area. Now would there be any financial insight by taking the integral of a stock price curve
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I suppose the expectation could be used to get at some time-weighted average price (TWAP) where we assume each instant of observation has infinitesimal and equal weight $\frac{dt}{T}$: $\bar{S}_T := \frac{1}{T} \int_{t=0}^T S_t dt$.
One problem with this is we don't often care that much about an average stock price. When we do care, we often look at:
- an average of settlement prices over a time period, used for hedging delivery of a commodity (like electricity) each day over that time period (cf Asian options); or,
- a volume-weighted average price (VWAP) as a benchmark for trading a large portfolio with no alpha.
Many trading engines can trade to a TWAP or VWAP objective -- though I have yet to meet anyone who had a good reason (or any reason) to trade to a TWAP objective.
Therefore, I do not see any value to computing this integral nor any insight it would give.
