Interpreting the Time Integral of a Simulated Stock Price Path

Summary

The document asks what insight can be gained by integrating a simulated stock price path generated under a Merton jump-diffusion model. The response interprets the integral, after division by the observation horizon, as a time-weighted average price: each instant receives equal weight. It cautions that an average stock price is not always a quantity of practical interest, so calculating the area under a simulated path alone may not answer a useful trading question.

It gives examples where averages do matter: averaging settlement prices over a period for delivery-related hedging, as in some commodity settings and Asian options, and volume-weighted average price as a benchmark for executing a large portfolio without an alpha objective. Trading systems may target TWAP or VWAP, though the response questions the rationale for TWAP in general. The discussion is conceptual; it does not derive a distribution for the integral, connect it to a specific strategy, or assess the simulation’s assumptions.

Key ideas

• Dividing the time integral of a price path by its duration gives a time-weighted average price.
• A path average is useful only when it corresponds to a relevant payoff, hedge, or execution objective.
• Averaged settlement prices can matter for delivery hedging and Asian option payoffs.
• VWAP can serve as an execution benchmark for large portfolios without an alpha objective.
• The response questions generic TWAP use and provides no quantitative analysis of the simulated integral.

Tags

# What can the area under a GBM jump curve tell you


# What can the area under a GBM jump curve tell you












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

## Answer by kurtosis (score 3)

https://quant.stackexchange.com/a/55974

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.