Interpreting the Time Integral of Price as Average Price Exposure

Summary

The document asks what the time integral of price should be called, drawing an analogy with displacement and its derivatives in physics. One response points toward financial option Greeks as a reference for price-related sensitivities, though it does not identify a specific standard name for the integral itself.

Another response explains that integrating price over time accumulates price across the interval; dividing that quantity by the interval length gives the time-average price. It notes that average prices are used in exotic options whose payoff depends on the path’s average, as in an average-price contract. The discussion is conceptual and supplies no pricing formula or market example. Its distinction between the accumulated integral and normalized average is useful, while terminology for the unnormalized quantity is left unresolved.

Key ideas

• Integrating price over time gives an accumulated quantity over the chosen interval.
• Dividing the integral by the interval length produces the time-average price.
• Some exotic options define payoffs using an average price over the contract period.
• The responses do not establish a standard financial name for the unnormalized integral.

Tags

# What's the first time-integral of price called?


# What's the first time-integral of price called?












In general I'm wondering about the names of time-derivatives of price.

E.g. in physics the first few time-derivatives of position are:

- f(x) = displacement

- f'(x) = velocity

- f''(x) = acceleration

And the first integral (anti-derivative) of displacement is called absement.

What would the equivalent financial terms be?

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/8015

Although I don't think that this is a question that fits in here, I will give you a reference.

You might want to have a look at the so called greeks, you find a first overview here:

http://en.wikipedia.org/wiki/Greeks_(finance)

## Answer by SBF (score 1)

https://quant.stackexchange.com/a/8040

Well, if you divide a time integral by the length of the time interval, you'll get the average (in time) price: $$ \frac{1}{t}\int_0^T x_t\mathrm dt $$ so at least on of the meanings of the integral itself is an average price time the length of the interval. In such a case, I think the normalized quantity (the integral divided by the length) is more meaningful. It is used e.g. in the exotic options whose payoff depends on the average price.

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