Angle Brackets as an Average Across Stochastic Paths
Summary
The document clarifies an angle-bracket notation appearing in a stochastic-volatility discussion. Although angle brackets often denote quadratic variation or quadratic covariation in stochastic calculus, the same symbols can represent an average in other contexts, including statistical physics. The intended meaning depends on how the notation is used in the surrounding equation.
For the cited volatility and option P&L expression, the quantity inside the brackets represents the P&L along a path, while the outer brackets indicate averaging that quantity across paths. This explains why the expression is not simply an equality between two time integrals: the notation is being used to describe an average over possible paths. The answer is brief and does not derive the underlying P&L expression or discuss how the path average is estimated in practice. Readers must still distinguish this use from quadratic variation by checking context.
Key ideas
- Angle brackets can denote different operations in different mathematical contexts.
- In stochastic calculus, they commonly denote quadratic variation or covariation.
- In the cited expression, they indicate an average of pathwise P&L across paths.
- The surrounding equation is needed to determine which interpretation applies.
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Full text
# How to comprehend this notation? # How to comprehend this notation? I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets. So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion: Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context? In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here? ## Answer by Magic is in the chain (score 4) https://quant.stackexchange.com/a/46099 In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book: And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant. In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.