How Angle Brackets Denote Expectations and Return Variance

Summary

The document asks how angular brackets are used in quantitative finance, especially in volatility writing and in a discussion of return increments. Its examples include brackets around a stochastic integral, statements that daily returns have zero mean and unit variance, and the claim that squared price changes scale with elapsed time. These examples point to the common use of brackets for an expectation or average, but the notation is not universally fixed: its precise meaning must be established by the author or surrounding definitions.

The question also raises whether brackets can stand for quadratic variation. Those are distinct concepts: expectation averages a random variable across outcomes, while quadratic variation describes accumulated squared changes along a process. The excerpt does not provide an authoritative answer to the question or specify the probability measure for its examples. Readers therefore need to consult the relevant text’s conventions before interpreting the notation, particularly when working with risk-neutral expectations or stochastic-process quantities.

Key ideas

• Angular brackets commonly indicate an expectation or average, but their meaning depends on context.
• The expectation’s probability measure should be identified before interpreting a financial formula.
• Quadratic variation and expectation describe different properties of a stochastic process.
• Statements about return means and variances use brackets as statistical averages in the cited examples.

Tags

# Angular bracket notation (physics)


# Angular bracket notation (physics)












In a few papers I have seen the following notation: $$ \langle X_t \rangle $$ Also, in Bergomi's book, at page 8, we have the following equality: $$ \biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\sigma}}}{dS^2}\sigma^2_t dt \biggr\rangle=\biggr\langle\int_o^Te^{-rt}S^2\frac{d^2P_{\hat{\sigma}}}{dS^2}\hat{\sigma}dt\biggr\rangle $$ What does the angular bracket symbol mean? Is it the expectation under the risk neutral measure? Does its meaning vary as a function of the contect, or does it always has the same precisely-defined meaning?

Edit: It seems the notation originates from physics. I'd be very grateful if someone can explain what this notation means in physics.

Edit: To give more context, I quote from Bergomi's book : Indeed, to a good approximation, the variance of returns scales linearly with their time scale, thus $\langle \delta S^2 \rangle$ is of order $\delta t$ and $\delta S$ is of order $\sqrt{\delta t}$. The book had not specified any dynamic for $(S_t)_{t\geq 0}$.

Edit: Related question What do angle brackets (⟨⟩ ) mean in mathematics/statistics (autocorrelations)?

Edit: Also on Bergomi's book, on p. 9, we are told: Let is write the daily return $r_i$ as: $$ r_i=\sigma_i\sqrt{\delta t}z_i $$ where $\sigma_i$ is the realized volatility for day $i$, and $z_i$ is centered and has unit variance: $\langle z_i\rangle=0$, $\langle z_i^2\rangle=1$.

It seems this notation has dual meaning: both quadratic variation and expectation.

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