Reading Partial-Derivative Notation in SVI Volatility Conditions

Summary

The document asks how to read derivative notation in conditions from a theorem on arbitrage-free SVI volatility surfaces. One condition uses a time derivative of a parameter and requires it to be nonnegative. Another bounds the derivative with respect to that parameter of its product with a function, using a lower and upper bound involving the function and a correlation parameter.

The accepted response confirms the questioner’s reading of the first expression: the compact notation denotes the partial derivative of the time-indexed parameter with respect to time. It notes that this shorthand is used in stochastic calculus and is applied here to ordinary calculus as well. The response does not explain the second inequality’s derivation or its role in ensuring arbitrage-free surfaces.

The note is therefore a narrow guide to interpreting derivative shorthand, rather than a treatment of SVI calibration or volatility-surface theory. Its explanation relies partly on surrounding text in the cited paper.

Key ideas

• The notation for the time derivative denotes the partial derivative of the parameter with respect to time.
• A partial-derivative symbol may be used as shorthand for a derivative of a time-indexed function.
• The second condition differentiates the product of the parameter and a function of that parameter.
• The response clarifies notation but does not derive the inequality or explain its financial implications.

Tags

# Need help with understanding the Mathematical notation in a research paper


# Need help with understanding the Mathematical notation in a research paper












Shown below is a snippet from the paper Arbitrage-free SVI volatility surfaces by Jim Gatheral and Antoine Jacquier (2013) (https://arxiv.org/pdf/1204.0646.pdf) .

The formulae shown below are on page 12, Theorem 4.1.

Is the first line basically saying "The partial derivative of theta with respect to t is always greater than equal to zero"?

In the second line what is the middle condition? Is that "Partial derivative of (theta * phi(theta) with respect to theta"?

Can somebody with math background please explain the notation to me:

- $\partial_t \theta_t \ge 0$ for all $t \ge 0$;

- $0 \le \partial_\theta (\theta \varphi(\theta)) \le \frac{1}{\rho^2}(1+\sqrt{1-\rho^2} )\varphi(\theta)$.

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/58192

From the words that follow (or precede) these equations in the paper it seems that your interpretation is correct. $\partial_t \theta(t)$ is just an abbreviation for $\frac{\partial \theta_t}{\partial t}$. Both should be read as "the partial [derivative] of theta t with respect to t". This usage is common in Stochastic Calculus and the author has decided to use the same notation for ordinary calculus. (Although slightly non-standard it does reduce the amount of writing you have to do, and is especially convenient when you are at the blackboard, speaking and writing at the same time).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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