0
$\begingroup$

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 $t \ge 0$;
  • $0 \le \partial_\theta (\theta \varphi(\theta)) \le \frac{1}{\rho^2}(1+\sqrt{1-\rho^2} )\varphi(\theta)$.
$\endgroup$
4
  • $\begingroup$ edit the question with a link to or title of the paper might help, but it would be better if you add a list of what the paper's own definitions are for the variables displayed here, since the other symbols requested depend on the context of those variables' definitions $\endgroup$ – develarist Sep 23 '20 at 19:02
  • $\begingroup$ Please reopen my question. Thanks. $\endgroup$ – JohnGa Sep 23 '20 at 23:42
  • $\begingroup$ Reopened but please improve your presentation further using LaTeX. $\endgroup$ – Bob Jansen Sep 24 '20 at 7:42
  • $\begingroup$ In what page are these equations in the paper? What is $\theta$ in the second equation? is it a real number? so in the first equation $\theta$ is a function and in the second a number? $\endgroup$ – user39119 Sep 24 '20 at 11:10
1
$\begingroup$

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).

$\endgroup$
3
  • $\begingroup$ Thanks for editing the post noob2. Now I know how to use LaTex. And also, thanks for the answer. $\endgroup$ – JohnGa Sep 24 '20 at 13:53
  • $\begingroup$ When you mean such notation is common in stochastic calculus - do you mean expressions like $dW_t$? $\endgroup$ – rubikscube09 Sep 26 '20 at 19:56
  • $\begingroup$ Yes, never a fraction like $\frac{dy}{dx}$ with two differentials one on top of the other, always everything written on one line with $dW_t,dX_t,dt$ etc. appearing by themselves. $\endgroup$ – noob2 Sep 26 '20 at 21:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.