Serving the Quantitative Finance Community

 
cdsharm75
Topic Author
Posts: 3
Joined: November 5th, 2020, 6:40 pm

Ito Isosymmetry

March 15th, 2021, 1:09 am

I recently came across a bunch of equations describing Ito Iso-symmetry. Since it's an advanced book, not a lot of context was given. I was wondering if anyone could explain the relevance of this concept and provide an example of where it is used. In particular, what would be the issue(s) if this condition/equation did not work.....

Thanks!
 
User avatar
bearish
Posts: 6525
Joined: February 3rd, 2011, 2:19 pm

Re: Ito Isosymmetry

March 15th, 2021, 11:03 pm

OK, it’s actually Ito isometry, and it works. As much as math works, that is. And I think you’re right, in that this a functional analytic perspective on an aspect of stochastic calculus, treating stochastic integrals as operators on a Hilbert space (or something pretty close to that — many people around here are more qualified in this area than I am). Practical applications of the Ito calculus in general are found all over mathematical finance, the most celebrated example being the Black-Scholes-Merton option pricing formula.
 
User avatar
Alan
Posts: 10656
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: Ito Isosymmetry

March 16th, 2021, 3:17 pm

Here's an example. Say you have a random variate X that evolves as a diffusion with stochastic volatility [$]\sigma_t[$]. The evolution equation is

[$] dX_t = \mu \, dt  + \sigma_t  \, dW_t[$]. 

The terminal value is [$]X_T[$], a random variable with a probability distribution, and say you want the (expected, time-0) variance of that distribution. 
Removing the mean, and integrating the SDE, we have

[$] Y_T \equiv X_T - X_0 - \mu \, T = \int_0^T \sigma_t dW_t [$].

The variance we want is

(*) [$] E_0[Y_T^2] = E_0[(\int_0^T \sigma_t dW_t)^2] = E_0[\int_0^T \sigma_t^2 dt][$],

using Ito's isometry.

For example, if [$]dX_t = dS_t/S_t[$], where [$]S_t[$] is the S&P 500 Index price, and [$]\mu = r[$], 
then (*) is (up to an annualization factor) --  the square of the VIX index for horizon T.

The really interesting thing, which takes a little more work, is that the far right-hand-side expression of (*), and so the VIX index, can be entirely computed from the t=0 prices of vanilla put and call options with option maturity T.
 
User avatar
katastrofa
Posts: 10161
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Ito Isosymmetry

March 16th, 2021, 10:47 pm

The only field where I came across isosymmetry in the context of stochastic differential equations was the theory of critical phase transitions in condensed matter physics. Otherwise, it's probably isometry, as bearish noted :-)
 
User avatar
bearish
Posts: 6525
Joined: February 3rd, 2011, 2:19 pm

Re: Ito Isosymmetry

March 16th, 2021, 11:19 pm

Ah - I just assumed it was a typo. If the question really is about Ito iso-symmetry, which apparently is a thing, I do not have the first idea. But I’m pretty sure it’s not prevalent in finance.
 
cdsharm75
Topic Author
Posts: 3
Joined: November 5th, 2020, 6:40 pm

Re: Ito Isosymmetry

March 24th, 2021, 2:43 am

OK, it’s actually Ito isometry, and it works. As much as math works, that is. And I think you’re right, in that this a functional analytic perspective on an aspect of stochastic calculus, treating stochastic integrals as operators on a Hilbert space (or something pretty close to that — many people around here are more qualified in this area than I am). Practical applications of the Ito calculus in general are found all over mathematical finance, the most celebrated example being the Black-Scholes-Merton option pricing formula.
Thanks! I don't know why I kept calling it "Iso-symmetry"! From what I've been able to read, it's a property that lets you calculate the variance of the stochastic process. I've barely scratched the surface though...tons more to go through.
 
cdsharm75
Topic Author
Posts: 3
Joined: November 5th, 2020, 6:40 pm

Re: Ito Isosymmetry

March 24th, 2021, 2:46 am

Thanks everyone - for the responses and the correction:) Also @Alan - thanks for the detail, will review it.