Ito vs. Stratonovich

[YADD]

Originally Posted by Khoa Tran

Is it just due to the fact that Stratonovich Calculus is the same as Deterministic Calculus?

For deterministic calculus, it does not matter where one assigns the integrand. All choices have to converge (by virtue of Rieman integral's definition). Ito- or Stratonovich- is irrelevant. Stratonovich- might improve convergence speed but the limit must be the same as Ito's. I hope you agree, don't you? There seems to have been some sign of confusion on your side about this since the begining.

The Ito/Stratonovich distinction only matters for stochastic integration.

Obviously, Schroedinger equation is modeled in a deterministic world.

I can't disagree more.

• Take Black-Scholes; it is a PDE in the stochastic world. So is Schrodinger's.
  
  
• Uncertainty principle is at the heart of quantum mech.

If the stock moves continuously but you hedge in a discrete manner...

It is this discrete hedge that I meant for PNL.

... then what happens to the stock price in the open interval dS doesn't mean anything to your PnL as it relies solely on the stock price at the current time, right?

Correct.

In the limit dS -> 0, PNL value would depend on the position along dS you attach delta. In other words, Ito- vs Stratonovich- does matter.

Since delta IN REALITY is determined at the beginning of dS, PNL is thus understood in the Ito-sense.

I find it fundamentally important. There should be a deep reason that financiers bet trillions of dollar on a "non-traditional" integral model, shouldn't it? It's not as trivial as simplifying a model. Ito's and Stratonovich's interpretations may lead to totally different results.

What makes you think Ito- is "non-traditional"?

Ask yourself: Why would any rational investor bet money in a Stratonovich-type model just because it sounds more "traditional" while disregarding that fact that his/her "profit and loss" is Ito-???

[drew]

I used to ask (myself) a lot of questions like you, Khoa, I used to go down to the very bottom of any formula, say, start from B-S and get to basic notions of probability, I used to (try to) work out the proof of any theorem on my own, etc. I think that lies in the heart of every math worshiper (I'm not a math PhD though). Fact is we quants never need all that in the real world (*), you would be amazed (and may I say, disappointed), from a math perspective, if you see approaches to practical problems.
I gather that you will be looking for a quant job when you finish you PhD (or even before that :-), sorry couldn't resist), relaxing your math mindset will probably be important IMHO.
Anyway, keep doing that when you still have the time !
(*) on a side note, paying attention to every (practical) detail is vital.

[ngotuan]

Thanks drew for saying what I want to say.

[Khoa Tran]

Thanks folks. I had a long long discussion with YADD over the phone last night. We didn't reach any consensus so I'm not going to talk about those directions (Schrodinger, delta hedge) anymore.

I had a fortune talking with Andy Majda (I deeply admire this scientist for his brilliance and the way he does Math) of the Courant institute when he came to Texas yesterday. I asked him this question. He helped me clear up this mess in less than 3 minutes. I've reached the bottom. What did he say? "There's no debate about this. The two integrals are equivalent (there's a theorem that connects the two). What matters is how you model your process. Read chapter 10 of Stochastic Differential Equations: Theory and Applications by Arnold."

Chapter 10 of this book is about "modeling with stochastic differential equations". It is only more than 10 pages but has everything that I was looking for. The book itself is thin and old (printed in 1973) but so elegant. Having looked through the chapters, I can say that it is even a better read than Oksendal's and should be reprinted. [A bit more details, note that the ito-lognormal model of stock prices gives rise to the right expectation. If we model with Stratonovich, the form won't won't be the same. So I agree with Andy, the bottom line is that Ito- and Stratonovich- are equivalent ways of looking at a process. The only advantage/disadvantage between the two would be which one an individual feels comfortable with.]

[ngtridung]

I thought YADD explanation using dynamic hedging is clear enough regarding this issue. However, since Khoa Tran and YADD’s phone conversation still hasn’t reached any consensus, I want to add a few comments and the references related to this issue:

1) The financial world we are living is discrete (the state variables (price and time) space is discrete, the trading opportunities are also discrete)
2) We use continuous stochastic process to approximate the discrete world (and then approximate the continuous process with the discrete process when we use numerical method)
3) Ito is a better choice to model applications where the underlying process is discrete in nature

Point (3) is what we have discussing in this thread. I think YADD’s dynamic hedging example explain this point clearly. I suggest you to read the paper: “A comparison of the Ito and the Stratonovich formulations of problem in finances” by Sethi and Lehoczky Journal of Economic Dynamics and Control, Volume 3, page 343-356. They used the same hedging example to demonstrate why Ito is a better fit than Stratonovich for the finance applications. The authors also claimed that Stratonovich’ integral is anticipating.

The book “Numerical Solution of Stochastic Differential equations”, by Peter E Kloeden (available in books.google.com) has a section (section 6.1) explain which application we should use Ito/Stratonovich.

[YADD]

Originally Posted by Khoa Tran

Thanks folks. I had a long long discussion with YADD over the phone last night. We didn't reach any consensus so I'm not going to talk about those directions (Schrodinger, delta hedge) anymore.

Khoa, apparently you mistook the purpose of my engaged communications on the topics. I (plus a few forum members) have been patiently elaborating the details to help you get to the bottom of things. It was no intention of mine to seek a consensus.

Having been in the industry sufficiently long, I am at peace with Ito- in particular and the current state of affairs of derivative modeling in general. My position from day one has been that Ito- vs Stratonovich- is a futile direction of thinking. The distinction between them is of the lowest priority in the list of my concerns or pondering.

If you have troubles comprehending some parts of my reasoning on the topics, I leave it as your task to straighten them up. If you have eventually found enlightenment from Majda, my pleasure to hear so.

Anyway, when it comes to the point that what I think is trash in Finance, then hell no, I won't look for a quant job!

Never say never.

drew and I each have rolled on this bumpy road. He sees an ugly side of the profession. I've witnessed some, but I have a glance of its good part as well. I guess myself have passed a point in life to look back and confidently say: never say never...

[Khoa Tran]

@ngtridung: hedging doesn't play any role in the explanation. You, as well as anh Hoang, may strongly disagree with me but I would say it to anybody else. The fact that we use Ito all boils down to the lognormal model of stock price interpreted in Ito sense (it's the model that you started with for whatever technique you use to delta hedge). The 2 integrals Ito- and Stratonovich- are "equivalent", thanks to the conversion theorem. This means you can express the same delta hedge or PnL in terms of Stratonovich integral, modulo a deterministic integral, if you wish. [*]

Why lognormal in Ito sense: it's because we believe that the random part of the stock price is similar to the white noise. I'd point anybody who's interested in this comparison to chapter 10 of the beautiful text by Arnold. The reason for the use of which integral in chapter "Applications of SDEs" of the book that you mentioned seems to rely on the "rule of thumb" that Arnold pointed out.


@anh Hoang: what you tried to say is not the bottom. In fact, I'm feeling that you missed the bottom. Hedging is NOT the reason to support Ito (check my argument above) and hedging is NOT the right way to think about Stratonovich's integral (if you ever wish to know Stratonovich's) [*]. Why no quants (including you) bothering Stratonovich. Deep down, I think it's due to the equivalence that I mentioned above, i.e. if you feel comfortable with one, why bother the other.

I'm sure that you have the same feeling that I have on you. It's fair enough; it's usual and there's nothing wrong with it. That's why I said no consensus.

[*] I realized that I missed YADD's point on hedging in some previous posts where I said the 2 summations being the same. Well, I mistakenly assumed that the stock price stays constant in the interval dS (or say another way the stock price was discretized synchronously with the hedge). Nevertheless, the fact that we can express PnL in terms of Stratonvich's integral, modulo a deterministic integral, doesn't make Ito the the principle of hedging. Why : because you had put yourself in Ito world. So, in a sense, Ito- and Stra- are 2 equivalent ways of looking at the same thing.

[YADD]

A mortal like myself doesn't use Stratonovich because of - from the conversion theorem - the additional quadratic term (which accounts for an extra dollar amount in his PNL). Neither is a such a mortal able to peek into future to set the delta. Since you happen to know how to exploit it, I suggest you keep the secret and use it in your investments.

You have reached to the bottom of things yourself. Congratulations!

Next time, do tell me and our forum fellows in advance to avoid wasting our time.