Ito vs. Stratonovich

Khoa Tran · 11-08-2007, 04:06 AM

In Finance, Ito's calculus seems to be the stochastic calculus to work with. I still don't understand why Ito's formulation is preferred to Stratonovich's formulation (*) even though the former is less natural in Physics?

Here are some common explanations that I've seen/heard:

Ito's integrals are martingales. This martingale property gives important computational advantage
Ito's integral leads to Feynman-Kac formula

They lead to my further questions:
1. Why do we need the martingale property? Why is it an important concept in Finance? Why does it give computational advantage? We know that there's a conversion formula that connects Ito's integral with Stratonovich's integral. Then why is Ito's calculus computationally easier? (**)
2. There's no Feynman-Kac for Stratonovich, of course, but there's another PDE (in divergence form) that corresponds to Stratonovich's formulation (we can name it Oppenheimer-Teller formula if we want).

Above all, those arguments are like saying "X has such and such property and is thus easier to work with". It is just as nonsense as saying "let's define $\int_a^bf(x)dx$ to be $f(a)+f(b)$ because it's easier to work with".

So, would someone please give some words of wisdom?

(*): in the discretization of a stochastic integral, Ito evaluates the integrand at the left end point of each infinitesimal interval while Stratonovich evaluates it at the mid-point. Because the integrand is evaluated at the mid-point, in a sense, it gives a better linear approximation of the integrand on that interval.
(**): on wiki, it's said that stratonovich's formulation has numerical advantage. This makes me even more confused as to point #1 above.

ngtridung · 11-09-2007, 05:26 AM

Quote:

Originally Posted by Khoa Tran

1. Why do we need the martingale property? Why is it an important concept in Finance?
.

The Martingale property is important because it can be used to price contingent claims that possess this property. For example, if $X_t$ is a martingale, then $X_t = E[X_T | f_t]$ where $X_T$ is a random variable (but not a process). From here, if the expectation (or integration) can be computed easily, we could have the analytical solution for $X_t$ . If not, we could use Monte Carlo simulation to simulate many sample paths and average them.

Quote:

Originally Posted by Khoa Tran

...Why does it give computational advantage? We know that there's a conversion formula that connects Ito's integral with Stratonovich's integral. Then why is Ito's calculus computationally easier? (**)
2. There's no Feynman-Kac for Stratonovich, of course, but there's another PDE (in divergence form) that corresponds to Stratonovich's formulation (we can name it Oppenheimer-Teller formula if we want).

To demonstrate why it makes more sense to use Ito’s integral than Strantonovich’s integral in finance, let’s see how we could solve a typical PDE numerically:

Consider the PDE: $\frac{\partial V}{\partial t} = f_1V + f_2\frac{\partial V}{\partial S}+...$

With a discretization of $0 = t_0 <t_1 <...< t_n = T$ and the approximation:

$\frac{ V_{t+\delta} - V_t }{\delta} = f_1V_t + ...$

In this numerical scheme, $V_{t_{k+1}}$ is a function of $(V_{t_1}, V_{t_2}..., V_{t_k})$ . This conforms to the Ito’s integral definition where:
$X_{t_{k+1}} = \sum_{j=1}^k X_{t_j} (W_{t_{j+1}}-W_{t_j})$ for any process X (actually the limitation of this quantity but approximated under this discretization).

If we would have used Stratonovich’s integral, we should have:
$X_{t_{k+1}} = \sum_{j=1}^k X_{(t_{j+1/2}} (W_{t_{j+1}} - W_{t_j})$

However, $V_{t_{n-1/2}}$ is not available through the discretization to update the $V_{t_n}$ (and using interpolation would be complicated for the implicit scheme). I guess in finance, people say the Stratonovich’s integral looks ahead into the future because of this reason. The “looking ahead in time” property is not desirable in finance since information is money.

Khoa Tran · 11-10-2007, 07:35 AM

First of all, I want to emphasize my main point the the original post that in principle, we should choose the model that best describes the nature rather than the model that is easy to solve but doesn't approximate the nature well. Many times, we have to choose a simple model because the perfect (or good) one is too complicated to derive meaningful results. This leads to a few questions:

Which one between Ito's formulation and Stratonovich's gives rise to better modeling? This question is related to a new topic that I'm going to create. My feeling tells me that Stratonovich's is the better one (and it is actually preferred in Physics).
Suppose that Stratonovich's formulation is better (which is arguable), does Ito's give an acceptable approximation? Again, it is related to my new topic on Stock's dynamics.
In Finance, how much advantage does Ito's formulation give? (I said in Finance because I learned that Physicists and Analysts find Stratonovich easier since Stochastic Calculus becomes "the same" as Ordinary Calculus.)

Now, let me discuss a bit more on your answers.

Quote:

Originally Posted by ngtridung

The Martingale property is important because it can be used to price contingent claims that possess this property. For example, if $X_t$ is a martingale, then $X_t = E[X_T | f_t]$ where $X_T$ is a random variable (but not a process). From here, if the expectation (or integration) can be computed easily, we could have the analytical solution for $X_t$ . If not, we could use Monte Carlo simulation to simulate many sample paths and average them.

In derivatives pricing, we have to price an option given the terminal payoff. So, I can see why the martingale notion gives rise to the importance of Risk-neutral Pricing concept. However, as I said somewhere, Risk-neutral pricing is not the only pricing approach. We can use replication arguments and PDE techniques.

So, can you give me a derivatives pricing example where the risk-neutral pricing approach is much easier than the PDE approach? Or a non-derivatives example where we need to compute the current value $X_t$ given the future value?

Quote:

To demonstrate why it makes more sense to use Ito’s integral than Strantonovich’s integral in finance, let’s see how we could solve a typical PDE numerically:

Consider the PDE: $\frac{\partial V}{\partial t} = f_1V + f_2\frac{\partial V}{\partial S}+...$

With a discretization of $0 = t_0 <t_1 <...< t_n = T$ and the approximation:

$\frac{ V_{t+\delta} - V_t }{\delta} = f_1V_t + ...$

In this numerical scheme, $V_{t_{k+1}}$ is a function of $(V_{t_1}, V_{t_2}..., V_{t_k})$ . This conforms to the Ito’s integral definition where:
$X_{t_{k+1}} = \sum_{j=1}^k X_{t_j} (W_{t_{j+1}}-W_{t_j})$ for any process X (actually the limitation of this quantity but approximated under this discretization).

If we would have used Stratonovich’s integral, we should have:
$X_{t_{k+1}} = \sum_{j=1}^k X_{(t_{j+1/2}} (W_{t_{j+1}} - W_{t_j})$

However, $V_{t_{n-1/2}}$ is not available through the discretization to update the $V_{t_n}$ (and using interpolation would be complicated for the implicit scheme). I guess in finance, people say the Stratonovich’s integral looks ahead into the future because of this reason. The “looking ahead in time” property is not desirable in finance since information is money.

I still don't quite understand how you correlate numerical schemes for a PDE to stochastic integration construction. (Btw, many many times, implicit schemes are preferred to explicit schemes because the latter ones though easier to compute are typically unstable -> worse approximation.)

I also don't think that "looking ahead in time" is a property of Stratonovich's formulation. This is just a theoretical construction of a stochastic integration and it is not relevant to the fact that you can't know $X_t$ ahead in time.

p/s: Next time, please put your formulas in the tex environment cuz tex not in its own environment is hard to read. It's very easy to do it. I did it for you in the previous post.

YADD · 11-12-2007, 11:01 AM

My answers will mostly be from hindsights. Not sure whether they will satisfy you. I'll try, though...

Quote:

Originally Posted by Khoa Tran

This leads to a few questions:[list=1][*]Which one between Ito's formulation and Stratonovich's gives rise to better modeling? This question is related to a new topic that I'm going to create. My feeling tells me that Stratonovich's is the better one (and it is actually preferred in Physics).

Perhaps you meant this situation: It is shown (see Schulman's Techniques and applications of Path integration) that to mathematically recover the Lagrangian of a charged particle in a magnetic field (in which case the wavefunction's phase factor couples with a vector field), the corresponding path integral must be taken at "mid-point" (Stratonovich's sense). [Not sure where else the mid-point condition is required. Will you tell me? It doesn't seem Ito- and Stratonovich- make much a difference except for this case.]

[I don't recall that Schulman elabrorated on the consequences of this mid-pointedness requirement in that case. Ok, ok, it is observed to be techincally required, but why is it so? Does this indicate that the quantum particle manages to "look" ahead of time then take the average? If this is indeed so, then the double-slit experiment might no longer be paradoxical since the electron - by the time it decides how to behave thru the slits - will have "been" to the screen and "learnt" if any slit - and which one - is covered. Anyhow, let's not stray into this metaphysical domain.]

Quote:

[*]Suppose that Stratonovich's formulation is better (which is arguable), does Ito's give an acceptable approximation? Again, it is related to my new topic on Stock's dynamics.

A derivative involves a hedging scheme which must be previsible. Precisely what previsibility means is math. I simply understand that the amounts of the underlying instruments for hedging (stocks, bonds, ...) must be known based on the info up to the instant. Profit and loss (PNL) is an integral of delta * dS. Since delta is determined at the beginning of dS, this path integral is in the Ito-sense.

Quote:

[*]In Finance, how much advantage does Ito's formulation give? (I said in Finance because I learned that Physicists and Analysts find Stratonovich easier since Stochastic Calculus becomes "the same" as Ordinary Calculus.)

Well, if my above understanding of PNL/previsible hedge holds, Ito or not Ito is no longer a matter of choice nor chance. It is a matter of principle.

Quote:

In derivatives pricing, we have to price an option given the terminal payoff. So, I can see why the martingale notion gives rise to the importance of Risk-neutral Pricing concept. However, as I said somewhere, Risk-neutral pricing is not the only pricing approach. We can use replication arguments and PDE techniques.

So, can you give me a derivatives pricing example where the risk-neutral pricing approach is much easier than the PDE approach?

I find - from a practitioner's standpoint - the martingale stuff sometime extremely misleading. One of my favorite interview questions is whether a trinomial world is complete. It is generally not, since there are three outcomes versus two hedging instruments (a stock and a bond) - in other words, 3 equations vs 2 unknowns. However, a great many candidates would answer "yes, but the hedge is not unique" since there are three risk-neutral probabilities (up, middle, down) versus two constraints (total probabilities = 1, and discounted stock price being a martingale)!

Risk neutrality approach is more an after-thought.

So why is it prevalent? Probably because it sounds fancy. Whenever (at least in practice) one uses martingale blah blah blah, one must beware of an implicit assumption that the market already is complete. After that, martingale may come to rescue in some calculations hard to do otherwise. Martingale and RN are neat, but - in my opinion - not at all important.

An example of RN versus PDE? How about a knockout option? In this option, one can use the reflection principle to get the analytical formula.

How about the MC method itself (which RN approach is very suggestive)?

Re your mentioning of PDE, I don't see why you conntrast PDE against Ito-? Remember that the BS eqn (derived from Merton's self replicating approach) is Ito- exactly because the previsible hedging argument I sketched above.

Quote:

Originally Posted by ngtridung

....
I guess in finance, people say the Stratonovich’s integral looks ahead into the future because of this reason. The “looking ahead in time” property is not desirable in finance since information is money.

Excellent!

I have some reservation, though. I don't think the choice of Ito- or Stratonovich- is relevant in solving a PDE. Don't they agree eventually when the lattice size vanishes? The fact that the BS eqn can be equivalently solved by explicit, implicit, and Crank-Nicolson schemes tells me so. Well, you guy are experts in PDE, let me know if I am wrong.

I think Ito- or Stratonovich- is for the SDE (asset price diffusions) and the PNL (accumulated wealth, an integral of delta * dS).

For SDE, Ito- is more natural as ngtridung mentioned right above. For PNL, Ito- is a must.

shinichi9htv · 11-13-2007, 06:21 AM

Thank you anh Hoang for this excellent post

shinichi9htv · 11-13-2007, 06:28 AM

Quote:

Originally Posted by YADD

A derivative involves a hedging scheme which must be previsible. Precisely what previsibility means is math. I simply understand that the amounts of the underlying instruments for hedging (stocks, bonds, ...) must be known based on the info up to the instant. Profit and loss (PNL) is an integral of delta * dS. Since delta is determined at the beginning of dS, this path integral is in the Ito-sense.

always believe that hedging is the only principle of pricing theory. Hedging and martingale approach are linked by martingale representation theorem, Martingale and PDE are linked by Feynman-Kac (as Khoa said)

YADD · 11-13-2007, 10:56 AM

Quote:

Originally Posted by shinichi9htv

Thank you anh Hoang for this excellent post

You're most welcome. Learning finance thru trial and error, many things I said can be errornous or even false, though I try my best to limit them and refine my understanding. Feel free to correct me

Quote:

Originally Posted by Khoa Tran

So, can you give me a derivatives pricing example where the risk-neutral pricing approach is much easier than the PDE approach?

Three more examples here... (though by no mean an exhaustive list)

A spread option of two lognormal stocks - the Margrabe formula. Direct (two-dimensional) integration of the final payoff is not awfully tough (it's relatively easy). However, the change-of-measure technique elegantly maps this two-dimensional problem into a one-dim problem! This same trick has been applied to convertible bond pricing as well.
Interest rate modelling. Handling the whole yield curve diffusion is no fun with PDE. (In fact I am not aware of any PDE approach for rates! Maybe there are some, but just too complicated.) Tree and MC algorithms are straightforward and standard for rates.
In M. Steele's novel-like text, there is one example involved a joint distribution of two BMs reducible to one-dim problem (Bachelier kicked that one.) Can't recall where that beautiful example is, chapter 10 or 11???

Khoa Tran · 11-14-2007, 07:08 AM

Quote:

Originally Posted by YADD

Perhaps you meant this situation: It is shown (see Schulman's Techniques and applications of Path integration) that to mathematically recover the Lagrangian of a charged particle in a magnetic field (in which case the wavefunction's phase factor couples with a vector field), the corresponding path integral must be taken at "mid-point" (Stratonovich's sense). [Not sure where else the mid-point condition is required. Will you tell me? It doesn't seem Ito- and Stratonovich- make much a difference except for this case.]

Well, I wouldn't say mid-point rule is required. It is just a naive belief that the mid-point rule gives a better approximation to nature. Why does that belief make sense? In the deterministic world, both Ito's integral and Stratonovich's integral agree in the limit. However, Stratonovich's gives a better approximation, i.e. it converges to the limit faster.

Quote:

A derivative involves a hedging scheme which must be previsible. Precisely what previsibility means is math. I simply understand that the amounts of the underlying instruments for hedging (stocks, bonds, ...) must be known based on the info up to the instant. Profit and loss (PNL) is an integral of delta * dS. Since delta is determined at the beginning of dS, this path integral is in the Ito-sense

Well, if my above understanding of PNL/previsible hedge holds, Ito or not Ito is no longer a matter of choice nor chance. It is a matter of principle.

As I wondered in the previous post, I'm not convinced yet that "looking ahead in time" is a property of Stratonovich's formulation. As I still understand, it is a purely theoretical construction of an integral and has nothing to do with future information.

Would you elaborate a bit more (possibly much more

) on the delta hedging argument? I still can't relate it to Ito's integral. As I see, since you hedge before dS, your hedge stays constant during dS. For this, Ito's formulation and Stratonovich's formulation agree. Or did I miss your point?

YADD · 11-14-2007, 12:27 PM

Quote:

Originally Posted by Khoa Tran

Well, I wouldn't say mid-point rule is required. It is just a naive belief that the mid-point rule gives a better approximation to nature. Why does that belief make sense? In the deterministic world, both Ito's integral and Stratonovich's integral agree in the limit. However, Stratonovich's gives a better approximation, i.e. it converges to the limit faster.

I recall it now. This problem is deep. I think it's worth a few lines.

Basically, a vector field (the curl of which is a magnetic field) couples with a charged particle thru the form $\int \vec A .\vec v \approx \int \vec A .d\vec x\$ where $\vec v$ is the particle's velocity. The question is whether $\vec A$ should be taken at the beginning or middle point of $d\vec x$ ?

Since the trajectory of the particle is stochastic, Ito- or Stratonovich- makes a whole lot difference. Schulman illustrates that only Stratonovic- gives rise to the correct Schodinger eqn of charged particle in a magmetic field in the continuum limit $dx \rightarrow 0$ . [He next goes on to show that for scalar potential Ito- or Stratonovic hardly differs.]

So, it's not simply a matter of fast convergence that Stratonovic is prefered. It is required!

Quote:

Would you elaborate a bit more (possibly much more

) on the delta hedging argument? I still can't relate it to Ito's integral. As I see, since you hedge before dS, your hedge stays constant during dS. For this, Ito's formulation and Stratonovich's formulation agree. Or did I miss your point?

Say, you dynamically hedge your option. At frequent occasions, you obsvered the stock at that instant and readjust the position (amounts of stock and bond) of your replicating portfolio. As stock moves, your portfolio gains or loses value. The change is delta * dS. The total PNL accumulated at expiry is integral of delta * dS interpreted in the Ito-sense. The position delta is set at the beginning of dS, not at the middle point of the interval where S will already have experienced a stochastic move.

The trajectory of the stock is stochastic, so Ito- or Stratonovich- does make a difference. [Recall Schulman's charged particle above.] Very likely that Stratonovich would lead to a different BS eqn.

That's for the PNL.

Regarding asset price diffusion (e.g., the lognormal SDE), the decision to go for Ito- or Stratonovich- has nothing to do with PNL. But very likely that if Stratonovich- is used, one might end up with another different BS eqn and a prob density distribution for S_T different than what is usually known.

Adfter all, almost everything said in finance is more or less assumptive and approximate. So why a fuss about Ito- or Stratonovich-?

For the case of charged particles, I would see your anxiety...

Khoa Tran · 11-16-2007, 03:07 AM

Quote:

Originally Posted by YADD

I recall it now. This problem is deep. I think it's worth a few lines.

Basically, a vector field (the curl of which is a magnetic field) couples with a charged particle thru the form $\int \vec A .\vec v \approx \int \vec A .d\vec x\$ where $\vec v$ is the particle's velocity. The question is whether $\vec A$ should be taken at the beginning or middle point of $d\vec x$ ?

Since the trajectory of the particle is stochastic, Ito- or Stratonovich- makes a whole lot difference. Schulman illustrates that only Stratonovic- gives rise to the correct Schodinger eqn of charged particle in a magmetic field in the continuum limit $dx \rightarrow 0$ . [He next goes on to show that for scalar potential Ito- or Stratonovic hardly differs.]

So, it's not simply a matter of fast convergence that Stratonovic is prefered. It is required!

Is it just due to the fact that Stratonovich Calculus is the same as Deterministic Calculus? Obviously, Schroedinger equation is modeled in a deterministic world.

Quote:

Say, you dynamically hedge your option. At frequent occasions, you obsvered the stock at that instant and readjust the position (amounts of stock and bond) of your replicating portfolio. As stock moves, your portfolio gains or loses value. The change is delta * dS. The total PNL accumulated at expiry is integral of delta * dS interpreted in the Ito-sense. The position delta is set at the beginning of dS, not at the middle point of the interval where S will already have experienced a stochastic move.

The trajectory of the stock is stochastic, so Ito- or Stratonovich- does make a difference. [Recall Schulman's charged particle above.] Very likely that Stratonovich would lead to a different BS eqn.

That's for the PNL.

When you say hedging at the beginning of dS, you're in a discrete world ( $dS>\epsilon>0$ ), aren't you? During this small interval dS, you do nothing (well, if you change your hedge during this interval, your hedge at the beginning doesn't mean anything). There are 2 ways of understanding PnL here:

If you interpret $\delta dS$ in a discretized setting, you no longer assume that the trajectory of the stock is continuously stochastic within the open interval dS. S simply jumps at the end of dS. Then the 2 models of integral agree.
If the stock moves continuously but you hedge in a discrete manner, 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?

Quote:

Regarding asset price diffusion (e.g., the lognormal SDE), the decision to go for Ito- or Stratonovich- has nothing to do with PNL. But very likely that if Stratonovich- is used, one might end up with another different BS eqn and a prob density distribution for S_T different than what is usually known.

Adfter all, almost everything said in finance is more or less assumptive and approximate. So why a fuss about Ito- or Stratonovich-?

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.

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Ito vs. Stratonovich

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