# What is Ito's lemma used for in quantitative finance?

Further to my question asked here: prior post

and which left some points unanswered, I have reformulated the question as follows:

What is Ito's lemma used for in quantitative finance? and when is it applicable?

I don't understand for instance if Ito's lemma is used for obtaining a SDE from a stochastic process or the converse: obtain a stochastic process from an SDE.

Furthermore vonjd's reply is a bit confuse to me: does he mean "Ito's lemma can

only

or

also

be used for processes with bounded quadratic variation?

If you are given a diffusion process $X_t$, and a $C^{1,2}$ transformation $Y_t=f(t,X_t)$ of the process $X_t$.

Then Itô's lemma gives you the SDE followed by the process $Y_t$ in terms of $dX_t$, and $dt$ and partial derivatives of $f$ up to order 1 in time and 2 in $x$.

If you are given the SDE followed by $X_t$ in terms of Brownian motion, drift, and diffusion term then you can write down the SDE of $Y_t$ in terms of Brownian motion, drift, and diffusion term.

This shows in particular that diffusions are stable by those type of transformations.

There is nothing more and nothing less in it.

Of course you can extend this lemma in various fancy and sophisticated ways.

Regards

• Itô's lemma is also applicable if $f$ is a $C^1$ function in time and space and also $C^2$ in space everywhere except in a countable set of points Jun 28 '11 at 12:53

A common way to use Ito's lemma is also to solve the SDEs.

The most classic example (I guess) is the geometric Brownian motion:

$$dX_t = \mu X_t dt + \sigma X_t dW_t$$

and this can be solved easily by applying Itô's lemma with

$$f(x)=\ln(x)$$

That's the BnB example:

$$f'(x)=\frac{1}{x}$$ $$f''(x)=-\frac{1}{x^2}$$

and by Itô:

$$d(ln(X_t))=\frac{1}{X_t} dX_t -\frac{1}{2X_t^2} d<X_t>$$ $$d(ln(X_t))=\mu dt + \sigma dW_t - \frac{\sigma^2}{2} dt$$ $$d(ln(X_t))=\mu dt + \sigma dW_t - \frac{\sigma^2}{2} dt$$ $$d(ln(X_t))=(\mu - \frac{\sigma^2}{2}) dt + \sigma dW_t$$

And then,

$$ln(X_t)-ln(X_0)=ln(\frac{X_t}{X_0})=(\mu - \frac{\sigma^2}{2})(t-0) + \sigma W_t$$ $$X_t=X_0 \exp^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t}$$

This means that $X_t$ is log-normally distributed...

Such model is used in the most common (and hence trivial) derivative pricing framework such as the Black and Scholes Model.

Another example is the Ornstein–Uhlenbeck process which can be solved using a different $f(x)$.

• This is a fantastic solution! My lecture slides apply Ito's to $e^{aX(t) + bt}$ to get the answer and it takes about 30 lines of working!
– Jase
Dec 5 '12 at 7:43

Buy copies of Brent Oksendal's "Stochastic Differential Equations An Introduction with Applications" and Thomas Bjork's "Arbitrage Theory in Continuous Time." These are well written graduate level textbooks. I can't promise it will be painless, but if you want to understand continuous time derivative pricing models these are a place to start.

Another option is to not worry about continuous time models and get a copy of Stanley R. Pliska's "Introduction to Mathematical Finance." It is a graduate textbook covering discrete time models. To use these models all you need to know is linear algebra and how to optimize linear equations using the simplex method. (not to be confused with the simplex numerical optimization algorithm.)

Bluntly put: Ito Integration can be viewed two ways. 1) As an incomplete Riemann Stieltjes Integral 2) An extended Lebesgue Integral.

If you have no idea what either of the above two things are, go with the descrete time models.

• Given the way the question is asked, I think he'd really struggle with Oksendal's book. "Elementary Stochastic Calculus With Finance in View" from Thomas Mikosch is a quite good introduction in my opinion. More accessible in any case...
– SRKX
Jul 15 '11 at 7:45