Using Itô’s Lemma to Show a Portfolio Is Self-Financing

Summary

The example considers a portfolio value defined as Brownian motion squared minus time. Applying Itô’s lemma to this function produces a drift term from the time derivative and the second derivative with respect to the Brownian state, alongside a stochastic term from the first derivative. The two drift contributions cancel, leaving the differential equal to twice Brownian motion multiplied by its increment.

The explanation shows how the result follows directly from the derivatives and the quadratic variation term in Itô’s lemma. It clarifies the calculation in the stated example, but the thread provides no broader construction of trading positions or discussion of portfolio constraints. The calculation by itself illustrates the differential identity; it does not develop a general test for whether an arbitrary portfolio is self-financing.

Key ideas

• For the stated function, the time derivative contributes a negative drift term.
• The second derivative in Itô’s lemma contributes an equal positive drift term for Brownian motion.
• The drift terms cancel, leaving only the stochastic differential term.
• The example explains one identity and does not establish a general self-financing portfolio method.

Tags

# Showing a portfolio is a self financing portfolio


# Showing a portfolio is a self financing portfolio












I'm having some trouble understanding how you are able to use Ito's lemma to show in the example I've attached it's equal to $d\pi = 2BdB$ because I can't collapse the $dt$ term I've attached the Ito's lemma I'm using and I can't grasp what is the $a(X,t)$ term or $b(X,t)$ term, any help will be much appreciated

## Answer by Vladimir Nabokov (score 1, accepted)

https://quant.stackexchange.com/a/37603

$$ d\Pi_t=d(B^2_t-t)=d(B^2_t)-dt=2B_tdB_t+dt-dt=2B_tdB_t $$

Your question is why?

Let $f(B_t, t) = B_t^2 -t=\Pi_t$, then Ito's Lemma tells us that:

$$ df = (\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2f}{\partial x^2})dt + \frac{\partial f}{\partial x}dB_t=(-1+1)dt+2B_tdB_t. $$

Make more sense now?

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