Itô’s Lemma and Self-Financing Black–Scholes Hedging Portfolios

# Black-Scholes Portfolio


# Black-Scholes Portfolio












In the black-scholes model, the hedging portfolio is given (in some textbooks) by $$\Pi_t = V_t - \Delta S_t,$$ i.e., the portfolio consits of a long position in the option $V$ and $\Delta$ units of short positions in the stock $S_t$.

How does we get the change in the portfolio value, i.e., how does we get $$d\Pi_t = dV_t - \Delta dS_t$$ In some textbook they argue it is due to Ito's lemma. I know Ito's lemma, but I don't know how to apply it to get the above.

My Idea: In some textbooks they start with a (replicating) portfolio given by $$\Pi_t = \alpha S_t + \beta B_t,$$ where $B_t$ is the risk-free asset. Then, by assuming that the replicating portfolio is self-financing, we have $$d\Pi_t = \alpha dS_t + \beta dB_t,$$ i.e., the change in portfolio value is due to changes in market conditions and not to either infusion or extraction of cash.

Does we have $$d\Pi_t = dV_t - \Delta dS_t$$ because of the self-financing assumption? But why do some books say that we get it by Ito's lemma? How do we apply Ito's lemma to $$\Pi_t = V_t - \Delta S_t \qquad ?$$

Kind regards

## Answer by j4bert0 (score 1)

https://quant.stackexchange.com/a/67918

The "hedging portfolio" argumentation is problematic. In this paper Peter Carr comments to "Is the Hedging Argument Given in the Black-Scholes Paper Correct?". I believe this answers to your question.

The other way of arguing the BS equation, using the "replicating portfolio", can be found, for example, from Shreve II. As you have written, you can assume the self-financing condition. It can be subsequently shown, that the BS equation implies the existence of self-financing portfolio. Ideas on how to do this can be found from this post.

## Answer by Yoda And Friends (score 0)

https://quant.stackexchange.com/a/67917

I am not sure I have fully understood your question.

Anyway, consider your financial derivative as a function of time and stock: $$V_t = v(t, \ S).$$ Using Ito's lemma, we can recover: $$dV_t = \frac{\partial v}{\partial t}dt + \frac{\partial v}{\partial S}dS_t + \frac{1}{2}\frac{\partial^2 v}{\partial S^2}d<S>_t.$$ Being $\frac{\partial v}{\partial S} = \Delta$ by definition, you get: $$d\Pi_t = \frac{\partial v}{\partial t}dt + \frac{1}{2}\frac{\partial^2 v}{\partial S^2}d<S>_t.$$ Note a perfect hedge is not possible due to the diffusion of $S_t$.

Is that what you are looking for?