How can Ito's Lemma be used to show that a delta-neutral portfolio is instantaneously risk-free?

Question

The lecture notes I am currently reading give the following example of a delta-neutral portfolio:

• minus one derivative (whose value at time $t$, when the value of the underlying is $S_t$, is denoted $f(t, S_t)$)
• $\Delta := \frac{\partial f}{\partial S_t}$ shares of the asset underlying the derivative

Following this example is a question which asks me to show that a delta-hedged portfolio with value $V(t, S_t)$ is instantaneously risk-free, if $S_t$ is a diffusion, by using Ito's Lemma. The first line of the solution of this questions states that:

Ito's Lemma tells us that: $$dV(t, S_t) = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S_t} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S_t^2} (dS_t)^2$$

Could anyone help me to understand how the above expression has been deduced?

Answer 1

Once you subtract the delta term from your portfolio (thus canceling the middle term on the RHS), the two terms remaining have no uncertainty - they are deterministic. This is because square of $dS$ is the quadratic variation of the process S, presumably a deterministic (as in, known at the time of placing the hedge) quantity, and of the order $dt$.

Thus, the portfolio that remains has a deterministic drift with no uncertainty (No dependence on the brownian motion), so it is instantaneously risk free.