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?


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.


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