Why are there two expressions for the Black-Scholes hedging portfolio

Question

I am new to derivatives pricing and am trying to understand why there are two different expressions for the Black-Scholes hedging portfolio. The first approach, used in books like Hull, stipulates that the hedging portfolio consists of being short one option and long $\frac{\partial V}{\partial S}$ shares at any time $t$, that is: $$\Pi_t=-V(t,S_t)+\frac{\partial V}{\partial S}S_t.$$ In other references such as Shreve, the hedge portfolio is presented via the following tautology: $$\Pi_t=\frac{\partial V}{\partial S}S_t+\left(\Pi_t-\frac{\partial V}{\partial S}S_t\right).$$ This might be a stupid question, but are these two expressions just different ways of expressing the same thing? Is the cash position in the second equation equivalent to reinvesting the price of the option if we are selling it in the first place (i.e. $-V(t,S_t)$)?Any comments or explanations would be greatly appreciated.

Answer 1

The first portfolio $\Pi^{(1)}_t$ is a self-financing hedging portfolio. It is typically what you get when you delta hedge an option position (here short hence the minus sign, but it could be long without loss of generality) with shares of the underlying asset. If the only source of risk comes from the randomness of the underlying asset price $S_t$, then one can claim that $\Pi^{(1)}_t$ evolves at the risk-free rate i.e. $d\Pi^{(1)}_t = r\Pi^{(1)}_t dt$, because applying the self-financing property along with Itô shows that $d\Pi^{(1)}_t$ is a deterministic quantity (the randomness reflected by the $dS_t$ term disappears) and should hence evolve at the risk-free rate under no arbitrage assumptions.

The second portfolio $\Pi^{(2)}_t$ is a self-financing replicating portfolio. It is composed of shares of the underlying and money placed/withdrawn from a risk-free money market account (or equivalently a position in zero-coupon bonds). Usually, $\Pi^{(2)}_t$ is used to dynamically replicate an option position $V_t$, in the sense that, for any infinitesimal period of time we want to make sure that $d(\Pi^{(2)}_t - V_t) = 0$.

The equations $d\Pi^{(1)}_t = r\Pi^{(1)}_t dt$ and $d\Pi^{(2)}_t - dV_t = 0$ are two different yet equivalent ways of deriving a pricing PDE under no arbitrage assumptions (at least in a complete single factor market).