In his book 'Stochastic Calculus for Finance II' Shreve uses the term: 'Short Option Hedging Portfolio' on page.156 (4.5.3). Can someone please explain this term with some kind of an example?
It is preventing me from understanding why Portfolio Value evolution is equated with Option Value Evolution to derive the Differential Equation of Black-Scholes-Merton.
From the book mentioned in the question, the short hedging option portfolio is used to replicate the option value with investment in the stock and money market account so that the portfolio value $X(t)$ at each time $t \in [0, T]$ agrees with the option value $c(t, S(t))$. That is, \begin{align*} c(t, S(t)) = \Delta(t) S_t + \Delta_t^1 B_t, \end{align*} where $B_t = e^{rt}$ is the money market account value, and \begin{align*} \Delta_t^1 = \frac{c- \Delta(t) S_t}{B_t} \end{align*} is the units invested in the money market account. We require that the change, over the infinitesimal interval $[t,\, t+dt]$, of the option value is from the change of the invested securities, that is, \begin{align*} dc(t, S_t) = \Delta(t)dS_t+\Delta_t^1 dB_t.\tag{1} \end{align*} In other words, it is self-financing.
From $(1)$, we obtain that \begin{align*} \left(\frac{\partial c}{\partial t}+\frac{1}{2}\frac{\partial^2 c}{\partial S^2}\sigma^2 S_t^2 \right)dt + \frac{\partial c}{\partial S}dS_t = \Delta(t)dS_t+r(c- \Delta(t) S_t) dt. \end{align*} Then \begin{align*} \Delta(t) = \frac{\partial c}{\partial S},\tag{2} \end{align*} and \begin{align*} \frac{\partial c}{\partial t}+rS_t\frac{\partial c}{\partial S} + \frac{1}{2}\frac{\partial^2 c}{\partial S^2}\sigma^2 S_t^2 = rc. \end{align*} Here, $(2)$ is called the delta hedging rule in the book.
Recognizing that there are both long and short hedge portfolios, the long portfolio involves combining a long stock position with $\beta$ (it is unknown) written call options on the stock.
The hedge is created by selling just enough calls to offset changes in the value of the stock position. This portfolio will involve a net investment of funds because the premium income received from writing the calls will not be sufficient to purchase the stock position.
Similarly, the short hedge portfolio involves shorting the stock, buying call options to hedge the position against upward changes in the stock price, and investing the balance of the funds in a risk-less asset.
Observe that, for both the long and short hedge portfolio, as the stock price changes the option hedge has to be continuously adjusted to maintain the hedge position.
In order to determine the number of call options to buy, let $V=\beta C-S$ be the value of the hedge portfolio. From the hedge portfolio construction: $$\frac{\partial V}{\partial S}=\beta\frac{\partial C}{\partial S}-1=0\implies \beta=\frac{1}{\frac{\partial C}{\partial S}}$$
Given this specification for $\beta$, the change in the value of risk-less hedge portfolio will earn the risk-less rate of interest (this follows from the assumption that interest ).
In terms of arbitrage portfolios, this condition is necessary in order to prevent the execution of arbitrage trades by either borrowing at the risk-less rate and buying the hedge portfolio or selling the hedge and investing the funds at the riskless rate.