Cash deposit in replicating portfolio for BS equation unnecessary?

Question

The book on Option Valuation Methods that I currently study (Higham 2013) constructs a replicating portfolio $\Pi = A(S,t)S + D(S,t)$ for deriving the BS PDE, where $D$ is a cash deposit. $D$ does not appear in the Black-Scholes equation, neither is it essential in the derivation of the equation, so why bother including it in our portfolio?

Higham does say: "We want to make the portfolio self-financing, that is, beyond time t=0 we do not want to add or remove money. This can be achieved by using the cash account to finance the updated $-$ the money needed for, or generated by, the asset rebalancing." So is he only using $D$ to give extra intuition?

Edit For example; $\Pi=V-\Delta S$ is also a valid portfolio for deriving the BS PDE.

Answer 1

The key point here is that the portfolio must be self-financing, namely the initial option premium $V_0$ should be enough to allow you to hedge it throughout its life. If not, the option price $V_0$ is either too low or too high.

Because the option is written on the asset $S$, buying or selling $S$ is how you neutralize the changes in value of the option: for example, if you are long a call option and you need to hedge it, you know its value will increase if the asset price increases, therefore you need to be short the asset in a quantity $\Delta$ to neutralize the gains you make on $V$ when $S$ goes up and vice versa.

However, the value of the asset holding $\Delta S$ will not always perfectly offset the value of the option $V$. The deposit account $D$ allows you to match things: you might withdraw or contribute to it if you need to modify your holding $\Delta S$ in order to neutralize movements in $V$.

This is because options are non-linear derivatives: linear derivatives such as forwards only need the underlying asset to be hedged because a price move in the asset has a linear impact on the derivative price, however in the case of options we know the price has a non-linear behavior to changes in the underlying. For example, if you are long a call option you have: $$\lim_{S\rightarrow+\infty}\Delta(S)=1$$ Therefore if the price of the asset increases we need to be able to drawn cash from a deposit to keep increasing our allocation $\Delta$ in order to hedge $V$.

There are multiple ways to express the hedging portfolio but they all can be pinned down to an equation of the form: $$a(t,S)V+b(t,S)S+c(t,S)D=0$$ Namely a position $a(t,S)$ in an option $V$ needs to be hedged with the asset $S$ which is bought or sold in a quantity $b(t,S)$; any required additional financing $c(t,S)$ must be borrowed or lent at a rate $r$.

Finally, note that the portfolio $V-(\Delta S+D)=0$ you mention in your edit yields the correct PDE but is not self-financing: see my answer Dynamic Delta Hedging And a Self Financing Portfolio. Indeed, suppose at some time $t$ you are required to change your allocation $\Delta$: where does the money come from? You need to have a coefficient assigned to $D$ so that any withdrawal/contribution to the deposit offsets any change in the asset's allocation:

$$V-\big(b(t,S)S+c(t,S)D\big)=0$$