# Cash deposit in replicating portfolio for BS equation unnecessary?

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.

## 1 Answer

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:

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