1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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:

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.