Where does the idea that “you need the cash to buy the shares” show up?
That’s the source of the minus sign in Drake's representation:
The Motion Animating The Equation
Portfolio component: cash loan
In the model, how do you borrow money to buy shares?
You sell a T-bill (zero coupon bond) with a face value of the probability-weighted strike.
The probability-weighted strike is the amount of cash we expect to receive at maturity from the shares we sell.
Strike * N(d2)
$125 * 28.8% = $36
If we sell a 1 year T-bill with a face of $36, then today we receive the present value of $36:
$36e^(-.10%) = $32.57
Portfolio component: shares
The delta-weighted share quantity tells us how much stock we need to own today to hedge the value of the stock conditional upon the strike being in-the-money:
S* N(d1)
$100 * .397 = $39.77
We need to own $39.77 worth of stock to be hedged against the possibility of the stock going in the money.
🔥
The value of the call option emerges
We borrow $32.57 today
We invest it in the stock.
We need more stock to cover the contingency that the call gets assigned. On average we need:
$39.77 - $32.57 = $7.20
The value of the call option is therefore the price that reflects the full cost to replicate its payoff!
⛏️
Decompose the p/l:
The loan cost the interest on the T-Bill:
$32.57 - $36= ($3.43)
In expectancy terms, I will be selling you $39.77 worth of shares for only $36:
$36 - $39.77 = ($3.77)
Net P/L = ($7.20)
The replicating portfolio will cost you $7.20 in expectancy, therefore that must be the value of the call option!
