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# Animating the equation

My recap of the replicating portfolio:

Where does the idea that “you need the cash to buy the shares” show up?

### 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.

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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!

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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!

The recap table: