how I understand the Black-Scholes formula

Paid subs will recall my story of Doug teaching Black-Scholes to my cohort at SIG back in 2001. Four hours in one day to explain the assumptions and four hours the next day to derive the equation. I tried to keep up but dropped off embarrassingly quickly.

I did that webinar to explain how I eventually came to understand the formula. The recording is paywalled but these are the slides for the talk.

Here’s the distilled version:

Start with what we know.

At expiry, a call option is worth the stock price minus the strike price (or zero if the call is “out-of-the-money”)

So today, the call price equals

“the current expected value of the stock given the call is exercised”

minus

“the discounted strike price”

[The strike price gets discounted for both the time value of money AND the probability of exercise.]

Let’s work through this with common sense.

You’re looking at a 1-year $50 strike call. The stock trades at $50 today, risk-free rate is 5%.

Say the call has a 50% chance of being in-the-money.

Let’s also assert that in the state of the world where the call gets exercised, the stock is on average $58*. That happens 50% of the time, so the expected value is 0.50 × $58 = $29.

*Think of this like rolling a die: given that you roll greater than 3, what’s the expected value? It’s 5 (the average of 4, 5, 6).

What about the discounted strike price?

The $50 strike discounted to present value is $50 × e^(-0.05) = $47.56. With a 50% exercise probability: 0.50 × $47.56 = $23.78.

The call value from our definition

“the current expected value of the stock given the call is exercised”

minus

“the discounted strike price”

maps to

$29 – $23.78 = $5.22.

The key insight: we can replicate a call option with a portfolio of stock and cash

You can replicate a call’s payoff by owning some amount of stock. This amount is more commonly referred to as the “delta” (or hedge ratio).

This delta changes as the stock becomes more or less likely to finish in-the-money. As the stock rises, you buy more shares to replicate the call’s potential payoff. As it falls, you sell shares since exercise becomes less likely. You’re buying high and selling low—creating negative cash flows. That sum of negative P&L should is what the option is worth.

You can either buy the option (pre-paying these cash flows) or manufacture it yourself through this delta hedging strategy.

In an arbitrage-free world, the option price must equal the present value of these replicating cash flows. If the option were priced with higher volatility than actual, you could short it, hedge with shares, and pocket the difference.

The self-financing part is elegant.

To replicate the call, you need to buy the “delta” quantity of shares. With what cash? You borrow it—specifically, you borrow $23.78 and use that cash to buy the shares today. This is why the strategy is self-financing: we’re simply borrowing against a future cash flow.

Why does this work?

At expiration, if the call gets exercised, you sell your stock at $50 to the call owner. With 50% exercise probability, your mathematical expectation is to receive $25 in one year. So you can borrow the present value of $25 today (ie $23.78), use that borrowed money to buy the shares, knowing you can repay the loan at expiry with the proceeds from selling those shares.

Notice why call values increase with interest rate:

a call is ultimately the difference in value between the number of shares you need to buy (delta shares) and the number of shares you can afford to buy via the loan. The higher the interest rate, the less you can borrow, the fewer shares you can buy, so the call value—which bridges that gap—increases.

In a sentence…

a call value represents the difference between how much stock you need to buy and how much you can afford to buy to achieve that hockey stick payoff.

Fwiw…

One of the webinar attendees says this diagram made it click. I’ve never seen it anywhere else and came up with it when I wrote A Visual Appreciation for Black-Scholes Delta