🧮 How options are priced: Black-Scholes, binomial trees and put-call parity

🎯 In 30 seconds: An option's price depends on six things: the stock price, the strike, time to expiration, interest rates, dividends, and volatility. Five of those you can look up. Volatility you cannot, which is why traders talk about implied volatility. The Black-Scholes formula turns the six inputs into a price, and a binomial tree shows the logic behind it: an option can be copied with shares and borrowed money, so it must cost the same as the copy. Put-call parity is the simplest consequence, and you can check it on any option chain.

The six inputs

Every pricing model, from the simplest to what market makers run, starts from the same list. Here is how each one moves the price of a call and a put, with everything else held still.

The Greek letters you met in The Greeks are just these sensitivities with numbers attached: delta for the stock price, theta for time, vega for volatility, rho for interest rates.

Volatility is the odd one out. The others are known today. Future volatility is not. So traders turn the question around: given the market price of an option, what volatility would the model need to produce it? That number is implied volatility, covered in Implied volatility.

Value today vs value at expiration

At expiration, an option is worth only its intrinsic value: the "hockey stick" you see in every payoff chart. Before expiration, it is worth more, because there is still a chance of a bigger payoff. The model curve sits above the hockey stick. The gap is time value.

The Black-Scholes formula

In 1973, Fischer Black and Myron Scholes published a formula for the price of a European option, one that can only be exercised at expiration. Robert Merton derived it another way. Scholes and Merton received the 1997 economics Nobel prize for it. Black had died in 1995.

For a European call on a stock with no dividends:

C = S x N(d1) - K x e^(-rT) x N(d2)

d1 = [ ln(S/K) + (r + σ²/2) x T ] / (σ x √T)
d2 = d1 - σ x √T

And for the matching put:

P = K x e^(-rT) x N(-d2) - S x N(-d1)

What the symbols mean, in plain words:

• S is the stock price, K the strike, T the time to expiration in years, r the risk-free interest rate per year (continuously compounded), σ (sigma) the volatility per year.
• ln is the natural logarithm. e^(-rT) is the discount factor: what $1 paid at expiration is worth today.
• N(x) is the chance that a standard bell-curve variable is below x. Spreadsheets have it built in: NORM.S.DIST(x, TRUE).

A plain reading of the call formula: you receive the stock if the option ends in the money (the first term), and you pay the strike, discounted to today, if it ends in the money (the second term). N(d2) is the model's risk-neutral probability that the call finishes in the money. N(d1) is the call's delta.

Worked example: pricing a call and a put

Hypothetical inputs: S = $100, K = $100, T = 0.5 years, r = 4%, σ = 25%, no dividends.

1. ln(S/K) = ln(1) = 0.
2. (r + σ²/2) x T = (0.04 + 0.03125) x 0.5 = 0.035625.
3. σ x √T = 0.25 x 0.7071 = 0.17678.
4. d1 = 0.035625 / 0.17678 = 0.2015. d2 = 0.2015 - 0.1768 = 0.0247.
5. N(d1) = 0.5799, N(d2) = 0.5099, N(-d1) = 0.4201, N(-d2) = 0.4901.
6. Discounted strike: K x e^(-0.04 x 0.5) = 100 x 0.98020 = $98.02.
7. Call = 100 x 0.5799 - 98.02 x 0.5099 = 57.99 - 49.98 = $8.01.
8. Put = 98.02 x 0.4901 - 100 x 0.4201 = 48.04 - 42.01 = $6.03.

Per contract, that is $801 for the call and $603 for the put. Both are pure time value, since the stock is exactly at the strike.

The binomial idea in one step

Black-Scholes can look like magic. A binomial tree shows where it comes from. Imagine the stock can only do one of two things over the next six months: go up 10% or go down 10%.

Step 1: copy the option. Buy 0.5 shares and borrow some money. If the stock goes to $110, the half share is worth $55. If it goes to $90, it is worth $45. Borrow exactly the amount that grows to $45 in six months: $45 x e^(-0.02) = $44.11. Then in the up case you have $55 - $45 = $10. In the down case $45 - $45 = $0. That is exactly what the call pays.

Step 2: price the copy. The copy costs 0.5 x $100 - $44.11 = $5.89. Since the copy and the call pay the same in every case, they must cost the same. If the call cost more, you could sell it, buy the copy, and pocket the difference with no risk.

Step 3: see the shortcut. The same answer comes from the formula in the figure, using p = (1.0202 - 0.90) / (1.10 - 0.90) = 0.601. That p is called a risk-neutral probability. It is not anyone's forecast. It is the weight that makes the stock grow at the risk-free rate on average.

Real models use hundreds of small steps instead of one big one. As the steps get smaller, the binomial price gets closer to the Black-Scholes price. Binomial trees have one big advantage: at each step you can check whether exercising early would be worth more than holding. That is how brokers and exchanges value American-style options, which Black-Scholes does not handle on its own.

Why is $5.89 lower than the $8.01 above? Different assumptions. A single up-or-down 10% step over six months is a much calmer stock than 25% volatility. The method is the point, not the number.

Put-call parity

For European options on a stock with no dividends, same strike, same expiration:

C - P = S - K x e^(-rT)

Why it must be true: a long call plus a short put pays exactly (stock price - strike) at expiration, whatever happens. If the stock is above the strike, the call pays S - K and the put expires. If it is below, the call expires and you pay K - S on the put, which is also S - K. Owning the stock and owing K at expiration pays the same. Two things that always pay the same must cost the same today.

Check it with the worked example.

It matches. That is not luck: the Black-Scholes call and put prices satisfy parity by construction.

What a violation would look like. Suppose a market showed the same call at $8.01 but the put at $5.50. Then C - P = $2.51, which is $0.53 more than $1.98. A trader could sell the call, buy the put, and buy the stock. That costs $100 - $8.01 + $5.50 = $97.49 per share, borrowed at 4%. In six months, the position is worth exactly $100, whatever the stock does, because either the call is assigned or the put is exercised at $100. Repaying the loan costs $97.49 x e^(0.02) = $99.46. Locked-in profit: $0.54 per share, $54 per contract, before costs. This trade is called a conversion.

In real markets, professional traders close gaps like this within moments, so you will rarely see one big enough to beat commissions and the bid-ask spread. The value of parity for you is as a sanity check: if the numbers on a chain do not roughly fit it, look for a dividend, a hard-to-borrow stock, or a stale quote.

Dividends and American options

• Dividends. A known dividend lowers the stock price on the ex-date, so it lowers calls and raises puts. A simple fix is to subtract the present value of expected dividends from S before using the formula.
• American options. Most US stock and ETF options can be exercised early. For a call on a stock with no dividend, early exercise is almost never worth it, so the European price is a good guide. For puts, and for calls just before a dividend, early exercise can be worth something. That is when the binomial method matters. Parity becomes a range instead of an exact equation.
• Index options like SPX are European style, so the formulas above fit them more closely. See Index options.

Try it

Pick a stock from your Watchlist that pays no dividend. On its option chain, find a call and a put with the same strike near the stock price, about two to three months out. Take the midpoint of the bid and ask for each. Look up a current short-term Treasury yield on the interest rates page. Then compute both sides of C - P = S - K x e^(-rT). Write the gap in your trading journal, and explain it in one line: early exercise value, bid-ask spread, or something else.

Sources

• Options Industry Council (OIC): Black-Scholes formula
• OIC: Put-call parity
• NobelPrize.org: The Sveriges Riksbank Prize in Economic Sciences 1997, press release
• OIC: Understanding options Greeks