How Options Are Priced: Binomial Trees, Black-Scholes and Put-Call Parity

Imagine someone offers you a bet: in one year you get $20 if a stock has risen, otherwise nothing. How much is that worth to you today? The obvious answer is: it depends on how likely a rise is. The surprising answer from financial mathematics is: no, it doesn’t matter. The fair price can be calculated without knowing where the stock is heading.

In the 1970s this insight was a revolution. It turned options trading – until then something close to gambling among dealers – into one of the largest financial markets in the world. In this lesson we rebuild it step by step.

A tree with two branches#

We start with the simplest possible model, the binomial model. A stock costs $100 today. In one year it will be at either $120 or $80. For simplicity, interest rates are zero for now.

A call with a $100 strike then pays either $20 (if the stock is at $120) or $0 (at $80).

Rebuilding the call#

The trick is called replication: we rebuild the same payoff from things whose price we know – shares and a loan.

1. The stock swings by $40 (between $120 and $80), the call by $20. So we need 0.5 shares per call. This ratio is the delta.
2. In one year 0.5 shares are worth $60 or $40. The call pays $20 or $0. The difference is exactly $40 in both worlds.
3. So we borrow $40 today and repay it in one year. What remains is $20 or $0 – exactly the call’s payoff.

Cost of this replica today: 0.5 × $100 = $50 for the shares, minus the $40 loan = $10. That is what the call must cost. If it cost $12, you could sell it, buy the replica for $10 and pocket $2 risk-free – traders would keep doing this until the price was back at $10. This logic is called no arbitrage.

Risk-neutral valuation#

There is a more elegant way to get the same result. We pretend that all investors are completely indifferent to risk. Then every investment would earn exactly the risk-free rate on average. For our stock at a 0% rate this means the probability q of a rise must make the expected price equal to $100.

The call is then the discounted expected value of its payoff: 0.5 × $20 + 0.5 × $0 = $10. The same result as the replica.

With interest the calculation changes only slightly. At 5% the stock must reach $105 on risk-neutral average. That gives q = (1.05 − 0.8) ÷ (1.2 − 0.8) = 0.625, and the call costs 0.625 × $20 ÷ 1.05 = $11.90. Higher rates therefore make calls a little more expensive – because the loan in the replica costs less.

From the tree to the Black-Scholes formula#

Two states in a year are of course unrealistic. But if you divide the year into many small steps – days, hours, seconds – and let the stock move up or down a little in each step, you get a tree with countless end points. In the limit of infinitely many steps, prices follow a lognormal distribution and the tree turns into a closed formula: the Black-Scholes formula.

S is the stock price, K the strike, T the time in years, r the interest rate, σ volatility and N the cumulative standard normal distribution. The formula looks intimidating but follows the same logic as our tree:

• N(d₁) is the delta – the number of shares in the replica. S × N(d₁) is the share part.
• K × e^−rT × N(d₂) is the loan part: the discounted strike, weighted by N(d₂), the risk-neutral probability that the call is exercised.

Of the five inputs, four can be read directly from the market. Only future volatility is unknown. In practice the formula is therefore usually turned around: from an option’s market price you calculate the volatility the market expects – the implied volatility. Traders often don’t even quote options in dollars but in volatility points.

Put-call parity#

One of the most beautiful relationships in finance needs no model at all. If you buy a call and sell a put with the same strike and expiry, at expiry you will always have bought exactly one share at price K – wherever the stock is: above K you exercise the call; below it the share is put to you through the put.

This combination must therefore cost as much as the stock today, minus the present value of the strike you only pay later:

Put-call parity holds regardless of any model, as long as you can borrow the stock and invest money at the interest rate. If the market deviates, arbitrageurs buy the cheap side and sell the expensive one – within fractions of a second.

Where the model reaches its limits#

Black-Scholes rests on simplifying assumptions: constant volatility, no price jumps, normally distributed returns, no transaction costs, continuous trading. Reality differs, and the market knows it:

• Volatility smile and skew: if Black-Scholes fitted perfectly, all options on a stock would have the same implied volatility. Since the 1987 crash, however, far out-of-the-money puts have been systematically “more expensive”. Investors pay for crash protection, and real return distributions have fat tails.
• Price jumps: with takeovers, profit warnings or bankruptcies a price jumps rather than moving in small steps. Perfect replication is then impossible.
• American options: most stock options can be exercised at any time, not only at the end. For them binomial trees are used, checking at every node whether early exercise is better.

Summary#

Options can be rebuilt from shares and a loan. Because the same thing must cost the same, their price follows from the stock price – without any forecast about direction. Risk-neutral valuation is the calculation trick behind it, the Black-Scholes formula the limit of infinitely many small steps, and put-call parity links calls and puts immovably. Whatever the model doesn’t know is captured in implied volatility – and in its lopsided smile.