Why a Payoff Isn't a Price

You can already draw a call’s payoff in your sleep: a flat line, a kink at the strike, then a 45° climb. That diagram answers exactly one question — if the stock ends at price X, what is this thing worth? It is a map of every possible future, laid out side by side. What it stubbornly refuses to tell you is the one number you actually have to write a cheque for: how much should the option cost today, before you know which of those futures shows up?

That gap — between a fan of “what it’ll be worth then” and a single “what it’s worth now” — is the entire subject of this topic. Pricing is the act of collapsing all of those possible futures into one number today. Let’s start with your instinct.

Payoff is a future; price is a number today

Hold the two ideas apart, because conflating them is the original sin of options pricing:

• A payoff is what the contract delivers at expiration, as a function of the final stock price. It’s the hockey stick. There’s one payoff per possible ending — a whole curve of them.
• A price (the premium) is what you pay now, in today’s money, before the dice are thrown. It’s a single scalar.

Here’s the call payoff again, just to anchor what we’re not pricing yet:

The payoff diagram is downstream of the stock price — feed it an ending price, it hands back a value. The premium is upstream of everything: it’s the cost of admission, paid while the future is still undecided. To turn a curve of futures into one number, you need a rule for weighing and combining those futures. Picking that rule is where pricing gets interesting — and where the obvious idea face-plants.

The naive idea, and why it dies

The tempting move is the one you make for any gamble: price = the expected payoff, discounted back to today. Take each possible payoff, weight it by how likely it is, add them up, then shave off a bit for the time-value of money. Clean. Intuitive. Wrong — or at least, useless in practice. Two leaks sink it.

Leak 1 — you don’t know the real probabilities. To average the payoffs you need the actual odds the stock rises versus falls. Nobody knows them. Two analysts will hand you two different numbers, and the price will swing with whatever each one guessed. A pricing rule that depends on a quantity nobody can observe isn’t a rule; it’s a vibe.

Leak 2 — and this one is fatal — you don’t know the discount rate. Risky cash flows must be discounted at a rate that includes a risk premium: the extra return investors demand for bearing that risk. But an option’s riskiness is a moving target. A call is wildly leveraged when the stock is near the strike and tame when it’s deep in the money — its effective risk changes as the stock moves. So there’s no single, stable discount rate to apply, and no one agrees on what the risk premium for this option even is. The naive formula needs two numbers you can’t pin down, one of which won’t sit still.

So we’re stuck — unless we can find a way to price the option without ever naming the real odds or the risk premium. Astonishingly, we can. The escape hatch isn’t probability at all. It’s plumbing.

No-arbitrage: the law of one price

Forget forecasting. Lean instead on a near-iron law of functioning markets: two things that deliver identical cash flows in every possible future must cost the same today. If they didn’t, you’d buy the cheap one, sell the expensive one, and walk away with risk-free money for nothing — and traders would do this until the prices snapped back together. That free-lunch trade is called arbitrage, and the assumption that it can’t persist is no-arbitrage, also known as the law of one price.

A toy example. Suppose one dollar bill traded for $0.99 at one bank and $1.01 at another. You’d borrow nothing, buy dollars at $0.99, sell them at $1.01, and pocket two cents per dollar, all day, with zero risk. That can’t last. Real markets aren’t this silly with dollar bills — but the principle is the lever that prices every derivative ever written.

The payoff: if we can build a portfolio of things we already know the price of (the stock and cash) that reproduces the option’s payoff in every future, then the option must cost exactly what that portfolio costs. Not approximately — exactly, or arbitrage appears. We’ve traded an impossible forecasting problem for a solvable construction problem.

Replication: building a call out of stock and cash

Let’s do it with the smallest possible universe — a one-period, two-outcome world. Today the stock is S₀ = $100. In one period it goes to exactly one of two places: up to $110 or down to $90. (Yes, real stocks have infinitely many endings; we’ll stack these simple steps into a fine lattice in the next lesson. The logic is identical and clearer with two.)

Price a call struck at $100. Its payoff in each future:

We want a portfolio — Δ shares of stock plus B dollars of cash (B negative means we borrow) — that pays exactly $10 in the up-state and $0 in the down-state. To keep the arithmetic clean we’ll set the interest rate r = 0 for now (so cash is just cash), then add discounting at the end.

Step 1 — find Δ, the share count. Δ is chosen so the portfolio’s swing between states matches the option’s swing. The option moves $10 (from $10 down to $0); the stock moves $20 (from $110 to $90). So:

$\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{10 - 0}{110 - 90} = \frac{10}{20} = 0.5 \text{ shares.}$

Half a share rises and falls by exactly half the stock’s $20 swing — i.e. $10 — which is precisely the option’s swing. The share leg now tracks the option’s sensitivity. (That Δ is the option’s delta, the first Greek; we devote a whole later lesson to it.)

Step 2 — find B, the cash. Match one state exactly and the other falls into line. In the up-state the portfolio must total $10:

$\Delta \cdot S_u + B = C_u \;\Rightarrow\; 0.5 \times 110 + B = 10 \;\Rightarrow\; 55 + B = 10 \;\Rightarrow\; B = -45.$

So we borrow $45. Sanity-check the down-state: 0.5 × 90 + (−45) = 45 − 45 = $0 — exactly the call’s down payoff. The portfolio “0.5 shares, $45 borrowed” mimics the call in both futures.

Step 3 — price the call. By no-arbitrage, the call must cost whatever this portfolio costs to assemble today:

$C_0 = \Delta \cdot S_0 + B = 0.5 \times 100 - 45 = 50 - 45 = \boxed{\$5.}$

The fair premium is $5. Charge $6 and someone sells you the call, builds the $5 portfolio, and banks a riskless $1. Charge $4 and the trade runs the other way. Only $5 leaves no free lunch on the table.

The punchline: the real odds vanish

Now the magic trick that names this whole topic. Take the replication price and grind the algebra — substitute Δ and B back in and rearrange. Out pops a formula that looks exactly like the naive “expected payoff, discounted” we threw away:

$C_0 = \frac{1}{1+r}\,\big[\,q\cdot C_u + (1-q)\cdot C_d\,\big], \qquad q = \frac{(1+r)\,S_0 - S_d}{S_u - S_d}.$

It’s a discounted expected payoff — but the weight q is not the real-world probability. The genuine odds of an up-move (your coin-flip, your analyst’s guess) cancelled out completely during the algebra. They are nowhere in the formula. In their place sits q, a manufactured number — the risk-neutral probability — defined purely by today’s price and the two future prices.

Plug in our numbers with r = 0: q = (100 − 90)/(110 − 90) = 10/20 = 0.5, and C₀ = 0.5·10 + 0.5·0 = $5. Same $5, derived a second way. Here q happened to equal one-half, but that’s a coincidence of the symmetric $10 moves — shift the up/down prices and q drifts away from the real odds entirely.

What q means: it’s the probability that would make the stock’s own expected return equal the risk-free rate — the odds in a hypothetical world where nobody charges a risk premium. We don’t believe that world is real. We just price as if it were, because in that world the impossible discount-rate problem evaporates: everything discounts at the clean, observable, risk-free rate r. You price the option without forecasting the stock. That is the headline of options pricing, and the next lesson gives q its full derivation.

Putting it together

A payoff diagram is a fan of futures; a price is one number today, and the bridge between them is the whole job of options pricing. The intuitive bridge — average the payoffs and discount — fails, because it needs the real odds (unknown) and a risk-adjusted rate (unknowable and moving). The way out is no-arbitrage: copy the option with stock and cash, and its price is forced to equal the copy’s cost. Crunch that, and the real odds vanish, leaving a clean risk-neutral valuation that discounts at the risk-free rate. You price the option without ever forecasting the stock.

Next up — The Binomial Model in Full — we give the risk-neutral probability q its formal derivation and stack these one-step worlds into a multi-period lattice that can price real options.