No-Arbitrage and Put-Call Parity: Option Pricing Before Any Model

Before Black–Scholes, before any distributional assumption, a large share of derivatives pricing follows from one principle: two positions with identical payoffs must have identical prices. If not, buy the cheap one, sell the expensive one, and collect free money — and free money gets collected until prices align. Interviews test this layer first because it requires no calculus, only clear thinking.

Forwards: pricing by replication

What's the fair forward price $F$ to agree today for delivery of a (non-dividend) stock at time $T$, with the stock at $S_0$ and interest rate $r$?

No expectations about the stock's drift are needed. Replicate: borrow $S_0$, buy the stock, hold to $T$. You deliver the stock and owe $S_0 e^{rT}$. That position is a forward, so

$$F = S_0 e^{rT}.$$

If the market forward were higher, sell it and run the replication (cash-and-carry arbitrage); lower, reverse it. The stock's expected return never appears — replication already prices the risk. Dividends reduce the carry cost ($F = S_0e^{(r-q)T}$); for commodities, storage costs add to it. This "price = cost of manufacturing the payoff" logic is the template for everything that follows.

Put-call parity

A European call $C$ (right to buy at strike $K$) and put $P$ (right to sell at $K$) on the same stock, strike, and expiry satisfy

$$C - P = S_0 - K e^{-rT}.$$

Proof by payoff table. Portfolio A: long call, short put. At expiry it pays $\max(S_T - K, 0) - \max(K - S_T, 0) = S_T - K$ in every state. Portfolio B: long stock, borrow $Ke^{-rT}$; at expiry, also $S_T - K$. Identical payoffs everywhere → identical prices today. No model, no volatility, no probabilities.

What interviews extract from parity:

• Synthetics: long call + short put = synthetic long forward. Any three of {call, put, stock, bond} manufacture the fourth. Desk vocabulary — "buying the combo," "conversion," "reversal" — is parity arbitrage.
• One volatility, not two: parity forces a call and its same-strike put to carry the same implied volatility. If someone quotes you a vol smile where they differ, that's not a view, it's an arbitrage.
• Instant consistency checks: given $S=100$, $K=100$, $r \approx 0$, and a call at \7, the put must be \7 too; if the market shows \5, you can state the trade (sell call, buy put, buy stock — lock in \2) in one breath.

Model-free bounds

Same style of argument bounds option prices with no assumptions at all: a call is worth at least its intrinsic forward value, $C \ge \max(0,\, S_0 - Ke^{-rT})$, and never more than the stock, $C \le S_0$; calls decrease in strike, and butterflies can't be negative — $C(K_1) - 2C(K_2) + C(K_3) \ge 0$ for equally spaced strikes, because the butterfly's payoff is non-negative everywhere. Violations of these are pure arbitrage regardless of any model. (That butterfly inequality is secretly the statement that the implied probability density is non-negative — the gateway to the Breeden–Litzenberger result.)

The interview version

"Why doesn't the expected return of the stock appear in the forward price?" Because the forward is priced by replication, not by forecast: the hedge (own the stock) eliminates the exposure the drift would compensate. Whoever holds the replicating portfolio bears the stock's risk and earns its return; the derivative's price is just the cost of the recipe. Internalizing this — prices come from hedging, not predicting — is the conceptual leap that makes Black–Scholes obvious rather than mysterious, and it's covered next in the binomial model.