Payoffs, and the One Relation That Needs No Model

An option is a contract giving its holder the right, without the obligation, to trade an asset at a fixed price on or before a fixed date.

Four terms specify it. The underlying is what may be traded. The strike is the fixed price. The expiry is when the right lapses. And the type is call for the right to buy, put for the right to sell.

The asymmetry between right and obligation is the whole product. The holder exercises only when it benefits them, so their payoff is never negative. That is worth something, which is why options cost money up front, and the entire subject is the question of how much.

One more distinction matters practically. A European option may be exercised only at expiry; an American option at any time up to it. Most index options are European, most single-stock options in the United States are American.

On the expiry date all uncertainty is gone and the value is a formula with no assumptions in it.

A call struck at K, with the underlying at S, is worth max(S - K, 0). If the asset is above the strike, exercise and capture the difference; otherwise let it lapse. A put is worth max(K - S, 0), the mirror image.

def call_payoff(S, K):
    return max(S - K, 0.0)

def put_payoff(S, K):
    return max(K - S, 0.0)

[call_payoff(s, 100) for s in (80, 100, 120)]   # [0.0, 0.0, 20.0]
[put_payoff(s, 100)  for s in (80, 100, 120)]   # [20.0, 0.0, 0.0]

The kink at the strike is the defining feature. Below it a call is worth nothing and is insensitive to the underlying; above it the call tracks the underlying one for one. Everything difficult about options comes from that kink being smoothed out before expiry, when it is still uncertain which side of it the asset will land on.

Prior to expiry an option trades above its immediate exercise value, and the excess has a name and a cause.

Intrinsic value is what the option would pay if expiry were now: max(S - K, 0) for a call. Time value is the market price minus that. A call struck at 100 with the underlying at 105, trading at 8, has 5 of intrinsic and 3 of time value.

Time value is not a fee for waiting. It is the value of the optionality itself: the chance that the underlying moves further in your favour, combined with the fact that it cannot move against you beyond the premium already paid.

Two consequences. Time value is largest at the money, where the outcome is most uncertain, and it decays to zero at expiry, when there is no chance left to pay for. An option that expires with the underlying exactly at the strike is worth nothing, however much it cost.

Now the relation that holds without any model of how the underlying behaves.

Build two portfolios, both European, same strike K and expiry T.

A: one call, plus cash equal to the present value of K. B: one put, plus one unit of the underlying.

At expiry, if S is above K: A exercises the call using the cash and holds the asset, worth S. B lets the put lapse and holds the asset, worth S. If S is below K: A lets the call lapse and holds cash K. B exercises the put, selling the asset for K, and holds K.

The two portfolios are worth the same in every state of the world. Two things with identical payoffs in all states must cost the same now, or one buys the cheap one and sells the dear one for a certain profit. So:

C  +  K * exp(-rT)   =   P  +  S

No distribution, no volatility, no assumption about drift.

The argument is a proof by contradiction and it uses one premise: risk-free profit from a zero-cost position does not persist.

That premise is weaker than it looks, which is why the conclusion is so strong. It does not require markets to be efficient in any broad sense, or prices to be right, or anyone to be rational. It requires only that if free money appears, somebody takes it.

This is the template for every derivatives result worth knowing. Construct two things with identical payoffs, argue they must have identical prices, and read off a relation. The next lesson applies exactly this template to a much harder case, where the replicating portfolio has to be rebalanced continuously rather than held.

Parity is not a trade so much as a constraint that makes several things possible.

Synthetic positions. Rearranging gives S = C - P + K*exp(-rT): long a call and short a put at the same strike behaves exactly like owning the underlying. Traders use this to take a position when the option market is more liquid than the cash market, or when holding the asset directly is awkward.

Data validation. A quoted call, put and forward that violate parity indicate a stale quote, a dividend not accounted for, or a borrow cost you have not modelled. It is one of the first checks on any option data set.

import math

def parity_gap(call, put, spot, strike, r, T):
    return (call + strike * math.exp(-r * T)) - (put + spot)

parity_gap(call=8.0, put=2.6, spot=105.0, strike=100.0, r=0.04, T=0.5)
#  about -1.58 : parity implies a put near 1.02, so 2.6 is unexplained

Implying the forward. Because parity holds for every strike, the observed call minus put across strikes reveals the forward price the market is using, including dividends and financing.

The proof assumed European exercise, and that assumption is load bearing.

For American options the two portfolios are no longer equivalent, because the put holder may exercise early and change the timing of cash flows. Parity weakens to an inequality, and the gap widens with dividends and interest rates.

Dividends shift the relation even in the European case: a holder of the underlying receives them and an option holder does not, so the present value of expected dividends enters the equation.

Borrow costs matter for the short leg. If the underlying is expensive or impossible to borrow, the arbitrage that enforces parity cannot be executed, and observed prices can sit outside the bound for as long as the constraint lasts.

That last point generalises. Every no-arbitrage relation is enforced by someone's ability to trade both sides. Where that ability is restricted, the relation becomes a guideline rather than a law.

A payoff diagram shows the value at expiry, and reading it as the description of the trade causes a specific and common error.

Buying a call is often described as cheap leverage: control 100 shares for a fraction of their cost, with losses capped at the premium. Both halves are true. What the diagram omits is that you paid time value, and time value goes to zero with certainty.

So the underlying rising is not sufficient. It must rise enough, and soon enough, to cover the time value paid. A call bought at 3 with the underlying at 100 and a strike of 100 needs the asset above 103 at expiry merely to break even, and a move to 102 that a shareholder would enjoy leaves the option holder down.

The last row is what the diagram cannot show, and it is why the next two lessons are about behaviour before expiry rather than at it.

Parity ties a call to a put. It does not price either of them.

Given a call price it produces the put price, and vice versa. Given neither, it says nothing. That is a real limitation, because the quantity a trader wants is the price of an option before expiry, and parity is silent on it.

The difficulty is precise. At expiry, value depends only on where the underlying is. Before expiry it depends on where the underlying might go, which means it depends on the distribution of future prices, and that is not observable.

So the problem seems to require forecasting: estimate how the asset will move, and price the option against that. That approach is possible and it is not what the market does.

The next lesson shows why it is not necessary. There is an argument that prices an option without anyone forecasting the asset's return at all, and it works by the same substitution used here: find something with an identical payoff.