LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A listed equity option is a contract that gives its holder a choice, at a cost called the premium, to buy or sell the shares later at a fixed strike. A call is the choice to buy. A put is the choice to sell. At expiration the call's gross payoff is the larger of spot minus strike and zero. The put's is the larger of strike minus spot and zero. The holder's gain subtracts the premium. The seller's gain flips that picture. A one-step example with a 50 percent chance of each outcome prices a call by discounting the two possible payoffs.
Why it matters
Later models, including the binomial tree associated with Cox, Ross, and Rubinstein, are built from this expiration arithmetic. If the payoff at a known share price is unclear, the tree has nothing to discount. Standardized contracts are what the lesson stays with: listed strikes and listed maturities, and a multiplier of 100 shares per contract. Over-the-counter and exotic structures are deferred. The seller is the person who took the other side, sometimes called the writer.
The idea in plain words
For the holder, a call's expiration value is the premium paid, with a minus sign, plus the positive part of spot minus strike. On a chart the lesson describes a slope of one once spot clears the strike, still below the axis until the intrinsic gain covers the premium, and above the axis after that. A put has the mirror shape: premium lost, plus the positive part of strike minus spot, with the climb happening as the share price falls. Owning a call is a bet on a rise. Owning a put is a bet on a fall. That directional reading is the lesson's plain summary of the charts.
The seller receives the premium and owes the positive part. For a call, if spot finishes above the strike, the seller delivers shares at the strike. A seller who does not already own them buys them at the expiration spot and sells them at the strike, and the gap is the loss. A put seller likewise owes the positive part of strike minus spot.
How to use it
The lesson's first case is a call the seller wrote a year ago on XYZ, premium $100, strike $50, and spot at expiration also $50. The positive part of 50 minus 50 is zero, so the holder does not exercise. The seller's result is the $100 premium.
The second case is the holder of a call on XYZ, again a $100 premium and a $50 strike, with the share at $51.50 at expiration. The raw gap is $1.50. The contract multiplier of 100 turns that into $150. Gain is $150 minus the $100 premium, which is $50. The lesson's point is the order of operations: positive part, then multiplier, then subtract the premium the holder paid.
The third case looks ahead. The strike is $105. In one year the share is $110 or $90 with equal probability. The riskless rate is 5 percent. In the high state the raw gap is $5, times 100 is a $500 payoff. Discounted one year at 5 percent, the lesson reports $476.20. That would be the fair price today if the high state were certain. In the low state, 90 minus 105 is negative, the positive part is zero, and the discounted value is zero. With equal chances, the lesson takes half of $476.20, which is $238.10, and half of zero. The sum, $238.10, is the premium it calls the fair price today, the expected discounted payoff. That weighting of two discounted expiration values is the seed of the binomial method.
What usually goes wrong
Subtracting the premium before the multiplier mixes a position premium with a per-share gap. In the second example only the $1.50 is multiplied by 100. The first example's $100 premium is not multiplied again.
A negative gross payoff is not what the holder receives. The holder declines to exercise, and the seller keeps the premium. The 50-50 price scores each state, discounts, then weights by one half. Averaging the share prices first, then scoring the strike, would miss the $500 the high state still pays. A gross payoff smaller than the premium is still a net loss for the holder.
Questions people ask
What is the holder's call gain at expiration?
The positive part of spot minus strike, scaled by the contract multiplier when the quote is per share, minus the premium paid. In the XYZ case that is $150 minus $100, or $50.
What does the seller of that call keep if spot finishes at the strike?
The premium. With spot and strike both $50, the lesson's seller keeps $100 and delivers nothing.
How did the 50-50 call get a premium of $238.10?
The high state pays $500, discounted to $476.20. The low state pays nothing. Half of $476.20 plus half of zero is $238.10, at a 5 percent riskless rate.