Calls and Puts
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An option is a contract granting a right without an obligation: to buy, or to sell, an underlying asset at a fixed price, on or before a fixed date. That asymmetry, the holder chooses, the writer must comply, is the source of everything distinctive about options: kinked payoffs, premiums, and an entire pricing discipline. This course builds that discipline from the contract up.
The right to BUY the underlying at the strike price K, on the expiry date (European style) or any time up to it (American style). Exercised only when the underlying S exceeds the strike, since buying below market is the only reason to use the right.
The right to SELL the underlying at the strike price K. Exercised only when the underlying is below the strike: the right to sell above market.
Payoffs at expiry
The two fundamental payoff functions. The max operator IS the option: the holder takes the favourable branch and walks away from the other.
Four positions, two premiums
Each contract has two sides, so there are four elementary positions: long call, short call, long put, short put. The buyer (long) pays a premium up front for the right; the writer (short) collects the premium and carries the obligation. The buyer's loss is capped at the premium; a call writer's loss is uncapped (prices have no ceiling) and a put writer's is capped only by the underlying reaching zero. The premium is the price of the asymmetry, and pricing it correctly is the subject of the modules ahead.
A 100-strike call bought for a premium of 4. At expiry with :
At the option expires worthless and the profit is : the whole premium, and no more. Breakeven sits at : the underlying must move past the strike by the premium's width before the position profits. Payoff and profit differ by the premium everywhere, and keeping the two words distinct prevents a family of errors.
Long call, strike 100, premium 4
The hockey stick. Profit is the payoff shifted down by the premium: flat at -4 below the strike, crossing zero at the breakeven, unbounded above.
Long put, strike 100, premium 4
The mirror image: gains grow as the underlying falls, capped only by the underlying reaching zero; above the strike the loss is the premium and nothing more.
Why options exist
The contract serves three broad uses, developed across this course. Insurance: a put on a held asset caps downside for a known premium, the financial form of an insurance policy. Leverage with limited liability: a call commits a small premium to a large notional exposure, with losses capped at the premium. And volatility trading: because option values respond to how MUCH prices move, not just which way, combinations of options (the strategies lesson) isolate the size of movement as a tradeable quantity, the theme the course's final module completes.
Common trap
Selling options is not the mirror image of buying them in risk terms. The writer's maximum gain is the premium, fixed and small, while the maximum loss is large or unbounded: a payoff profile of frequent small wins and rare large losses. Strategies that sell options look steadily profitable in calm samples for exactly the reason they are dangerous, the loss tail has not yet been sampled. The asymmetry of the contract survives in every statistic computed about it.
Quick check
A 50-strike put is held at expiry with the underlying at 43. What is the payoff?
Quick check
A 100-strike call is bought for 6. What is the profit if the underlying expires at 103? Answer as a signed number.