Pay-off

A pay-off is just the answer to one question: at each possible expiry price, how much do I end up with?

Drawn as a graph, it is a pay-off diagram, and the shape of that graph is the single clearest way to tell instruments apart. A futures pay-off is a straight line — every rupee the commodity moves is a rupee to you, in either direction, and the buyer's line is the exact mirror of the seller's.

An option pay-off has a kink. Below the strike a long call earns nothing more however far the price falls, because the buyer simply walks away; above it the line rises one-for-one. That bend is the optionality, and it is what the premium buys.

For futures the arithmetic is two subtractions and nothing else. Buy at Rs 100, and at an expiry price of Rs 110 you gain Rs 10; at Rs 90 you lose Rs 10. The seller's numbers are the same with the signs reversed, because one side's gain is always the other side's loss.

Options break that symmetry, and the workbook's risk-reward table is the examinable summary:

The put rows are the ones candidates get wrong. A long put cannot earn without limit, because the commodity cannot fall below zero — the most it can deliver is the strike, less what was paid for it.

Long futures pay-off  = ST - F
Short futures pay-off = F - ST

where ST is the spot price at expiry and F the price at which the futures contract was traded.

For options, at expiry:

Long call  = max(ST - X, 0) - premium
Long put   = max(X - ST, 0) - premium

Futures, the linear case. A trader expects gold to rise and in April buys one 1 kilogram gold futures contract at Rs 50,000 per 10 grams — a contract value of Rs 50,00,000. In May the June futures reach Rs 55,000 per 10 grams and he squares off.

Pay-off = (55,000 - 50,000) x 100 units of 10 g = Rs 5,00,000

Had gold fallen to Rs 45,000 instead, the pay-off would have been minus Rs 5,00,000. Same size, opposite sign — that is what linear means.

Options, the kinked case. Zinc, strike Rs 180, call premium Rs 10, put premium Rs 8, on the workbook's zinc lot of 5 MT = 5,000 kilograms:

Below Rs 180 the loss stops dead at the premium — Rs 50,000 on the lot, whether zinc lands at Rs 160 or at Rs 40. Above Rs 190 it runs. The call seller's column is this one inverted, and his upside stops at the Rs 50,000 he collected.

Chapter 3 (Commodity Futures), section 3.8, for the long and short futures pay-off formulae and their tables, and Chapter 4 (Commodity Options), section 4.3, for the option pay-off profiles and the risk-reward table. Expect to be handed a strike, a premium and a settlement price and asked for the net position of one of the four parties, and expect "maximum loss to the buyer of an option is ____" (the premium).

• A long put is not unlimited upside. Its ceiling is the strike price less the premium, because the underlying cannot go below zero.
• Pay-off is measured against the settlement price, not the last traded price. The workbook is careful that LTP and DSP/FSP are different numbers.
• Premium is sunk. The buyer of a call that expires worthless loses the premium regardless of how far out of the money it finished; the workbook's zinc table shows the same −Rs 10 at 160 and at 140.
• Futures pay-offs are symmetric; option pay-offs are not. A question that says "the return would fall by the same amount for the same fall in price" is describing a future, not an option.
• The option buyer's gain equals the option seller's loss in every cash-settled case — the workbook says so twice in the same table.
• Adding CTT, brokerage and GST turns a small positive pay-off negative. That is exactly why a contrary instruction exists.