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FRM Part I · FRM Exam Part I

Properties of Options for FRM Part I

Properties of options covers what drives option values and the no-arbitrage rules that limit them: payoffs, price factors, upper and lower bounds, put-call parity, early exercise of American options, and spreads. Solve questions by naming the rule, plugging in the numbers, and checking the result against the bounds.

What this chapter covers

This chapter explains how options behave before you price them with a model. You learn the payoff of calls and puts, what pushes prices up or down, and the limits that no sensible price can break. Those limits come from no-arbitrage logic: if a price breaks a bound, you could lock in a risk-free profit.

The chapter ends with trading strategies. Spreads, straddles and strangles combine basic options into new payoff shapes. You need to read a payoff diagram, find the profit at a given price, and find the break-even points.

It links to much of the rest of Part I. The ideas of replication and no-arbitrage return in forwards and futures, and the factors behind option prices lead into the Greeks and the Black-Scholes-Merton model in valuation. Put-call parity also helps with questions on the cost of carry and on synthetic positions.

Questions on this chapter are usually short and rule-based, so they reward candidates who know the relationships cold. With 100 questions in 4 hours, you have limited time for each one. Put-call parity, the bounds and spread payoffs can be solved in under two minutes once you have practised them. The same ideas also support later topics, so time spent here pays off beyond this chapter. GARP publishes no weightage in marks, so treat the whole chapter as worth mastering.

Properties of Options: topics in the order to study them

  1. 1Option Basics and Payoff ProfilesEverything else builds on the payoffs of long and short calls and puts, so start here.
  2. 2Factors Affecting Option PricesOnce you know the payoffs, learn the direction in which each input moves call and put values.
  3. 3Upper and Lower Bounds on Option PricesBounds use the payoffs and the price factors, and they set up the arbitrage logic used in parity.
  4. 4Put-Call ParityParity is the most tested relationship and relies on the bound logic and present values you just practised.
  5. 5Early Exercise of American OptionsYou need bounds and parity first to see why early exercise can pay for puts, and for calls only in certain cases.
  6. 6Option Trading Strategies and SpreadsStrategies combine the basic payoffs, so they come last and double as a review of the whole chapter.

How to prepare Properties of Options

Aim to know the rules well enough to apply them without notes, then drill numbers until they are quick.

  1. Draw the payoff and profit diagrams for long and short calls and puts from memory until you can do it without hesitation.
  2. Make a table of the six inputs (spot price, strike, time to expiry, volatility, risk-free rate, dividends) and write the direction of their effect on European and American calls and puts. Learn the reason for each, not just the sign.
  3. Write each bound in plain words and a formula, then test it with numbers. Check that a quoted price lies inside the bounds before you use any other method.
  4. Practise put-call parity both ways: find a missing price, and spot an arbitrage and its trades. For European options on a non-dividend stock use c + K·e^(−rT) = p + S₀. Remember to adjust S₀ for the present value of known dividends.
  5. Learn why early exercise of an American call on a non-dividend stock is never optimal, and when it can be sensible for a put or for a dividend-paying stock.
  6. For each spread, write the construction, the maximum profit, maximum loss and break-even. Then solve timed practice questions and review every wrong answer.

Common mistakes in Properties of Options

  • Confusing payoff with profit.

    Fix: Write payoff first, then adjust for the premium. Mark break-even as the strike plus or minus the premium.

  • Using the wrong sign for the effect of a factor, especially for rates and time.

    Fix: Think through the logic. A higher rate lowers the present value of the strike, which helps calls and hurts puts. Check whether the option is European or American.

  • Forgetting to discount the strike in put-call parity.

    Fix: Always write K·e^(−rT), or K ÷ (1 + r)^T if the question uses annual compounding, and use the compounding convention given in the question.

  • Ignoring dividends in bounds and parity.

    Fix: Read the question for dividends. If they are known, reduce S₀ by their present value before applying the formula.

  • Applying European results to American options.

    Fix: Parity is an equality only for European options. For American options you get inequalities, and early exercise can matter for puts and for calls on dividend-paying stocks.

  • Misreading the construction of a spread.

    Fix: Write each leg with its sign and strike, sum the payoffs at a few key prices, and then identify the maximum gain, loss and break-evens.

Last-day revision: Properties of Options

  • Long call payoff = max(S_T − K, 0); long put payoff = max(K − S_T, 0).
  • Profit equals payoff minus the premium paid for a long position. The seller's profit is the reverse.
  • A higher spot price raises call values and lowers put values.
  • A higher strike lowers call values and raises put values.
  • Higher volatility raises both calls and puts.
  • Put-call parity for European options with no dividends: c + K·e^(−rT) = p + S₀.
  • With known dividends, subtract their present value from S₀ in the parity formula.
  • A European call is worth at least max(S₀ − K·e^(−rT), 0) on a non-dividend stock.
  • An American option is worth at least as much as the matching European option.
  • An American call on a non-dividend stock should not be exercised early.
  • A bull call spread has limited profit and limited loss; a straddle profits from large moves in either direction.
  • If a price breaks a bound or parity, form the arbitrage: buy the cheap side, sell the dear side.

Properties of Options practice questions

Properties of Options in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Properties of Options: frequently asked questions

What is the most important formula in Properties of Options?

Put-call parity is the most useful, because it links European call and put prices through the stock price and the present value of the strike. It also lets you find missing prices and spot arbitrage. Learn it with its dividend adjustment.

Do I need a financial calculator for this chapter?

You mostly need a calculator with an exponential function for discounting. Questions are usually short, so the arithmetic is simple once you know the formula. Practise the key presses so you do not lose time.

Is it ever optimal to exercise an American call early?

For a stock that pays no dividends, early exercise is not optimal, because the option is worth more kept than exercised. With dividends, early exercise can be sensible just before an ex-dividend date. American puts can be worth exercising early, especially when deep in the money.

How should I study option spreads?

Break each spread into its legs, compute the payoff at a few stock prices, and note the maximum profit, maximum loss and break-even. Compare the spreads by the view they express, such as bullish, bearish or expecting a big move.