This is the final lesson of the course. And, like the lesson on futures pricing and hedging, it was designed as a reunion: present value, arbitrage, beta and the yield curve return to explain how much a right is worth and why swapping CDI for a fixed rate may be the most prudent decision for a finance department.1. You’ve already bought an option — and may not even know itThink about your car insurance. You pay a premium to the insurer, which does not refund it. In return, you receive a right: to be compensated if you have an accident. And what if the year passes without a crash? The policy expires unused, and you have lost only the premium.A stock put option works exactly the same way. You pay the premium to the person who sold the option and receive the right to sell the stock at a set price. If the stock does not fall, the option, as Assaf Neto puts it, "expires worthless" — and you lose only the premium.That is what distinguishes an option from a futures contract: with futures, both sides have an obligation; with an option, one side has the right and the other has the obligation — and the premium pays for that asymmetry.2. The vocabulary: call, put, holder and writer2.1 The four rolesThere are two types of options and two sides to each one — four combinations:Call (call option) — the holder, who buys the option and pays the premium, has the RIGHT to buy the asset at the strike price. Profits when the price rises. The writer, who sells the option and receives the premium, has the OBLIGATION to sell at the strike price if the holder exercises. Profits if the asset does not rise.Put (put option) — the holder has the RIGHT to sell the asset at the strike price. Profits when the price falls. The writer has the OBLIGATION to buy at the strike price if the holder exercises. Profits if the asset does not fall.The rest of the vocabulary: the premium is the price of the option; the strike price is the agreed price for buying or selling; the expiration is the deadline for exercising the right; and the underlying asset is what is being traded — stocks, indexes, dollars, interest rates.A rule of thumb that answers half the questions on an exam: who PAYS the premium has the RIGHT; who RECEIVES the premium has the OBLIGATION.2.2 European × AmericanA European option can be exercised only on the expiration date. An American option can be exercised at any time up to expiration. Note that the name has nothing to do with the country where the option is traded.2.3 How to read an option code on B3The code has three parts: the underlying asset, a letter and a number. The letter indicates the type and expiration month: A through L are calls (January through December); M through X are puts (January through December). The number identifies the series, associated with the strike price. The standard contract size is 100 shares.Using a fictional ticker: XPTOC32 is a March call (C is the third letter between A and L); XPTOO32 is a March put (O is the third letter between M and X).2.4 Who posts collateralThe holder does not post margin: the worst that can happen is losing the premium, which has already been paid. The person who posts collateral is the writer, because they carry the obligation. Alternatively, the writer can leave their own shares in custody — this is known as a covered writer.On B3, the code XPTOO32 identifies a...Answer: March put — the letters M through X are puts; M is January, N February and O March. Anyone who chooses "October" is counting O as the 15th letter of the alphabet without subtracting the 12 letters used for calls.In an options transaction, who must post collateral with the exchange?Answer: The writer — they received the premium and assumed the obligation to buy or sell if exercised. The holder’s risk is limited to the premium already paid.3. Payoff and profit: the four charts that appear on every examWe will use a single example throughout the lesson: XPTO stock is trading at R$ 30. There is a call with a R$ 30 strike and a R$ 2,00 premium, and a put with a R$ 30 strike and a R$ 0,97 premium (you will soon find out where that number came from). All results are per share, at expiration.3.1 Long call × short callIf the stock finishes below R$ 30, the holder does not exercise — no one buys at 30 what is worth less in the market — and loses only the premium: −R$ 2,00. Above R$ 30, the holder exercises and earns R$ 1 for every R$ 1 increase.Long call profit = max(S − K; 0) − premiumWith the stock at R$ 38: (38 − 30) − 2 = R$ 6,00 per share. The breakeven point is strike + premium = 30 + 2 = R$ 32. Assaf Neto’s example follows the same logic: a call with a $ 33,60 strike and a $ 2,10 premium is profitable to exercise only above $ 35,70.Summary for the call holder: maximum loss = the premium; gain = unlimited. The writer is the mirror image: maximum gain = the premium; loss = unlimited if the stock soars. That is why the writer posts collateral.3.2 Long put × short putThe put holder exercises only if the stock falls below R$ 30: selling at 30 something worth less in the market.Long put profit = max(K − S; 0) − premiumWith the stock at R$ 22: (30 − 22) − 0,97 = R$ 7,03 per share. The breakeven point is strike − premium = 30 − 0,97 = R$ 29,03. The maximum gain is limited (R$ 29,03 if the stock falls to zero), because the price cannot drop below zero. Above R$ 30, the put expires worthless: a loss of R$ 0,97.The put writer pockets the premium if the stock does not fall — and is required to buy at R$ 30 if it does.Exam shortcut: a long call and a short put profit from a RISE; a long put and a short call profit from a FALL. And the holder’s loss is always limited to the premium.An investor bought 100 XPTO calls with a R$ 30 strike and a R$ 2,00 premium. At expiration, the stock is trading at R$ 35. What is the profit on the trade?Answer: R$ 300 — the investor exercises: buys at 30 what is worth 35, earning R$ 5 per share, less the R$ 2 premium = R$ 3 per share, times 100 shares. The R$ 500 answer forgets the premium paid.An investor bought 100 XPTO puts with a R$ 30 strike and a R$ 0,97 premium. At expiration, the stock is trading at R$ 31. What is the result?Answer: Loss of R$ 97 — with the stock above the strike, no one sells at 30 what is worth 31: the put expires worthless and the holder loses only the premium (0,97 × 100).4. How much is a premium worth4.1 Intrinsic value + extrinsic valueEvery premium has two components:Intrinsic value — what the option would be worth if exercised RIGHT NOW. For a call: max(S − K; 0). For a put: max(K − S; 0).Extrinsic value (or time value) — everything else. It is the price of the POSSIBILITY that the option will become more valuable by expiration.Three calls with a R$ 30 strike, three months from expiration (illustrative premiums):Stock at R$ 27 — out of the money: zero intrinsic value, R$ 0,60 premium, all of it time value. Pure hope.Stock at R$ 30 — at the money: zero intrinsic value, R$ 2,00 premium, all of it time value — and the highest of the three, because this is where uncertainty matters most.Stock at R$ 33 — in the money: R$ 3,00 in intrinsic value + R$ 1,20 in time value = R$ 4,20 premium.For a put, it is the reverse: it is in the money when the stock is BELOW the strike. And at expiration, time value falls to zero — only intrinsic value remains.4.2 What affects the premiumAssaf Neto lists five factors. Without a complicated formula, what the exam tests is the direction of each one:Asset price rises — call rises; put falls.Higher strike — call falls (the right to buy at a high price is worth less); put rises (the right to sell at a high price is worth more).More time — generally, both rise: more time for things to work out. For American options, always.Volatility rises — both rise. The holder benefits from a favorable swing and, when it moves against them, loses no more than the premium.Interest rates rise — call rises; put falls. The strike is paid in the future, so its present value is lower.Anyone looking for the complete formula can find the binomial and Black & Scholes models cited by Assaf Neto and discussed in detail in Hull (Options, Futures, and Other Derivatives).The expected volatility of a stock increases significantly, with everything else unchanged. What happens to the premiums of that stock’s calls and puts?Answer: Both rise — volatility is not direction, but the size of the move. Greater fluctuation increases the holder’s potential gain in both directions, while the loss remains limited to the premium; that is why the writer charges more for both.4.3 Put-call parityEuropean calls and puts with the same strike and expiration are tied together by an arbitrage relationship:C + PV(K) = P + SOn the left, a call plus a fixed-income investment that pays K at expiration. On the right, the stock plus a put — the stock with insurance. At expiration, both portfolios are worth the greater of S and K, in any scenario. If they are worth the same in the future, they must cost the same today: this is the logic of the "two routes" in the futures lesson.The calculation for our example: S = K = R$ 30, interest rate of 15% per year, 63 business days (three months), call at R$ 2,00.PV(K) = 30 / (1,15)^(63/252) = 30 / 1,15^0,25 = R$ 28,97P = C + PV(K) − S = 2,00 + 28,97 − 30 = R$ 0,97That is where the put premium came from. If the market charges a different price, the arbitrageur buys the cheaper side, sells the more expensive one and pockets the difference — there they are again.“A call option should be purchased when prices are expected to rise; when a decline is expected, a put option is recommended.”- Assaf Neto, Mercado Financeiro (adapted)5. Strategies: designing risk with optionsBy combining options with stocks — and options with other options — investors choose the shape of the risk they want to carry.5.1 Protective put: portfolio insuranceYou own the stock, bought at R$ 30, and buy a put with a R$ 30 strike for R$ 0,97. Below R$ 30, every cent the stock loses is gained by the put: the loss stops at R$ 0,97 — that is the floor. On the upside, you continue to gain from the stock, but R$ 0,97 lower: that is the cost of the insurance.Compare this with hedging using futures: a futures contract locks in the price and gives up the upside; a put protects against a decline and KEEPS the upside — but charges a premium.5.2 Covered call: income now, with a capYou own the stock at R$ 30 and WRITE a call with a R$ 32 strike, receiving R$ 1,20. The maximum gain becomes (32 − 30) + 1,20 = R$ 3,20 — above R$ 32, you are assigned and must deliver the stock. The protection is limited: the premium only cushions the decline down to R$ 28,80 (30 − 1,20). You give up any upside above R$ 32 in exchange for immediate income.5.3 Bull call spread: a cheaper bet on an increaseYou BUY the R$ 30 call for R$ 2,00 and SELL the R$ 34 call for R$ 0,60. Net cost: R$ 1,40. Maximum gain: the distance between the strikes minus the cost = 4 − 1,40 = R$ 2,60. Maximum loss: the R$ 1,40 paid. It is a cheaper bet on an increase — but with a cap.5.4 Straddle: a bet on volatilityYou BUY the 30 call and the 30 put, paying 2,00 + 0,97 = R$ 2,97. You profit if the stock moves a LOT, in either direction: below R$ 27,03 or above R$ 32,97. The worst-case scenario is for the stock to remain at the strike. It is a bet on volatility, not direction — typical ahead of quarterly earnings or an election.An investor buys the R$ 30 call for R$ 2,00 and sells the R$ 34 call for R$ 0,60. If the stock expires at R$ 40, what is the profit per share?Answer: R$ 2,60 — the long 30 call is worth 10; the short 34 call costs 6; the net is 4 (the distance between the strikes), less the 1,40 cost. Above 34, one call offsets the other: the profit cannot exceed this amount. The R$ 8,00 is the result of anyone who forgets about the short call.6. Swaps: exchanging reference rates without exchanging principal6.1 What is a swapIn English, swap means exchange. Two parties exchange the returns on two reference rates — DI for a fixed rate, DI for the dollar, IPCA for DI — on the same reference amount, the notional. Each party has an active leg (the reference rate it receives) and a passive leg (the one it pays).The principal is NOT exchanged: at maturity, the return on each leg is calculated, and only the difference changes hands. It is an over-the-counter contract, tailored to the parties' needs and registered with B3 — part of the same family as a forward contract.6.2 DI × fixed-rate swap: turning a DI-linked debt into a fixed-rate debtCase adapted from Assaf Neto: a company has a R$ 6 million debt indexed to DI, maturing in 81 business days. Its assets, however, earn a fixed rate. If CDI rises, the debt grows while the assets do not keep pace.Solution: a swap in which the company RECEIVES DI and PAYS a fixed rate of 16,5% per year on the same R$ 6 million.Fixed-rate leg: 6.000.000 × (1,165)^(81/252) = R$ 6.301.882Scenario 1 — CDI accumulated at 18% per year: the debt rises to 6.000.000 × (1,18)^(81/252) = R$ 6.327.850. The company RECEIVES the difference through the swap: R$ 25.968. Final cost: 6.327.850 − 25.968 = R$ 6.301.882.Scenario 2 — CDI accumulated at 15% per year: the debt remains at 6.000.000 × (1,15)^(81/252) = R$ 6.275.687. But the company PAYS R$ 26.196 through the swap. Final cost: 6.275.687 + 26.196 ≈ R$ 6.301.882 (the R$ 1 difference is only rounding).In both scenarios, the cost is the same: the DI-linked debt has effectively become a fixed-rate debt at 16,5%. It is a hedge — the company gives up the benefit of a decline in CDI to avoid the cost of a rise.6.3 DI × dollar swap: turning a dollar debt into CDINow for the most classic use. The company owes US$ 1 million at 4% per year for one year and has revenue in reais. The dollar is at R$ 5,20; CDI is at 15% per year.Without protection, the cost in reais depends on the exchange rate: with the dollar at R$ 5,00, it pays 1,04 × 5,00 = R$ 5,20 million; with the dollar at R$ 6,00, it pays R$ 6,24 million.It enters into a swap with a notional of R$ 5,2 million (US$ 1 million at R$ 5,20): it RECEIVES the change in the dollar + 4% per year — exactly what the debt costs — and PAYS 100% of CDI.Dollar at R$ 6,00: the dollar leg is worth 5,2 million × (6,00/5,20) × 1,04 = R$ 6,24 million; the CDI leg, 5,2 million × 1,15 = R$ 5,98 million. It receives R$ 260 thousand, offsetting the increase in the debt. Final cost: R$ 5,98 million.Dollar at R$ 5,00: the dollar leg is worth R$ 5,20 million; it pays R$ 780 thousand on the swap. Final cost: R$ 5,98 million again.The dollar debt has become a CDI debt. And notice the exchange rate at which the two scenarios break even: 5,98 / 1,04 = R$ 5,75 — exactly the one-year futures dollar we calculated using interest-rate parity in the futures lesson. It all connects.When the Central Bank offers foreign-exchange swaps to the market, it plays the role of this bank: it pays the change in the exchange rate and receives interest, offering protection against a rise in the dollar without spending reserves. In the reverse transaction — the Central Bank pays interest and receives the change in the exchange rate — the instrument is called a reverse foreign-exchange swap.A company has dollar-denominated debt and revenue in reais. To eliminate its exchange-rate risk with a DI × dollar swap, it should...Answer: Receive the change in the dollar and pay DI — the dollar-based active leg rises along with the debt and offsets it; what remains is a CDI payment, which matches the revenue in reais. It is the golden rule of hedging: the hedging position is the opposite of the natural exposure.7. Common exam traps"The writer of a call option acquires the right to sell the asset at the strike price." WRONG. The writer received the premium: they have the OBLIGATION to sell if the holder exercises. The right belongs to whoever paid the premium — exam questions constantly switch "right" for "obligation.""The buyer of a put option is betting on an increase in the underlying asset." WRONG. The put holder profits from a DECLINE: they sell at the strike price something worth less in the market. Shortcut: call holders and put writers like an increase; put holders and call writers like a decline."An American option is one traded in the United States." WRONG. An American option is one that can be exercised at any time up to expiration; a European option can be exercised only at expiration — on any exchange in the world."In a swap, the parties exchange the principal amounts at the beginning and end of the contract." WRONG. The notional is only the calculation base: at maturity, only the difference between the legs changes hands.8. Tying it all together — and completing the courseThe essential points from the lesson:Option: the holder pays the premium and has the RIGHT; the writer receives the premium, has the OBLIGATION and posts collateral.Payoff: a long call profits from an increase (break-even = strike + premium); a long put profits from a decline (break-even = strike − premium).Premium = intrinsic value + time value; volatility and time to expiration increase both premiums; the parity C + VP(K) = P + S links calls and puts.Strategies: protective put (floor) · covered call (income, with a cap) · bull call spread (cheap upside, with a cap) · straddle (volatility).Swap: exchange of reference rates on a notional, with only the difference changing hands — a DI-linked debt becomes fixed-rate, and a dollar debt becomes CDI-linked.And this is where the entire course comes full circle. We started with the map of the financial system, learned the language of Selic, CDI and the yield curve, priced bonds and managed their risks — from fixed-rate government bonds to duration and convexity —, valued companies and assessed the price of risk — from the Gordon model to CAPM — and ended by managing time and uncertainty with derivatives: futures, hedging, options and swaps.If there is one idea running through everything, it is this: every financial asset is a future cash flow viewed through a rate. Even today's option: the premium is the present value of an uncertain cash flow. The difference between instruments lies in who promises the cash flow, how certain it is and what price is paid for the risk.“There is no such thing as a free lunch: in financial markets, every expected return above the risk-free rate comes with a price in risk.”- Market maxim, inspired by Milton Friedman — the phrase from the first lesson, now with the entire course behind itTo go further: Hull, Options, Futures and Other Derivatives — the binomial model, Black & Scholes and the "Greeks." Good luck on the exam, good work — and good investing!Exercise for classroom discussionDr. Paulo's insurance policy. Let's think it through together — this is the final case of the course.Dr. Paulo owns 1.000 shares of XPTO, bought at R$ 30 (R$ 30 thousand). He is worried about the quarterly earnings, due in three months, but does not want to sell: he believes in the company over the long term and does not want to pay taxes now. A 20% plunge would be more than he could bear. In the market, there is a put with a R$ 30 strike at R$ 0,97 and a call with a R$ 32 strike at R$ 1,20, both expiring in three months.Discuss with your group: (a) how much does the protective put cost, and what floor does it provide? (b) does selling the 32 call protect against a 20% plunge? What does he give up? (c) and what about both together — buying the put and writing the call (the so-called collar)? Is it free insurance? (d) why might selling an index future, as in the hedging lesson, fail to solve his problem?Guidance for the instructor: (a) Buy 1.000 puts and pay R$ 970. The maximum loss is R$ 970 — equivalent to selling at R$ 29,03 — and he keeps all the upside, less the premium. (b) Write 1.000 32 calls and receive R$ 1.200, but the protection only extends down to R$ 28,80: with a 20% plunge, the stock falls to R$ 24 and he loses (24 − 30 + 1,20) × 1.000 = R$ 4.800. In exchange, he gives up everything above R$ 32 (maximum gain of R$ 3.200) and may also be assigned and have to deliver the shares — precisely what he did not want. (c) With the collar, he receives 1,20 and pays 0,97: that leaves R$ 0,23 per share (R$ 230). Below R$ 30, the put guarantees a minimum result of +R$ 230; above R$ 32, the call caps the gain at R$ 2.230. It looks like free insurance — but R$ 30 thousand invested in CDI at 15% per year would earn about R$ 1.067 in the quarter. The collar guarantees far less than that in the worst case and also cuts off the upside: there is no such thing as a free lunch; protection costs potential upside. (d) An index future hedges SYSTEMATIC risk, not XPTO-specific risk; if the concern is the company's own earnings, the index will not solve the problem. A put on the stock itself protects against the specific risk, but charges a premium.Exercises1) (Original - CESGRANRIO-style) An investor purchased a call option on a stock with a strike price of R$ 40,00, paying a premium of R$ 3,00 per share. On the expiration date, the stock is trading at R$ 47,00. Ignoring transaction costs, the investor’s result per share is:A) loss of R$ 3,00.B) profit of R$ 7,00.C) profit of R$ 4,00.D) profit of R$ 10,00.E) profit of R$ 3,00.2) (Original - CESGRANRIO-style) The holder of a put option with a strike price of R$ 25,00, purchased for a premium of R$ 1,50, only begins to make a profit if, at expiration, the stock’s spot price is below:A) R$ 26,50.B) R$ 25,00.C) R$ 1,50.D) R$ 23,50.E) R$ 22,00.3) (Original - FGV-style) A stock is trading at R$ 48,00. A European call option on the stock, with a strike price of R$ 50,00, is worth R$ 3,00, and the present value of the strike price, discounted at the risk-free rate to expiration, is R$ 46,00. According to put-call parity, the fair price of a European put option with the same strike and expiration is:A) R$ 1,00.B) R$ 5,00.C) R$ 3,00.D) R$ 2,00.E) R$ 7,00.4) (Original - CESGRANRIO-style) A put option with a strike price of R$ 20,00 is trading at a premium of R$ 3,40, while the underlying stock is trading at R$ 17,00. The intrinsic and extrinsic (time) values of this option are, respectively:A) R$ 0,00 and R$ 3,40.B) R$ 3,40 and R$ 0,00.C) R$ 3,00 and R$ 0,40.D) R$ 0,40 and R$ 3,00.E) R$ 3,00 and R$ 3,40.5) (Original - FGV-style) With all else held constant, an increase in the expected volatility of the underlying asset causes the premiums of call and put options, respectively, to:A) increase and decrease.B) decrease and increase.C) decrease and decrease.D) increase and increase.E) have no effect, because volatility does not affect European options.6) (Original - FGV-style) A company has debt indexed to the CDI and assets earning a fixed rate. To eliminate the mismatch through a DI × fixed-rate swap, the company should enter into a transaction in which it:A) pays DI and receives a fixed rate.B) receives DI and pays a fixed rate.C) receives DI and pays the exchange-rate variation.D) exchanges the principal of the debt for that of a fixed-rate security.E) buys put options on the CDI.7) (Original - CEBRASPE-style, judge the statement) Judge the following statement: "The covered call strategy, in which an investor who owns the shares writes call options on them, fully protects the portfolio against declines in the stock price, at the cost of limiting gains if the price rises."( ) Correct ( ) Incorrect8) (Original - CEBRASPE-style, judge the statement) Judge the following statement: "In a swap transaction, the parties exchange the returns of reference indexes calculated on a notional amount, with only the difference between the two legs being settled at maturity and no exchange of principal."( ) Correct ( ) IncorrectAnswer Key1) C — Exercise: (47 − 40) − 3 = R$ 4,00 per share. Answer B (R$ 7,00) ignores the premium paid; answer A would be the result if the option expired worthless.2) D — Put breakeven point = strike − premium = 25,00 − 1,50 = R$ 23,50. Answer A adds the premium, as if it were a call.3) A — P = C + PV(K) − S = 3,00 + 46,00 − 48,00 = R$ 1,00. Answer B (R$ 5,00) uses the full strike price (R$ 50,00) instead of its present value—the classic parity mistake.4) C — Put intrinsic value = max(K − S; 0) = 20 − 17 = R$ 3,00; the remainder of the premium, 3,40 − 3,00 = R$ 0,40, is time value.5) D — Volatility measures the size of a move, not its direction: it increases the potential gain for the holder in both directions, while the loss remains limited to the premium. That is why it raises both call and put premiums.6) B — By receiving DI, the swap’s active leg tracks and offsets the debt; by paying a fixed rate, the effective cost matches the fixed-rate assets. Answer A would double the company’s exposure to the CDI.7) Incorrect — The premium received only cushions the decline down to the purchase price minus the premium (in the example from the lesson, R$ 28,80). Below that level, the investor loses money on the stock. Full protection against a decline requires buying a put, not writing a call.8) Correct — That is the definition of a swap: the notional amount serves only as the calculation base, and only the net balance between the two legs changes hands.
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