Introduction to Derivatives Ch. 2 PDF

Introduction to Derivatives Ch. 2 Derivative securities Sophie Moinas Master 1 Finance, Toulouse School of Management 2024-2025 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivat...

Introduction to Derivatives Ch. 2 Derivative securities Sophie Moinas Master 1 Finance, Toulouse School of Management 2024-2025 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Topics Covered in Chapter 1 Introduction to financial derivatives 1 Definition 2 Uses 3 Derivative markets 4 The value chain of derivatives 5 Arbitrage pricing What is next? Chapter 2: Main derivatives and how to use them to implement a hedging strategy ⇒ Economic agents’ perspective (firms, households, asset managers, etc.): how to use derivatives to hedge or design a trading strategy Then (Chapters 3 to 5), we will review the simplest instruments (forward, European options) one by one and understand how to price them S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 2 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Outline of Chapter 2: Derivative securities 1 2.1. Long and short positions 2 2.2. Forward contracts 3 2.3. Options 4 2.4. Combining derivatives Reading: Ch 2.1 to 2.5, Ch 3.1, Ch. 4.1 to 4.3 Chapter’s objectives: explain how these derivative instruments may be used to manage risks or design directional strategies, that is, Be able to write the payoffs at maturity of forward contracts, European call or put options Be able to describe hedging/directional strategies with Forward contracts and options Recommended problems: 2.1-2.5, 2.13-2.14, 3.1-3.2, 4.1-4.9, 4.11-4.12 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 3 / 53 2.1. Long and short positions 2.1. Long and short positions a. Risk exposure 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Derivatives and Risk management Risk is part of any business Obviously, there is uncertainty about competitors or consumers; this is the core of the business But some risks may be independent of the business strategy Examples A farmer is exposed to weather conditions and uncertainty about his production and the future price at which he will be able to sell When a company engages in international business, it becomes exposed to variations in currencies The role of a corporate risk manager is: To identify and possibly quantify risk exposure And if the risk is large, to hedge it S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 4 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A long position What is the risk exposure if one holds a long position in an underlying asset? Oil barrels, for instance One holds a long position in oil when one already holds the underlying asset, or when one is going to produce it so that one needs to sell it in the future In this case, profits increase with the underlying asset’s price at the maturity date (when you will need to sell), and one may lose if this price is too low. Example below: TotalEnergies is about to produce one oil barrel and needs to sell it at date T (say, on Dec 31st); its production cost is USD 30/barrel. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 5 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A long position Strategy: long one oil barrel Payoff Oil Price Payoff Profit 40 Profit = 30 Payoff – 0 20 cost to 10 build the 20 10 30 position 0 40 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 50 60 -20 at date T 70 -30 80 -40 90 100 It is costly to produce oil / one may have entered one’s long position by buying the asset. Profit is then obtained with a translation of the payoff – But this does not change risk exposure S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 6 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A long position Strategy: long one oil barrel Payoff Oil Price Payoff Profit 40 Profit = Payoff – 0 0 0 - cost 30 cost to 10 10 10 - cost 20 build the 20 20 20 - cost 10 position 30 30 30 - cost 0 40 40 40 - cost 50 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 80 -40 90 100 It is costly to produce oil / one may have entered one’s long position by buying the asset. Profit is then obtained with a translation of the payoff – But this does not change risk exposure S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 6 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A short position One holds a short position in oil when one needs to buy an asset in the future Either for business purposes Or because one “sold short” In this case, profits decrease with the underlying asset’s price at the maturity date (when one will need to buy), and one may lose if this price is too high. Example below: Today, Air France lines sells airplane tickets for a flight that will be departing at date T (say, on Dec 31st) at a unit price of USD 30 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 7 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A short position: Payoffs vs. Profits Strategy: short one oil barrel Payoff Oil Price Payoff Profit 40 0 30 10 20 20 10 30 0 40 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 50 60 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 8 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives A short position: Payoffs vs. Profits Strategy: short one oil barrel Payoff Oil Price Payoff Profit 40 0 0 revenue - 0 30 10 -10 revenue - 10 20 20 -20 revenue - 20 10 30 -30 revenue - 30 0 40 -40 revenue - 40 50 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 Profit 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 8 / 53 2.1. Long and short positions b. Hedging 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Risk exposure and hedging Companies are exposed to risk because they have a long or a short position in an underlying asset Long position ˜ T = Q(S̃T − c0 ), Profit where S̃T is the (unknown, uncertain) price of the underlying asset at date T (in the future) → Loss if the price of the underlying asset decreases Short position ˜ T = Q(π0 − S̃T ) Profit → Loss if the price of the underlying asset increases Should they do something to avoid this risk? If so, what could they do to protect themselves? S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 9 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Why do companies hedge? Indeed: Shareholders would adjust their portfolio to diversify their risks optimally So why pay hedging costs? But there are frictions in reality, so hedging enables To reduce transaction costs (e.g., issuance or solvency costs) To manage earnings, to reduce cash flows’ and earnings’ volatility To make companies’ evaluation easier Agency problems So, companies design and stick to a “hedging policy.” To minimize management time and cost To minimize discretion spent on hedging decisions To align risk management with business objectives To avoid coordination costs or failures across departments S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 10 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Insurance When one needs to protect oneself against a risk, one usually looks for an insurance contract. Insurance policies are such that: the client pays a premium the insurance company compensates the client for (part of) the loss incurred after a particular event The insurance premium accounts for: The likelihood of the event Administrative costs (e.g., litigation fees) Adverse selection costs (what would you think about the clients of a medical insurance promising “no medical exam needed, no questions asked”?) Moral hazard costs (-Sorry to hear about your house burning down...-Shh, that’s tomorrow...) S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 11 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Risk neutral pricing In a competitive environment, the risk premium would be such that the expected profit of the insurance company is equal to zero: Insurance premium = E(LOSS — the customer accepts the contract) Different contracts to “separate” good and bad risks (example: levels of deductibles) Besides, the insurance company designs contracts to make sure that customers exert effort in avoiding taking too much risk: Deductibles Health insurance: discount if a customer quits smoking, agrees to have a preventive medical examination, walks more than 10,000 steps a day, etc. Insurance companies are willing to acquire risk because : They have an advantage in evaluating the probability and the magnitude of a loss They have a large customer base, and risks are (usually) independent: it is very improbable that all customers would experience the same car accident S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 12 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives So why do we need derivatives? Business risks Possible huge quantities of risk at stake Businesses often have customized needs Risk are not independent ⇒ Consequently, risk-neutral pricing does not seem possible If an insurance company is left with residual risk, it will require a risk premium Basic idea of derivatives markets: “splitting” the risk into small pieces that are sold on the market Many investors are going to absorb the risk This enables one to lower the risk premium one needs to pay to get rid of this risk Markets make risk-sharing more efficient Diversifiable risks vanish, non-diversifiable risks are reallocated to those most willing to hold it S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 13 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Standard Derivatives Main families: 1 Forward, futures, and swaps (linear payoffs) A forward contract is an agreement to buy or sell an asset at a future date T at a price that has been fixed at the time both counterparts enter the agreement. A futures is essentially an exchange-traded forward contract. A swap is an agreement calling for an exchange of a stream of payments (swaps are conceptually similar to a basket of forward contracts). 2 Options (non-linear payoffs) A (European) option is a financial contract that gives the owner the right but not the obligation to trade the underlying asset at a specified price K at a future date T. ⋆ K is called the strike price and T is the maturity date. ⋆ A call option is a right to buy, and a put option is a right to sell. Chapter 2 shows how to use these standard derivatives, taking their prices as given. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 14 / 53 2.2. Forward contracts 2.2. Forward contracts a. Definition 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Forward Contracts The simplest derivative contract is a forward contract a futures contract is basically a forward contract traded on an exchange a swap is conceptually a bundle of forward contracts A forward contract is an agreement between two parties for the deferred delivery of an asset at a pre-determined price. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 15 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Forward Contracts A forward contract is an agreement between two parties for the deferred delivery of an asset. The contract specifies: the quantity and type of asset delivery time and place (or cash settlement) the pre-determined price, F Both the buyer and the seller are obligated to carry out the transaction. Terminology: if one agrees to be the buyer (i.e., if one buys forward), one is said to have a “long position in the forward” if one agrees to be the seller (i.e., if one sells forward), one is said to have a “short position in the forward” S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 16 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives The Price of a Forward Contract The “price” of a forward contract is the amount the long side pays to the short side at the delivery date. It is the only monetary transfer that takes place in a forward contract Indeed, notice that when the contract is signed, both counterparties agree on a future transfer, and that’s it! other than collateral (aka margin) Notations: Ft,T = the forward price for a contract initiated at time t for delivery at time T (i.e., the price for deferred delivery) St = the spot price at time t (i.e., the price for immediate delivery) S̃T = the price of the underlying asset at a future time T (unknown at date t) S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 17 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Futures Contracts Futures and forward contracts are conceptually very similar Futures are standardized contracts. They may not be customized Standardization enables futures to be exchange-traded contracts. forward contracts are arranged between pairs of individuals/firms and often traded “Over-The-Counter” (i.e., OTC) futures contracts are exchange-traded; Consequently, futures have lower transaction costs and lower counterparty risk S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 18 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Swap Contracts A swap is an agreement calling for an exchange of a stream of payments. Interest Rate Swaps (IRS) Commodity swaps 1st month 2nd month 3rd month … Expiration Today date X S1 X S2 X S3 X ST At each intermediary date (e.g., a month), one counterpart pays (receives) a fix price X in exchange for an asset whose price is variable/floating S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 19 / 53  2.2. Forward contracts b. Hedging a position with a forward contract 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Hedging a long position by shorting a forward contract To understand how forward contracts are useful to hedge risk, let us derive the payoff as a function of the price of the underlying asset (oil barrels in our example) Consider that one is long in oil, Strategy: long one oil barrel Payoff Oil Price Payoff 40 0 0 30 10 10 20 20 20 10 30 30 0 40 40 50 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 80 -40 90 100 ☞ What is the final payoff if one additionally takes a short position in a forward at a price F0,T =e60? S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 20 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Hedging a long position by shorting a forward contract To understand how forward contracts are useful to hedge risk, let us derive the payoff as a function of the price of the underlying asset (oil barrels in our example) Consider that one is long in oil, ☞ What is the final payoff if one additionally takes a short position in a forward at a price F0,T =e60? Entering a forward contract on oil completely Strategy: long oil + sell one eliminates risk compared to an oil position forward at 60 à You secure a payoff of 60 Total payoff Payoff Oil Price Long Oil after hedge 0 0 60 60 10 10 60 50 20 20 60 30 30 60 40 40 40 60 30 50 50 60 20 60 60 60 10 70 70 60 Oil Price 0 80 80 60 at date T -10 10 20 30 40 50 60 70 80 90 100 90 90 60 100 100 60 -20 -30 Long oil Long oil + short forward S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 20 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Payoffs of a long position + short Forward Strategy: long oil + sell one forward at 60 Payoff Short Total payoff Oil Price Long Oil Forward after hedge 60 0 0 60 60 50 10 10 50 60 40 20 20 40 60 30 30 30 30 60 20 40 40 20 60 10 50 50 10 60 60 60 0 60 0 70 70 -10 60 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 80 80 -20 60 -20 at date T 90 90 -30 60 -30 100 100 -40 60 Long oil Long oil + short forward Short Forward On this graph, one can decompose the hedged position into: initial position + hedge S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 21 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Payoffs of a short forward contract (without the initial position) When one sells forward, one receives F0,T today, but one needs to buy the underlying asset at date T to deliver it. the net payoff at the maturity date is F0,T − ST What if one already holds the underlying asset? F0,T − ST can be viewed as an opportunity cost of payoff: If one had not entered the forward contract, one could have sold the underlying asset at a price ST Now, one has to sell at price F0,T , because one entered the forward agreement The opportunity gain is also F0,T − ST , which can be positive ⇒ Notice that this writing enables us to identify the payoff of a forward contract irrespective of its settlement and of one’s initial position S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 22 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Comments on hedges with forward contracts Payoff of a (short) forward contract at the maturity date 60 50 40 30 20 10 0 -10 10 20 30 40 50 60 70 80 90 100 Price of the underlying -20 Forward asset at -30 price Ft,T maturity ST Once one signed a (short) forward contract, one is obliged to sell at a price Ft,T S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 23 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Comments on hedges with forward contracts Payoff of a (short) forward contract at the maturity date 60 50 40 30 J 20 10 0 -10 10 20 30 40 50 60 70 80 90 100 Price of the underlying -20 Forward asset at -30 price Ft,T maturity ST Once one signed a (short) forward contract, one is obliged to sell at a price Ft,T Sometimes, it is profitable, because ST < Ft,T S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 23 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Comments on hedges with forward contracts Payoff of a (short) forward contract at the maturity date 60 50 40 30 20 10 0 -10 10 20 30 40 50 60 70 80L90 100 Price of the underlying -20 Forward asset at -30 price Ft,T maturity ST Once one signed a (short) forward contract, one is obliged to sell at a price Ft,T Sometimes, it is profitable, because ST < Ft,T Sometimes, one wish one had not signed the contract S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 23 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Comments on hedges with forward contracts Payoff of a (short) forward contract at the maturity date 60 50 40 30 20 10 0 -10 10 20 30 40 50 60 70 80L90 100 Price of the underlying -20 Forward asset at -30 price Ft,T maturity ST Once one signed a (short) forward contract, one is obliged to sell at a price Ft,T Sometimes, it is profitable, because ST < Ft,T Sometimes, one wish one had not signed the contract But remember that situations in which one is unhappy with one’s hedge correspond to situations where the payoffs of the initial position are high! S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 23 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Long Forward contract Payoffs Short Forward Contract F 0,T L ST Payoffs Long Forward Contract L F 0,T ST Payoffs of the buyer and the seller are perfectly symmetric Seller: F0,T − ST Buyer: ST − F0,T One buys a forward contract to hedge when one initially holds a short position (one expect −ST , e.g. Air France’s exposure to oil prices) S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 24 / 53 2.3. Options  2.3. Options a. European call option 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European Call Option A European call option is a financial contract that gives the owner the right but not the obligation to buy the underlying asset at a specified price K , at the future date T stated in the contract. The buyer of a call option pays an option’s premium to the seller of this call. We denote by Ct the premium/price of the (call) option at date t. The price (or premium) of a call option C0 (K , T ) is the amount of money that the buyer of the option agrees to give to its seller at time 0 to receive the payoff promised by the option at the maturity date T. In Chapter 2, we take the option premium as given. Later, we will price this option premium. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 25 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Exercise Exercise: the act of paying the strike price to buy the asset Expiration or maturity date T : the date by which the option must be exercised or become worthless Strike or exercise price K : the amount paid by the option buyer for the asset if he/she decides to exercise Exercise style: specifies when the option can be exercised European-style: can be exercised only at the expiration date American-style: can be exercised at any time before expiration Bermudan-style: Can be exercised during specified periods S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 26 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a call ☞ Let us see what are the payoffs of a long position (=buy) in a call option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Payoff at Strategy: buy one call on oil (K=60) maturity date Oil Price Payoff 40 0 30 10 20 20 10 30 0 40 50 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 27 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a call ☞ Let us see what are the payoffs of a long position (=buy) in a call option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Payoff at When ST < K, exercising the call option is Strategy: buy one call on oil (K=60) maturity not profitable: You would buy at K something that is worth date ST < K: not worth it! If you need the underlying asset, buy it at price ST J Oil Price Payoff Or: if you exercise, buy at K and immediately 40 resell at ST, you incur a loss! 0 0 30 10 0 20 20 0 10 30 0 0 40 0 50 0 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 0 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 27 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a call ☞ Let us see what are the payoffs of a long position (=buy) in a call option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) When ST > K, exercising the call option is Payoff at profitable: Strategy: buy one call on oil (60) maturity You buy at K something that is worth ST < K date Or: You exercise, buy at K and immediately resell at ST Oil Price Payoff 40 0 0 30 10 0 20 20 0 10 30 0 0 40 0 50 0 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 0 -20 at date T 70 10 -30 80 20 -40 90 30 100 40 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 27 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European call option: Buyer’s Payoffs The payoff of the buyer of a call option at the maturity date T writes: max(ST − K , 0) = (ST − K )+ The buyer of a European call option needs to pay a premium to the seller immediately: C Pricing a call option = finding C → in chapter 4, we (as financial engineers / writers / market makers) will focus on options’ payoffs S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 28 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European call option: Buyer’s Payoffs vs. profits Risk managers looking for a hedging instrument will usually focus on the Profits of their positions They need to take into account the hedging cost in their strategy It will also matter to make their choice between various hedging strategies (e.g., options vs. forwards) To compute this net profit, one needs to take the option premium into account Be careful: the premium is paid at date t, the option may be exercised at the maturity date T → compounded value S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 29 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a Call ☞ Now, let us see the payoffs of a short position (=sell) in a call option on an oil barrel with a strike price of K=60 and a maturity date T (say, Dec. 31st) When one sells an option, one sells a right The seller is left with an obligation! The buyer will find it profitable to exercise a European call option if the underlying asset value exceeds the strike price K. At the maturity date T : ST > K : S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 30 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a Call ☞ Now, let us see the payoffs of a short position (=sell) in a call option on an oil barrel with a strike price of K=60 and a maturity date T (say, Dec. 31st) When one sells an option, one sells a right The seller is left with an obligation! The buyer will find it profitable to exercise a European call option if the underlying asset value exceeds the strike price K. At the maturity date T : ST > K :The buyer finds it profitable to exercise the call option. The seller is obliged to sell at K an asset that is worth ST : he incurs a loss −(ST − K ) ST ≤ K : S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 30 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a Call ☞ Now, let us see the payoffs of a short position (=sell) in a call option on an oil barrel with a strike price of K=60 and a maturity date T (say, Dec. 31st) When one sells an option, one sells a right The seller is left with an obligation! The buyer will find it profitable to exercise a European call option if the underlying asset value exceeds the strike price K. At the maturity date T : ST > K :The buyer finds it profitable to exercise the call option. The seller is obliged to sell at K an asset that is worth ST : he incurs a loss −(ST − K ) ST ≤ K :The buyer does not find it profitable to exercise. No gain or loss for the seller. The payoff of the seller of a call option at the maturity date writes: − max(ST − K , 0) = −(ST − K )+ S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 30 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a call ☞ Now, let us see the payoffs of a short position (=sell) in a call option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Strategy: sell one call on oil (K=60) Payoff at maturity date Oil Price Payoff - (ST − K)+ 40 0 30 10 20 20 10 30 0 40 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 50 60 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 31 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a call ☞ Now, let us see the payoffs of a short position (=sell) in a call option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Strategy: sell one call on oil (K=60) Payoff at maturity date Oil Price Payoff - (ST − K)+ 40 0 0 30 10 0 20 20 0 10 30 0 0 40 0 50 0 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 0 -20 at date T 70 -10 -30 80 -20 -40 90 -30 100 -40 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 31 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European call option: Seller’s Payoffs vs. profits At the maturity date, the payoff of the seller of a call option is always negative − max(ST − K , 0) However, the seller of a call option immediately receives a premium C when he sells the option S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 32 / 53  2.3. Options b. European put option 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European Put Option A European put option is a financial contract that gives the owner the right but not the obligation to sell the underlying asset at a specified price K (= the strike price) at the future date T (= the maturity or expiration date) stated in the contract. The buyer of a put option pays an option’s premium to the seller of the contract. We denote by Pt the premium/price of the (put) option at date t. The price (or premium) of a put option is the amount of money that the buyer of the option agrees to give to its seller at time 0 to receive the payoff promised by the option at the maturity date T. In Chapter 2, we take the option premium as given. Later, we will price this option premium. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 33 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a put ☞ Let us see the payoffs of a long position (=buy) in a put option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Payoff at maturity Strategy: long one put on oil (60) date 60 50 Oil Price Payoff 40 0 30 10 20 20 10 30 0 40 50 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 34 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a put ☞ Let us see the payoffs of a long position (=buy) in a put option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Payoff at When S < K, exercising the put option T maturity is profitable: Strategy: long one put on oil (60) date You sell at K something that is worth ST 60 >K 50 Or: You exercise, sell at K and Oil Price Payoff 40 immediately buy at ST 0 60 30 10 50 20 20 40 10 30 30 0 40 20 50 10 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 -20 at date T 70 -30 80 -40 90 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 34 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a put ☞ Let us see the payoffs of a long position (=buy) in a put option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec 31st) Payoff at When ST > K, exercising the put option maturity is not profitable: Strategy: long one put on oil (60) You would sell at K something that you date 60 can sell ST > K on the spot market: not worth it! J 50 Oil Price Payoff Or: if you exercise, sell at K and 40 immediately buy at ST, you incur a loss! 0 60 30 10 50 20 20 40 10 30 30 0 40 20 50 10 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 0 -20 at date T 70 0 -30 80 0 -40 90 0 100 0 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 34 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European put option: Buyer’s Payoffs vs. profits The payoff of the buyer of a put option at the maturity date T writes: max(K − ST , 0) = (K − ST )+ The buyer of a European put option needs to pay a premium to the seller at date t: Pt Consequently, to compute the net profit, one needs to take this premium into account. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 35 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a put ☞ Now, let us see the payoffs of a short position (=sell) in a put option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec. 31st) Strategy: sell one put on oil (K=60) Payoff at maturity date Oil Price Payoff 40 0 30 10 20 20 10 30 0 40 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 50 60 -20 at date T 70 -30 80 -40 90 -50 100 -60 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 36 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling a put ☞ Now, let us see the payoffs of a short position (=sell) in a put option on an oil barrel with a strike price of K = 60 and a maturity date T (say, Dec. 31st) Strategy: sell one put on oil (K=60) Payoff at maturity date Oil Price Payoff 40 0 -60 30 10 -50 20 20 -40 10 30 -30 0 40 -20 50 -10 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 60 0 -20 at date T 70 0 -30 80 0 -40 90 0 -50 100 0 -60 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 36 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives European put option: Seller’s Payoffs vs. profits At the maturity date, the payoff of the seller of a put option is always negative − max(K − ST , 0) However, the seller of a put option receives a premium Pt when he sells the option at date t S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 37 / 53  2.3. Options c. Hedging a Position with Options 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a Put to Hedge a Long position (e.g., TotalEnergies) Consider that one is long on oil: one needs to sell in the future One would like to hedge against a decrease in prices / one would like to secure the price at which one will sell in the future Buying a put enables one to hedge against a decrease in prices, but keeping the benefit of a price increase ☞ Let us see the payoffs associated with the combination of a long position in oil and a long position in a put option on oil with an exercise price K = 60 Payoff at Strategy: long one oil and buy one put (60) maturity date Payoff Payoff Payoff Oil Price Oil Put Portfolio 60 0 0 50 10 10 40 20 20 30 30 30 20 40 40 50 50 10 60 60 0 70 70 10 20 30 40 50 60 70 80 90 100 Oil Price 80 80 at date T 90 90 100 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 38 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a Put to Hedge a Long position (e.g., TotalEnergies) Consider that one is long on oil: one needs to sell in the future One would like to hedge against a decrease in prices / one would like to secure the price at which one will sell in the future Buying a put enables one to hedge against a decrease in prices, but keeping the benefit of a price increase ☞ Let us see the payoffs associated with the combination of a long position in oil and a long position in a put option on oil with an exercise price K = 60 Payoff at Strategy: long one oil and buy one put (60) maturity date Payoff Payoff Payoff Oil Price Oil Put Portfolio 60 0 0 60 60 50 10 10 50 60 40 20 20 40 60 30 30 30 30 60 20 40 40 20 60 50 50 10 60 10 60 60 0 60 0 70 70 0 70 10 20 30 40 50 60 70 80 90 100 Oil Price 80 80 0 80 at date T 90 90 0 90 100 100 0 100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 38 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a Call to hedge a short position (e.g., Air France) Consider that one is short on oil : one needs to buy in the future One would like to hedge against an increase in prices / one would like to secure the price at which one will buy in the future Buying a call enables one to hedge against an increase in prices, but keeping the benefit of a price decrease ☞ Let us see the payoffs associated with the combination of a short position in oil and a long position in a call option on oil with an exercise price K = 60 Payoff at Strategy: short oil and buy one call maturity date Payoff Payoff Payoff Oil Price Oil Call Portfolio 0 0 0 10 -10 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 20 -20 -20 at date T 30 -30 -30 40 -40 50 -50 -40 60 -60 -50 70 -70 -60 80 -80 90 -90 100 -100 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 39 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Buying a Call to hedge a short position (e.g., Air France) Consider that one is short on oil : one needs to buy in the future One would like to hedge against an increase in prices / one would like to secure the price at which one will buy in the future Buying a call enables one to hedge against an increase in prices, but keeping the benefit of a price decrease ☞ Let us see the payoffs associated with the combination of a short position in oil and a long position in a call option on oil with an exercise price K = 60 Payoff at Strategy: short oil and buy one call maturity date Payoff Payoff Payoff Oil Price Oil Call Portfolio 0 0 0 0 0 10 -10 0 -10 -10 10 20 30 40 50 60 70 80 90 100 Oil Price 20 -20 0 -20 -20 at date T 30 -30 0 -30 -30 40 -40 0 -40 50 -50 0 -50 -40 60 -60 0 -60 -50 70 -70 10 -60 -60 80 -80 20 -60 90 -90 30 -60 100 -100 40 -60 S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 39 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives The use of derivatives So far, we have seen how one can use derivatives to hedge a risky exposure But derivatives can also be helpful for other purposes, e.g., in the implementation of a directional strategy. Examples of information acquisition The chief economist expects an increase in the Fed fund rates in the next meeting The financial analyst in charge of the Telecom industry expects an increase in the demand for 4G The role of an asset manager is to design strategies that would take advantage of these pieces of information: To identify which security would pay off if prices move in the expected direction and possibly quantify risk exposure S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 40 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Summary: Hedging with options One can buy or sell an option to buy or to sell SELLING an option will never enable one to hedge How does one hedge with options?→ BUY PUTS or CALLS So who sells options and why? S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 41 / 53  2.3. Options d. Who sells options and why? 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling options At date T, the writer of a call (resp. put) is obligated to sell (resp. to buy) the underlying at the price K if the buyer of the option exercises it. The option premium received by the writer at date 0 is compensation for this obligation. S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 42 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Who sells options and why? Speculators may be ready to take positions. Market makers Match demand / supply Hold inventory to intermediate trades Institutions who write / issue options usually “make” the (secondary) market for these securities Market makers hedge their position (covered writing / naked writing) Key aspect of this course (Chapter 4): how to hedge because price = cost of hedging S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 43 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Selling options At date T, the writer of a call (resp. put) is obligated to sell (resp. to buy) the underlying at the price K if the buyer of the option exercises it. The option premium received by the writer at date 0 is compensation for this obligation. From the writer’s point of view, we will focus on payoffs at date T, not on profits ⇒ We aim at finding the option premium (and not at comparing hedging strategies). S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 44 / 53 2.4. Combining derivatives 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Combinations of puts & calls Cap: a call option is combined with a position in the underlying asset Goal: to insure against an increase in the price of the underlying asset Floor: a put option is combined with a position in the underlying asset Goal: to insure against a fall in the price of the underlying asset Option spread: a position consisting of only calls or only puts, in which some options are purchased and some written Examples: bull spread, bear spread, box spread, ratio spread Collar: the purchase of a put option and the sale of a call option with a higher strike price (same underlying asset, same expiration date) Example: zero-cost collar Options can be used to create positions that are non-directional with respect to the underlying asset Examples: Straddles, Strangles, Butterfly spreads S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 45 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Cap One can use a call option to hedge a short position Cash flow at expiration:−ST (short position) + max(ST − K , 0) Profit at expiration: −ST (short position) + max(ST − K , 0) - Future Value (Ct ) Payoff at maturity date Call 0 -10 10 20 30 40 50 60 70 80 90 100 Oil Price -20 at date T -30 -40 -50 -60 There is a cap on your losses Short position S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 46 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Floor One can use a put option to hedge a long position Cash flow at expiration:+ST (long position) + max(K − ST , 0) Profit at expiration: +ST (long position) + max(K − ST , 0) - Future Value (Pt ) Payoff at maturity Long position date There is a floor on your profits 60 50 40 30 20 Put 10 0 Oil Price 10 20 30 40 50 60 70 80 90 100 at date T S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 47 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Different strategies S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 48 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Bull spread buy a call and sell an otherwise identical call with a higher strike price S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 49 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Collar Collar resembles a short forward: Buy put, sell Call S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 50 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Straddle Buy put, Buy Call same Strike S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 51 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Strangle Buy put, Buy Call with a higher strike price S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 52 / 53 2.1. Long and short positions 2.2. Forward contracts 2.3. Options 2.4. Combining derivatives Topics Covered in Chapter 2 Definition, payoffs of standard derivatives Forward contracts European Call and Put options Derivatives as hedging instruments How to hedge a position with standard derivatives What is next? How to price these derivatives? S. Moinas - M1 Finance - 2024-2025 Introduction to Derivatives - Ch.2 53 / 53