Corso di Matematica Finanziaria 3 Introduzione ai Derivati
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Corso di Matematica Finanziaria 3 Introduzione ai Derivati Docente: Arturo Leccadito Università della Calabria Slides taken from Hull’s web-site http://www.rotman.utoronto.ca/∼hull/ofodslides/ Arturo Leccadito (Unical) Introduzione ai Derivati 1 / 36
Derivatives Markets A derivative is an instrument whose value depends on the values of other more basic underlying variables. Examples of Derivatives: Futures Contracts Forward Contracts Options Swaps Derivatives Markets: Exchange traded: Contracts are standard there is virtually no credit risk Over-the-counter (OTC): Contracts can be non-standard and there is some small amount of credit risk Arturo Leccadito (Unical) Introduzione ai Derivati 2 / 36
Types of Derivatives Forward/Futures Contract: Agreement to buy or sell an asset for a certain price at a certain time. A forward contract is traded OTC, whereas a futures contract is traded on an exchange. It costs nothing to take either a long or a short position. Swaps: Agreement to exchange cash flows at specified future times according to certain specified rules Options: ◮ A call (put) option is an option to buy (sell) a certain asset by a certain date for a certain price (the strike price) ◮ An American option can be exercised at any time during its life whereas a European option can be exercised only at maturity ◮ If a futures/forward contract gives the holder the obligation to buy or sell the asset, an option gives the holder the right to buy or sell the asset. Arturo Leccadito (Unical) Introduzione ai Derivati 3 / 36
Payoffs from Forward Contracts The payoff from a long position in a forward contract on one unit of an asset is ST − K , where K is the delivery price and ST is the spot price of the asset at maturity T . Similarly the payoff from a short position in a forward contract on one unit of an asset is K − ST . Profit Profit K K ST ST Long Position Short Position Arturo Leccadito (Unical) Introduzione ai Derivati 4 / 36
Forward Prices The forward price for a contract is the delivery price that would be applicable to the contract if it were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero) Some notation: S0 : Spot price today T: Time until delivery date F0,T : Futures or forward price today r: Risk-free interest rate for maturity T For any investment asset, i.e. an assets held by significant numbers of people purely for investment purposes (Examples: stocks, bonds, gold, silver), that provides no income (for instance, no dividends) and has no storage costs F0,T = S0 erT assuming interest rates are measured with continuous compounding. Arturo Leccadito (Unical) Introduzione ai Derivati 5 / 36
If F0,T > S0 erT an arbitrage opportunity arises from the following strategy (Remember the rule: “Buy the cheap, sell the expensive”!!): 1. Sell the forward; 2. Borrow the amount S0 at the rate r ; 3. Buy the stock. The result is a riskless profit of F0,T − S0 erT at time T . 0 T 1. 0 F0,T − ST 2. S0 −S0 erT 3. −S0 ST 0 F0,T − S0 erT Arturo Leccadito (Unical) Introduzione ai Derivati 6 / 36
Similarly, if F0,T < S0 erT an arbitrage opportunity arises from the following strategy: 1. Buy the forward; 2. Short sell the stock; 3. Invest the proceeds of the short sale at the rate r . The strategy results in a riskless profit of S0 erT − F0,T at time T . 0 T 1. 0 ST − F0,T 2. S0 −ST 3. −S0 S0 erT 0 S0 erT − F0,T Arturo Leccadito (Unical) Introduzione ai Derivati 7 / 36
Payoffs from Options The payoff from a long position in a call on one unit of an asset is max{ST − K , 0}, where K is the exercise price and ST is the spot price of the asset at maturity T . Similarly the payoff from a short position in a call on one unit of an asset is − max{ST − K , 0} = min{K − ST , 0}. Payoff Payoff K K ST ST Long Call Short Call Arturo Leccadito (Unical) Introduzione ai Derivati 8 / 36
The payoff from a long position in a put on one unit of an asset is max{K − ST , 0}, where K is the exercise price and ST is the spot price of the asset at maturity T . Similarly the payoff from a short position in a put on one unit of an asset is − max{K − ST , 0} = min{ST − K , 0}. Payoff Payoff K K ST ST Long Put Short Put Arturo Leccadito (Unical) Introduzione ai Derivati 9 / 36
Terminology Moneyness: At-the-money option: zero cash flow if immediately exercised (S0 = K ) In-the-money option: positive cash flow if immediately exercised (S0 > K for a call and S0 < K for a put) Out-of-the-money option negative cash flow if immediately exercised (S0 < K for a call and S0 > K for a put) Intrinsic value: the maximum of zero and the value the option would have if it were exercised immediately Time value: the difference between option value and intrinsic value. Time value is the part of the option’s value that derives from the possibility of future favorable movements in the stock price. Arturo Leccadito (Unical) Introduzione ai Derivati 10 / 36
American vs European Options Notation: c : European call option price C : American Call option price p : European put option price P : American Put option price S0 : Stock price today ST : Stock price at option maturity K : Strike price T : Life of option r : Risk-free rate for maturity T σ : Volatility of stock price An American option is worth at least as much as the corresponding European option: c ≤C p ≤ P. Arturo Leccadito (Unical) Introduzione ai Derivati 11 / 36
Upper and Lower Bounds for Option Prices For options on non-dividend-paying stocks the upper bounds are: c ≤ S0 and C ≤ S0 −rT p ≤ Ke and P ≤ K and the lower bounds: c ≥ max{S0 − K e−rT , 0} p ≥ max{K e−rT − S0 , 0} Arturo Leccadito (Unical) Introduzione ai Derivati 12 / 36
Put-Call Parity The relationship between p and c is known as Put-Call Parity. Suppose we combine a long position in a call and a short position in a put (same stock, strike and maturity). Both options are European. The final payoff is max{ST − K , 0} − max{K − ST , 0} = ST − K . Note that this is the payoff of a forward contract. It follows that the present value of the payoff is simply S0 − K e−rT . Thus for options on non-dividend-paying stocks c − p = S0 − K e−rT . Arturo Leccadito (Unical) Introduzione ai Derivati 13 / 36
Formally, we first assume c − p > S0 − K e−rT and consider the strategy 1. Sell the call; 2. Buy the put; 3. Buy the stock; 4. Borrow the amount K e−rT at the rate r . The result is a riskless profit of c − p − S0 + K e−rT at time 0. 0 T 1. c − max{ST − K , 0} 2. −p max{K − ST , 0} 3. −S0 ST 4. K e−rT −K c − p − S0 + K e−rT −[ST − K ] + ST − K = 0 Arturo Leccadito (Unical) Introduzione ai Derivati 14 / 36
Similarly, one could build an arbitrage strategy when c − p < S0 − K e−rT . The put-call parity justifies the lower bounds for option prices. Since p > 0 and c > 0 we have c = p + S0 − K e−rT ≥ S0 − K e−rT p = c + K e−rT − S0 ≥ K e−rT − S0 . Arturo Leccadito (Unical) Introduzione ai Derivati 15 / 36
Early Exercise Usually there is some chance that an American option will be exercised early. An exception is an American call on a non-dividend paying stock that should never be exercised early. Recall that c ≥ S0 − K e−rT . For r > 0 and T > 0 this implies c > S0 − K and as C ≥ c, we have C > S0 − K . Now, if it were optimal to exercise early, C would be equal to the intrinsic value S0 − K . We deduce that it can never be optimal to to exercise early. Arturo Leccadito (Unical) Introduzione ai Derivati 16 / 36
The intuition behind the result is as follows. Suppose the owner of the American call wants to hold the stock after the call’s maturity. He should not exercise early because No income is sacrificed Payment of the strike price is delayed (investor would lost the interest paid on the strike price in the case of an early exercise) Holding the call provides insurance against stock price falling below strike price On the other hand, if the owner plans to exercise and sell the stock, he would be better off selling the option than exercising it. Arturo Leccadito (Unical) Introduzione ai Derivati 17 / 36
Swaps A swap is an agreement to exchange cash flows at specified future times. Usually the calculation of the cash flows involves the future values of one or more market variables. A forward can be viewed as a simple example of a swap. At maturity the buyer pays K and receives the market value of the asset S. Whereas a forward leads to the exchange of cash flows on just one future date, swaps typically involve exchanges on several future dates. The most common type of swap is a “Plain Vanilla” Interest Rate Swap: B agrees to pay A cash flows equal to interest at a fixed rate on a notional principal and A agrees to pay B cash flows equal to interest at a floating rate on the same notional principal. Arturo Leccadito (Unical) Introduzione ai Derivati 18 / 36
Example Consider an agreement initiated on March 5, 2004 by Microsoft to receive 6-month LIBOR (rate of interest offered by banks on deposits from other banks in Eurocurrency markets) and pay a fixed rate of 5% per annum every 6 months for 2 years on a notional principal of $100 million. Cash flows (Millions of Dollars) are reported in the following table. Note that the first exchange of payments takes place on September 5, 2004 and every payment is based on the LIBOR of the previous period. LIBOR Floating Fixed Net Cash Flow 05 Mar 04 4.20% 05 Sept 04 4.80% 2.1 -2.5 -0.4 05 Mar 05 5.30% 2.4 -2.5 -0.1 05 Sept 05 5.50% 2.65 -2.5 0.15 05 Mar 06 5.60% 2.75 -2.5 0.25 Arturo Leccadito (Unical) Introduzione ai Derivati 19 / 36
Typical Uses of an Interest Rate Swap Converting a liability from ◮ fixed rate to floating rate ◮ floating rate to fixed rate Some companies may have a comparative advantage when borrowing in fixed-rate markets, other companies may have a comparative advantage when borrowing in floating-rate markets. It makes sense for a company to go to the market where it has a comparative advantage, but this may imply to borrow floating when it wanted to borrow fixed (or vice versa). Thus the swap is used to transform a floating rate loan to a fixed rate loan (or vice versa). Arturo Leccadito (Unical) Introduzione ai Derivati 20 / 36
Example Suppose AAACorp wants to borrow floating BBBCorp wants to borrow fixed and they have been offered the rates Fixed Floating AAACorp 4.00% 6-month LIBOR + 0.30% BBBCorp 5.20% 6-month LIBOR + 1.0% Since the difference between the two fixed rates is bigger than the difference between the two floating rates, BBBCorp has a compara- tive advantage in the floating-rate market and AAACorp has a com- parative advantage in the fixed-rate market. A possible swap is: 3.95% 4% AAACorp BBBCorp LIBOR+1% LIBOR Arturo Leccadito (Unical) Introduzione ai Derivati 21 / 36
Example cont’d At the end of the day AAACorp ends up paying LIBOR+ 4% – 3.95% = LIBOR +0.05% which is 0.25% less than the floating rate it was offered BBBCorp ends up paying LIBOR+ 1% – LIBOR + 3.95% = 4.95% which is 0.25% less than the fixed rate it was offered Arturo Leccadito (Unical) Introduzione ai Derivati 22 / 36
Ways Derivatives are Used To hedge risks. For instance ◮ A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract ◮ An investor owns 1000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 1000 put options. The value of his holding is always above $27500 ($26500 when the cost of the option is taken into account) Arturo Leccadito (Unical) Introduzione ai Derivati 23 / 36
Two more hedging examples: ◮ Due to an exceptionally high number of new employees, pension fund A will receive £10 million in pension contributions one year from now. The fund will want to invest this amount of money in the FTSE 100 Index, but it is worried the price of the index may become over-inflated over the next year. They decide to buy at-the-money European Call options expiring in one year time. The fund this way acquires the right (but not the obligation) to buy shares of the index in one year time at the current price. Effectively the found has bought protection against price increases. ◮ Fund B has £20 million invested in the same index but has to make a payment in one year time, due to an exceptionally high number of retiring employees. The fund could be in trouble if the index price falls in one year. Hence they decide to buy at-the-money European Put options maturing in one year time. This way the fund acquires protection against price decreases. Arturo Leccadito (Unical) Introduzione ai Derivati 24 / 36
To speculate (take a view on the future direction of the market) ◮ An investor with $4,000 to invest feels that Amazon.com’s stock price will increase over the next 2 months. The current stock price is $40 and the price of a 2-month call option with a strike of 45 is $2. He could buy 100 shares or 2000 call options. Profit and losses from the strategies are illustrated in the following table for two different price scenarios. Price after 2 months Strategy 30 50 Buy shares -1000 1000 Buy call options -4000 6000 This property is known as Financial Leverage. Arturo Leccadito (Unical) Introduzione ai Derivati 25 / 36
Positions in an Option & the Underlying Profit Profit K K ST ST a) Long Stock + Short Call b) Short Stock + Long Call Profit Profit K ST K ST c) Long Stock + Long Put d) Short Stock + Short Put Arturo Leccadito (Unical) Introduzione ai Derivati 26 / 36
Bull Spread Using Calls A spread trading strategy involves taking a position in two or more options of the same type (i.e. two or more calls or two or more puts). A trader who enters a bull spread is hoping that the stock price will increase. Profit K1 K2 ST Arturo Leccadito (Unical) Introduzione ai Derivati 27 / 36
Bull Spread Using Puts Profit K1 K2 ST Arturo Leccadito (Unical) Introduzione ai Derivati 28 / 36
Bear Spread Using Calls A trader who enters a bear spread is hoping that the stock price will decrease. Profit K1 K2 ST Arturo Leccadito (Unical) Introduzione ai Derivati 29 / 36
Bear Spread Using Puts Profit K2 K1 ST Arturo Leccadito (Unical) Introduzione ai Derivati 30 / 36
Box Spread A Box Spread is combination of a bull call spread and a bear put spread. If all options are European a box spread is worth the present value of the difference between the strike prices. This is because the payoff is for K1 < K2 : max{ST − K1 , 0} − max{ST − K2 , 0} | {z } payoff of a bull call spread + max{K2 − ST , 0} − max{K1 − ST , 0} | {z } payoff of a bear put spread = max{ST − K1 , 0} − max{K1 − ST , 0} | {z } =ST −K1 − (max{ST − K2 , 0} − max{K2 − ST , 0}) | {z } =ST −K2 = K2 − K1 . Arturo Leccadito (Unical) Introduzione ai Derivati 31 / 36
Butterfly Spread Using Calls A trader who enters a butterfly spread is hoping that the stock price will stay close to K2 (typically the option with strike K2 is at-the-money). A significant price move leads to a small lost. Profit K1 K2 K3 ST Arturo Leccadito (Unical) Introduzione ai Derivati 32 / 36
Butterfly Spread Using Puts Profit K1 K2 K3 ST Arturo Leccadito (Unical) Introduzione ai Derivati 33 / 36
Combinations A combination involves investing in both calls and puts on the same stock. A bottom straddle involves buying a call and a put with the same strike K and maturity. Payoff: |ST − K |. A top straddle is the reverse position. Profit K ST Arturo Leccadito (Unical) Introduzione ai Derivati 34 / 36
Strip & Strap Strip: long position in a call and two puts (same strike, same maturity). The trader hopes that there will be a big stock move (with a decrease more likely than an increase). Strap: long position in two calls and a put (same strike, same maturity). The trader hopes that there will be a big stock move (with an increase more likely than a decrease). Profit Profit K ST K ST Strip Strap Arturo Leccadito (Unical) Introduzione ai Derivati 35 / 36
A Strangle Combination Similar to a straddle, but the call strike price, K2 , is bigger than the put strike price, K1 . The price has to move farther in a strangle than in a straddle for the buyer to make a profit. Profit K1 K2 ST Arturo Leccadito (Unical) Introduzione ai Derivati 36 / 36
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