Finansiell ekonomi 723g28 Linköpings University Options and Futures Finansiell ekonomi 723g28 Linköpings University

What is a Derivative? A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Examples: futures, forwards, swaps, options, exotics… Obs: you may jump to slide #21 to start direct with options.

How Derivatives are Used To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit To change the nature of a liability To change the nature of an investment without incurring the costs of selling one portfolio and buying another 4

Options vs. Futures/Forwards A futures/forward contract gives the holder the obligation to buy or sell at a certain price at a certain date in the future An option gives the holder the right, but not the obligation to buy or sell at a certain price at a certain date in the future

Foreign Exchange Quotes for GBP, (£) May 24, 2010 The forward price may be different for contracts of different maturities (as shown by the table) Bid Offer Spot 1.4407 1.4411 1-month forward 1.4408 1.4413 3-month forward 1.4410 1.4415 6-month forward 1.4416 1.4422

Long position and short position The party that has agreed to buy has a long position The party that has agreed to sell has a short position 10

Example On May 24, 2010 the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 1.4422 This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010 What are the possible outcomes?

Svenska termer Forwards och Terminer • Spotkontrakt: en överenskommelse mellan två parter att utbyta något idag för ett specificerat pris, spotpriset. à vista marknad. • Terminskontrakt: en överenskommelse (skyldighet) mellan två parter att utbyta något för ett specificerat pris, terminspriset, vid en specifik framtida tidpunkt, lösendagen.

Profit from a Long Forward Position (K= delivery price=forward price at the time contract is entered into) Payoff diagram Profit Price of Underlying at Maturity, ST K 14

Profit from a Short Forward Position (K= delivery price=forward price at the time contract is entered into) Profit Price of Underlying at Maturity, ST K 15

Futures Contracts Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract a forward contract is traded over the counter (OTC) (Skräddarsydd) a futures contract is standardized and traded on an exchange. CME Group NYSE Euronext, BM&F (Sao Paulo, Brazil) TIFFE (Tokyo)

Key Points About Futures They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity

Margins A margin is cash or marketable securities deposited by an investor with his or her broker The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract

Pricing of forward Guld (commodities): F = (1 + rf + s) · S0 Finansiella tillgångar: F = (1 + rf) · S0 S0 is the spot price. S is the storage cost rf is risk free interest rate F is the forward price

Examples of Futures Contracts Agreement to: Buy 100 oz. of gold @ US$1400/oz. in December Sell £62,500 @ 1.4500 US$/£ in March Sell 1,000 bbl. of oil @ US$90/bbl. in April Oz: ounce Bbl: barrel 9

Example : An Arbitrage Opportunity? Suppose that: The spot price of gold is US$1,400 The 1-year forward price of gold is US$1,500 The 1-year US$ interest rate is 5% per annum Q: What should be the 1-year forward price? Is there an arbitrage opportunity? 18

The Forward Price of Gold If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S = 1400, T = 1, and r =0.05 so that F = 1400(1+0.05) = 1470 20

Hedging Examples An investor owns 1,000 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 10 contracts 16

Value of Microsoft Shares with and without Hedging

Some Terminology Open interest: the total number of contracts outstanding equal to number of long positions or number of short positions Settlement price: the price just before the final bell each day used for the daily settlement process Trading Volume : the number of trades in one day

Forward Contracts vs Futures Contracts Private contract between 2 parties Exchange traded Non-standard contract Standard contract Usually 1 specified delivery date Range of delivery dates Settled at end of contract Settled daily Delivery or final cash settlement usually occurs prior to maturity FORWARDS FUTURES Some credit risk Virtually no credit risk Contract usually closed out 16

The right but not the obligation… Options The right but not the obligation… Chapter 23 PPT Outline Calls and Puts Option Values and Profit Real Options Black-Scholes Pricing Model 2

Options A call option is an option to buy a certain asset by a certain date for a certain price (the strike price) A put option is an option to sell a certain asset by a certain date for a certain price (the strike price)

Option Obligations: the writer of the option 5

American vs. European Options An American option can be exercised at any time during its life A European option can be exercised only at maturity The time value will be lost when you exercise prematurely.

Option Value: Example Option values given an exercise price of $720 Exercise Price – The price at which the underlying security can be purchased (call option) or sold (put option). The exercise price is determined at the time the option contract is formed. Also known as the strike price. What are the payoff limits for call option buyers? Sellers? What are the payoff limits for put option buyers? Sellers? 7

Call option value (buyer) given a $720 exercise price. $120 720 840 Share Price 8

$20 call option (buyer) given a $720 exercise price Call Option Profit $20 call option (buyer) given a $720 exercise price Call option value $100 720 840 Share Price 8

Call Option Value Call option payoff (seller) given a $720 exercise price. 720 840 Call option $ payoff $-120 Share Price 10

$20 call option (seller) given a $720 exercise price: Call Option Profit $20 call option (seller) given a $720 exercise price: 720 840 $-100 Call option $ payoff $-120 Share Price 10

Call Option: Example How much must the stock be worth at expiration in order for a call holder to break even if the exercise price is $50 and the call premium was $4?      

Put option value (buyer) given a $720 exercise price: $120 600 720 Share Price 9

$30 put option (buyer) given a $720 exercise price: Put Option Profit $30 put option (buyer) given a $720 exercise price: Put option value $90 600 720 Share Price 9

Put Option Value Put option payoff (seller) given a $720 exercise price. Share Price -$120 Put option $ payoff 600 720 11

$30 put option (seller) given a $720 exercise price. Put Option Profit $30 put option (seller) given a $720 exercise price. -$90 Share Price Put option $ payoff 600 720 11

Put Options: Example What is your return on exercising a put option which was purchased for $10 with an exercise price of $85? The stock price at expiration is $81.          

(Stock price - exercise price) or 0 Options Value Stock Price Upper Limit Lower Limit (Stock price - exercise price) or 0 which ever is higher 21

Option Value

Option Value Point A -When the stock is worthless, the option is worthless. Point B -When the stock price becomes very high, the option price approaches the stock price less the present value of the exercise price. Point C -The option price always exceeds its minimum value (except at maturity or when stock price is zero). The value of an option increases with both the variability of the share price and the time to expiration.

Option Value Components of the Option Price 1 - Underlying stock price 2 - Strike or Exercise price 3 - Volatility of the stock returns (standard deviation of annual returns) 4 - Time to option expiration 5 - Time value of money (discount rate) 22

Call Option Value 22

Put-Call Parity: No Dividends Consider the following 2 portfolios: Portfolio A: call option on a stock + zero-coupon bond (or a deposit) that pays K at time T Portfolio B: Put option on the stock + the stock

Values of Portfolios are the same at expiration (förfalldag) ST > K ST < K Portfolio A Call option ST − K Zero-coupon bond K Total ST Portfolio B Put Option K− ST Share

The Put-Call Parity Result Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S0

Ex: put-call parity Suppose that What are the put option price? c + Ke -rT = p + S0 p = c-S0 +Ke -rT =3-31+30*EXP(-0,1*0,25) = 1,259 c= 3 S0= 31 T = 0.25 r = 10% K =30

Bounds for European and American Put Options (No Dividends)

Synthetic options Two or more options combines together creates exotic options

Option Value: profit diagram for a straddle Straddle - Long call and long put - Strategy for profiting from high volatility Long call Long put Straddle Position Value Share Price 18

Option Value Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Straddle An investor may take a long straddle position if he thinks the market is highly volatile, but does not know in which direction it is going to move. 19

Exotic options: a butterfly option A long butterfly position will make profit if the future volatility is lower than the implied volatility. The spread is created by buying a call with a relatively low strike (x1), buying a call with a relatively high strike (x3), and shorting two calls with a strike in between (x2).

Long Call Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months 30 20 10 -5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($)

Short Call Profit from writing one European call option: option price = $5, strike price = $100 -30 -20 -10 5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($)

Long Put Profit from buying a European put option: option price = $7, strike price = $70 30 20 10 -7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($)

Short Put Profit from writing a European put option: option price = $7, strike price = $70 -30 -20 -10 7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($)

Payoffs from Options What is the Option Position in Each Case? K = Strike price, ST = Price of asset at maturity Payoff Payoff ST K Payoff

The Black-Scholes-Merton Formulas

Real options max(VT −D, 0) With the limited liability of the modern corporations, the shareholders´ equity can be regarded as a real option on the assets of the firm. The shareholder value of equity value is max(VT −D, 0) where VT is the value of the firm and D is the debt repayment required. Thus the company can be considered as a call option on the firm value V at the strike price of D.

Options on Real Assets Real Options - Options embedded in real assets Option to Abandon Option to Expand 25

Options on Financial Assets Executive Stock Options Warrants Convertible Bonds Callable Bonds Executive Stock Options – Long term call options given to executives as part of their compensation package. Warrants - Right to buy shares from a company at a stipulated price before a set date. Convertible Bond - Bond that the holder may exchange for a specific number of shares. Callable Bond - Bond that may be repurchased by the issuer before maturity at specified call price. 26