Options Forwards Bonds and No Arbitrage Futures Mc

Options, Forwards, Bonds and No. Arbitrage Futures (Mc. Donald, Chapters 1 -2, Berk-Demarzo, Chapter 8)

Short-Selling Example: Suppose you short-sell 100 IBM shares for $90 a share. After 90 days, you close your position at a time in which as share costs $92. If you pay a lease fee of $0. 50 per share, what is your return over the 90 day period? 3 -2

Transactions Costs • Buying and selling a financial asset – Brokers: commissions – Market-makers: bid-ask (offer) spread • Example: Buy and sell 100 shares of XYZ – XYZ: bid = $49. 75, offer = $50, commission = $15 3 -3

Forward Contracts • Definition: a binding agreement (obligation) to buy/sell an underlying asset in the future, at a price set today • Futures contracts are the same as forwards in principle except for some institutional and pricing differences. • A forward contract specifies – The features and quantity of the asset to be delivered – The delivery logistics, such as time, date, and place – The price the buyer will pay at the time of delivery 3 -4

Payoff on a Forward Contract • Payoff for a contract is its value at expiration • Payoff for – Long forward = Spot price at expiration – Forward price – Short forward = Forward price – Spot price at expiration • Example 2. 1: S&R (special and rich) index: – Today: Spot price = $1, 000, 6 -month forward price = $1, 020 – In six months at contract expiration: Spot price = $1, 050 • Long position payoff = $1, 050 – $1, 020 = $30 • Short position payoff = $1, 020 – $1, 050 = ($30) 3 -5

Long Position of Crude Oil March 2009 Contract Spot Crude January, 2009: $73 Future March, 2009: $75 Payoff at expiration $75 Crude price at expiration -$75 3 -6

Short Position of Crude Oil March 2009 Contract Spot Crude January, 2009: $73 Future March, 2009: $75 Payoff at expiration $75 Crude price at expiration 3 -7

Additional Considerations • Type of settlement – Cash settlement: less costly and more practical – Physical delivery: often avoided due to significant costs • Credit risk of the counter party – Major issue for over-the-counter contracts • Credit check, collateral, bank letter of credit – Less severe for exchange-traded contracts • Exchange guarantees transactions, requires collateral 3 -8

Equation 2. 1 -2. 2 3 -9

Payoff Diagram for Forwards • Long and short forward positions on the S&R 500 index Figure 2. 2 Long and short forward positions on the S&R 500 index. 3 -10

Table 2. 1 Payoff after 6 months from a long S&R forward contract and a short S&R forward contract at a forward price of $1020. If the index price in 6 months is $1020, both the long and short have a 0 payoff. If the index price is greater than $1020, the long makes money and the short loses money. If the index price is less than $1020, the long loses money and the short makes money. 3 -11

Forward Versus Outright Purchase Figure 2. 3 Comparison of payoff after 6 months of a long position in the S&R index versus a forward contract in the S&R index. Forward payoff Bond payoff • Forward + bond = Spot price at expiration – $1, 020 + $1, 020 = Spot price at expiration 3 -12

Figure 2. 4 Payoff diagram for a long S&R forward contract, together with a zero-coupon bond that pays $1020 at maturity. Summing the value of the long forward plus the bond at each S&R index price gives the line labeled “Forward + bond. ” 3 -13

Call Options • A non-binding agreement (right but not an obligation) to buy an asset in the future, at a price set today • Preserves the upside potential, while at the same time eliminating the unpleasant downside (for the buyer) • The seller of a call option is obligated to deliver if asked 3 -14

Examples • Example 2. 3: S&R index – Today: call buyer acquires the right to pay $1, 020 in six months for the index, but is not obligated to do so – In six months at contract expiration: if spot price is • $1, 100, call buyer’s payoff = $1, 100 – $1, 020 = $80 • $900, call buyer walks away, buyer’s payoff = $0 • Example 2. 4: S&R index – Today: call seller is obligated to sell the index for $1, 020 in six months, if asked to do so – In six months at contract expiration: if spot price is • $1, 100, call seller’s payoff = $1, 020 – $1, 100 = ($80) • $900, call buyer walks away, seller’s payoff = $0 • Why would anyone agree to be on the seller side? 3 -15

Definition and Terminology • A call option gives the owner the right but not the obligation to buy the underlying asset at a predetermined price during a predetermined time period • Strike (or exercise) price: the amount paid by the option buyer for the asset if he/she decides to exercise • Exercise: the act of paying the strike price to buy the asset • Expiration: the date by which the option must be exercised or become worthless • Exercise style: specifies when the option can be exercised – European-style: can be exercised only at expiration date – American-style: can be exercised at any time before expiration – Bermudan-style: Can be exercised during specified periods • Premium – the cost of the option to the option buyer. • Settlement type - either money settlement or delivery of good. 3 -16

Equations 2. 3 -2. 6 3 -17

Table 2. 2 Closing prices, daily volume, and open interest for S&P 500 options, listed on the Chicago Board Options Exchange, on August 14, 2007. The S&P 500 index closed that day at 1426. 54. 3 -18

Payoff/Profit of a Purchased Call • Payoff = Max [0, spot price at expiration – strike price] • Profit = Payoff – future value of option premium • Examples 2. 5 & 2. 6: – S&R Index 6 -month Call Option • Strike price = $1, 000, Premium = $93. 81, 6 -month risk-free rate = 2% – If index value in six months = $1100 • Payoff = max [0, $1, 100 – $1, 000] = $100 • Profit = $100 – ($93. 81 x 1. 02) = $4. 32 – If index value in six months = $900 • Payoff = max [0, $900 – $1, 000] = $0 • Profit = $0 – ($93. 81 x 1. 02) = – $95. 68 3 -19

Diagrams for Purchased Call • Payoff at expiration Figure 2. 5 The payoff at expiration of a purchased S&R call with a $1000 strike price. • Profit at expiration Figure 2. 6 Profit at expiration for purchase of 6 -month S&R index call with strike price of $1000 versus profit on long S&R index forward position. 3 -20

Table 2. 3 Payoff and profit after 6 months from a purchased 1. 000 strike S&R call option with a future value of premium of $95. 68. The option premium is assumed to be $93. 81 and the effective interest rate is 2% over 6 months. The payoff is computed using equation (2. 3) and the profit using equation (2. 4). 3 -21

Payoff/Profit of a Written Call • Payoff = – max [0, spot price at expiration – strike price] • Profit = Payoff + future value of option premium • Example 2. 7 – S&R Index 6 -month Call Option • Strike price = $1, 000, Premium = $93. 81, 6 -month risk-free rate = 2% – If index value in six months = $1100 • Payoff = – max [0, $1, 100 – $1, 000] = – $100 • Profit = – $100 + ($93. 81 x 1. 02) = – $4. 32 – If index value in six months = $900 • Payoff = – max [0, $900 – $1, 000] = $0 • Profit = $0 + ($93. 81 x 1. 02) = $95. 68 3 -22

Figure 2. 7 Profit for writer of 6 -month S&R call with strike of $1000 versus profit for short S&R forward. 3 -23

Put Options • A put option gives the owner the right but not the obligation to sell the underlying asset at a predetermined price during a predetermined time period • The seller of a put option is obligated to buy if asked • Payoff/profit of a purchased (i. e. , long) put – Payoff = max [0, strike price – spot price at expiration] – Profit = Payoff – future value of option premium • Payoff/profit of a written (i. e. , short) put – Payoff = – max [0, strike price – spot price at expiration] – Profit = Payoff + future value of option premium 3 -24

Equations 2. 7 -2. 10 3 -25

Put Option Examples • Examples 2. 9 & 2. 10 – S&R Index 6 -month Put Option • Strike price = $1, 000, Premium = $74. 20, 6 -month risk-free rate = 2% – If index value in six months = $1100 • Payoff = max [0, $1, 000 – $1, 100] = $0 • Profit = $0 – ($74. 20 x 1. 02) = – $75. 68 – If index value in six months = $900 • Payoff = max [0, $1, 000 – $900] = $100 • Profit = $100 – ($74. 20 x 1. 02) = $24. 32 3 -26

Figure 2. 8 Profit on a purchased S&R index put with strike price of $1000 versus a short S&R index forward. 3 -27

Profit for a Long Put Position • Profit table Table 2. 4 Profit after 6 months from a purchased 1000 -strike S&R put option with a future value of premium of $75. 68. 3 -28

Figure 2. 9 Written S&R index put option with strike of $1000 versus a long S&R index forward contract. 3 -29

Figure 2. 10 Profit diagrams for the three basic long positions: long forward, purchased call, and written put. 3 -30

Figure 2. 11 Profit diagrams for the three basic short positions: short forward, written call, and purchased put. 3 -31

Uses of Derivatives The two most common cited reason for the use of derivatives • Risk management - Hedging • Speculation – Leveraging 3 -32

Using Options to Enhance Risk (Speculation) The price of an IBM is $100, a three month option of IBM with an exercise price of $100 (a naïve value of zero) costs $10. If you have $100 to invest, compare the payoff of buying a share of IBM compared to the purchase of 10 options. 3 -33

Options and Insurance • Homeowner’s insurance as a put option Figure 2. 12 Profit from insurance policy on a $200, 000 house. 3 -34

Table 2. 6 Forwards, calls, and puts at a glance: a summary of forward and option positions. 3 -35

Option and Forward Positions: A Summary Figure 2. 13 The basic profit diagrams: long and short forward, long and short call, and long and short put. 3 -36

6. Bond Cash Flows, Prices, and Yields Terminology • Bond certificate • Maturity date, term • Coupon • Face value (principal, par value) • Coupon rate • Zero-coupon bond - Treasury bills • Traded at discount (premium), pure discount bonds • Yield to maturity (YTM) 3 -37

Pricing a bond Assume a 10% coupon bond with face value of $1000 and maturity in 5 years. The YTM is 11%, what is the price of the bond? 3 -38

Equation Yield to Maturity of a Coupon Bond 3 -39

Relation between coupon rate and YTM • If YTM=Coupon rate Bond priced at PAR • If YTM>Coupon rate Bond priced at discount • If YTM<Coupon rate Bond priced at premium 3 -40

Interest rate risk 8 -41

Yields for Different Maturities The following table summarizes prices of various default-free zero-coupon bonds (expressed as a percentage of face value): Maturity (years) 1 2 3 4 5 Price (per $100 face value) 95. 51 91. 05 86. 38 81. 65 76. 51 a. Compute the yield to maturity for each bond - r 1’ r 2’ r 3’ r 4, r 5. b. Plot the zero-coupon yield curve. c. Is the yield curve upward sloping/downward sloping or flat? 3 -42

Computing forward rates A forward interest rate (or forward) rate is an interest rate that we can guarantee today for a loan or investment that will occur in the future. Using the prices below of the 1 -3 years default-free zero-coupon bonds, find f 1, 1 f 2, 1 f 1, 2. Notation is fstart, length Maturity (years) 1 2 3 Price (per $100 face value) 95. 51 91. 05 86. 38 YTM of zero coupon (r 1, r 2, r 3) 4. 70% 4. 80% 5. 00% 3 -43

Yields for Different Maturities Solve using the prices below of the various defaultfree zero-coupon bonds, Maturity (years) 1 2 Price (per $100 face value) 95. 51 91. 05 YTM of zero coupon (r 1, r 2) 4. 70% 4. 80% You need to borrow $1000 a year from now for a period of one year. How can you secure a fixed borrowing rate for the loan and what will the borrowing rate be? 3 -44

The Yield Curve and Bond Arbitrage Assume zero-coupon yields on default-free securities are as summarized in the following table: Maturity 1 2 3 4 5 YTM of zero coupon 4. 00% 4. 30% 4. 50% 4. 70% 4. 80% Consider a five-year, default-free security with annual coupon payments of 5% and a face value of $1000. a. Without doing any calculation, determine whether this bond is trading at a premium or at a discount. Explain. b. What is the YTM on this bond? c. If the YTM on this bond increased to 5. 2%, what would the new price be? 3 -45

No Arbitrage Question You are given the following information on various government bonds, each with a different coupon rate. a. b. c. d. What is the two year spot interest rate r 2 ? What is the three year spot interest rate r 3 ? What is the four year spot interest rate r 4 ? What is the two year forward rate of year 1 (f 1, 2 )? 3 -46

Figure 8. 3 Corporate Yield Curves for Various Ratings, February 2009 Source: Reuters 3 -47