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**Options, Forwards, Bonds and No-Arbitrage Futures**

(McDonald, Chapters 1-2, Berk-Demarzo, Chapter 8)

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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?

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**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

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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

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**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)

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**Long Position of Crude Oil March 2009 Contract**

Spot Crude January, 2009: $73 Future March, 2009: $75 Payoff at expiration Crude price at expiration $75 -$75

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**Short Position of Crude Oil March 2009 Contract**

Spot Crude January, 2009: $73 Future March, 2009: $75 Payoff at expiration Crude price at expiration $75 -$75

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**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

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Equation

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**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.

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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.

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**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, = Spot price at expiration

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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.”

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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

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**Examples Example 2.3: S&R index Example 2.4: 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 $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?

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**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.

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Equations

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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, The S&P 500 index closed that day at

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**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

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**Diagrams for Purchased Call**

Payoff at expiration Profit at expiration Figure 2.5 The payoff at expiration of a purchased S&R call with a $1000 strike price. 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.

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**Table 2. 3 Payoff and profit after 6 months from a purchased 1**

Table 2.3 Payoff and profit after 6 months from a purchased strike S&R call option with a future value of premium of $ 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).

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**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

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Figure 2.7 Profit for writer of 6-month S&R call with strike of $1000 versus profit for short S&R forward.

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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

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Equations

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**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

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Figure 2.8 Profit on a purchased S&R index put with strike price of $1000 versus a short S&R index forward.

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**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.

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Figure 2.9 Written S&R index put option with strike of $1000 versus a long S&R index forward contract.

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Figure Profit diagrams for the three basic long positions: long forward, purchased call, and written put.

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Figure Profit diagrams for the three basic short positions: short forward, written call, and purchased put.

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

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**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.

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**Options and Insurance Homeowner’s insurance as a put option**

Figure Profit from insurance policy on a $200,000 house.

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**Table 2.6 Forwards, calls, and puts at a glance: a summary of forward and option positions.**

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**Option and Forward Positions: A Summary**

Figure The basic profit diagrams: long and short forward, long and short call, and long and short put.

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**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)

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**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 Compute the yield to maturity for each bond. Plot the zero-coupon yield curve. Is the yield curve upward sloping/downward sloping or flat?

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**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 f1,1 f2,1 f1,2. Notation is fstart,length Maturity (years) 1 2 3 Price (per $100 face value) 95.51 91.05 86.38 YTM 4.70% 4.80% 5.00% Calculate the forward rates for years 1-3.

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**Yields for Different Maturities**

Solve using the prices below of the various default-free zero-coupon bonds, Maturity (years) 1 2 Price (per $100 face value) 95.51 91.05 YTM 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?

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**Equation Yield to Maturity of a Coupon Bond**

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**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 Zero coupon YTM 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. Without doing any calculation, determine whether this bond is trading at a premium or at a discount. Explain. What is the YTM on this bond? If the YTM on this bond increased to 5.2%, what would the new price be?

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**Figure 8.3 Corporate Yield Curves for Various Ratings, February 2009**

Source: Reuters

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