Section 1: Advanced Derivatives

Section 1: Advanced Derivatives Options, Futures, and Other Derivatives (6th edition) by Hull Chapter Mechanics of Futures Markets (Sections.7-.10 only) 3 Chapter 5 Determination of Forward and Futures Prices 7 Chapter 7 Swaps 17 Chapter 8 Mechanics of Options Markets (Sections 8.3-8.13 only) 9 Chapter 1 Wiener Processes and Itô s Lemma 36 Chapter 14 Options on Stock Indices, Currencies, and Futures 41 (Sections 14.1-14.3 and 14.7-14.9 only) Chapter 1 Credit Derivatives 49 Chapter Exotic Options 56 The Oxford Guide to Financial Modeling (FET-106-07) Chapter 5: Interest Rate Derivatives: Interest Rate Models 6 Chapter 6: Implied Volatility Surface: Calibrating the Models 71 The J. P. Morgan Guide to Credit Derivatives (FET-100-07) 8 Options, Futures, and Other Derivatives (6th edition) by Hull Chapter 31 Real Options 96 Equity-Indexed Life Products by Ho and Sham (FET-101-07) 10 Hedging with Derivatives in Traditional Insurance Products by Larry Rubin (FET-103-07) 106 Investment Guarantees by Mary Hardy (FET-104-07) Chapter 6: Modeling the Guarantee Liability 108 Chapter 1: Guaranteed Annuity Options 11 Options, Futures, and Other Derivatives (6th edition) by Hull Chapter 15 The Greek Letters 116 Investment Management for Insurers by Babbel and Fabozzi Chapter 11: The Four Faces of an Interest Rate Model 16 Handbook of Mortgage Backed Securities by Fabozzi (FET-105-07) Chapter 3: Valuation of Mortgage-Backed Securities 130 Variable Product Hedging Practical Considerations (FET-107-07) 135 JAM July 007 Section 1 Page 1 of 135

Section 1: Advanced Derivatives Learning Outcomes a. Define the cash flow characteristics of complex derivatives including exotic options, credit derivatives, interest rate derivatives, swaps, and other non traditional derivatives b. Evaluate the risk/return characteristics of complex derivatives c. Identify embedded options in assets and liabilities d. Define option adjusted spread analysis and its limitations e. Evaluate the impact of embedded options on risk/return characteristics of assets and liabilities JAM July 007 Section 1 Page of 135

Options, Futures, and Other Derivatives (Sixth Edition) by Hull Chapter : Mechanics of Futures Markets (Sections.7 to.10 only).7 Types of Traders and Types of Orders Types of Traders 1. Commission Brokers Charge a commission to trade for a client. Locals Trade for own account Classification of Traders 1. Hedgers. Speculators a. Scalpers (profit from small price changes by only holding a position for a few minutes) b. Day traders (hold position for less than one day) c. Position traders (profit from major movements over longer periods of time) 3. Arbitrageurs Types of Orders 1. Market order Trade carried out immediately at market price. Limit order Only execute trade at specified price or one more favorable to the investor 3. Stop order or stop-loss order Becomes a market order when the price gets to the specified price or worse Designed to stop the losses 4. Stop-limit order Becomes a limit order when the price gets to the specified price or worse JAM July 007 Section 1 Page 3 of 135

5. Market-if-touched (MIT) order Also known as a board order Becomes a market order when the price gets to the specified price or better Designed to lock in profits 6. Discretionary order or market-not-held order Like a market order, but can be delayed at the broker s discretion to get a better price 7. Time conditions Unless otherwise specified, orders will expire at the end of the day Time-of-day orders specify an earlier termination Open or good-till-canceled orders are open until executed Fill-or-kill orders must be executed immediately or cancelled.8 Regulation Since 1974 futures in the United States have been regulated by the Commodity Futures Trading Commission (CFTC) Contracts must have some useful economic purpose to be approved The CFTC looks out for the public interest by making sure the prices and outstanding positions are made public The CFTC also licenses individuals that offer services to the public in futures trading and follows up on complaints The National Futures Association (NFA) is responsible for self-regulating portions of the industry Other bodies like the Securities and Exchange Commission (SEC) and Federal Reserve Board occasionally get involved Trading Irregularities Cornering the market occurs if an investor group takes a large long position in a futures position and also limits the supply of the underling commodity As the futures contract approaches maturity, the short party will have difficulty purchasing the underlying commodity to deliver and thus drive the price up Regulators could increase the margin requirements to discourage this activity Other irregularities involve traders on the floor For example, front running takes place when the traders use knowledge from customer orders to first make trades in their own account JAM July 007 Section 1 Page 4 of 135

.9 Accounting and Tax Accounting Changes in market value on futures contract are recognized immediately unless it qualifies as a hedge Hedge contracts recognize gains and losses in the same period as the item being hedged recognizes gains and losses FAS 133 (issued by the Financial Accounting Standards Board) and IAS 39 (issued by the International Accounting Standards Board) require derivatives to be held at fair value and define parameters for hedge accounting Tax In the United States, corporate tax payers must pay the ordinary income tax rate on capital gains; also, loss deductions are restricted via carry back and carry forward limitations Noncorporate taxpayers pay a lower tax rate on long-term capital gains and can carry forward capital losses indefinitely Futures contracts are closed out on the last trading day of the year, which generates a gain or loss The gain or loss is deemed 60% long-term for noncorporate taxpayers Gains or losses from hedge transactions are treated as ordinary income in the same period the income or loss is recognized for the hedged items For tax purposes, a hedge transaction must either reduce the risk of price changes for a property held or a borrowed amount for the taxpayer.10 Forward vs. Futures Contracts Summary Comparison Forward Private contract between two parties Not standardized Usually one specified delivery date Settled at end of contract Delivery or final cash settlement usually takes place Some credit risk Futures Traded on an exchange Standardized contract Range of delivery dates Settled daily Contract is usually closed out prior to maturity Virtually no credit risk JAM July 007 Section 1 Page 5 of 135

Profits from Forward and Futures Contracts Forward contracts recognize the entire gain at the end of the contract; futures contracts recognize the gains and losses daily Foreign Exchange Quotes Futures prices are always quoted as the number of US dollars per unit of foreign currency Forward prices are quoted like spot prices This is dollars per foreign currency for the British pound, euro, Australian dollar, and the New Zealand dollar For all other currencies it is foreign currency per dollar Recommended Questions and Problems You can certainly work all the practice problems, but the following are most beneficial for exam preparation. 4, 5, 14 Solutions to Recommended Questions and Problems 4) 006: ( $19.10 $18.30)( 1000 ) = $800 007: ( $0.50 $19.10)( 1000 ) = $1400 The total profit is $00. Since it is a futures contract (and thus marked-to-market), the profits and losses would be realized daily. A hedger would pay taxes on all the profit in 007. A speculator would pay taxes on $800 in 006 and $1400 in 007. 5) A stop order to sell at $ is a market order to sell once a price of $ or less is reached. It could be used to limit the losses in a long position. A limit order to sell at $ is an order to sell at a price of $ or higher. It can be used to lock in the gain on a long position. 14) This is a stop order and limit order combined. As soon as a bid or offer is placed below $0.30, a limit sell order is created at $0.10. The limit sell will be executed provided it can be done at a price of $0.10 or higher. JAM July 007 Section 1 Page 6 of 135

Options, Futures, and Other Derivatives (Sixth Edition) by Hull Chapter 5: Determination of Forward and Futures Prices 5.1 Investment Assets vs. Consumption Assets Investment assets are held by many investors for investment purposes (e.g. stocks, bonds) A consumption asset is held primarily for consumption (e.g. copper, oil) 5. Short Selling Short selling involves selling a borrowed asset (i.e. one that is not owned) The broker borrows shares from another client to lend to the investor wanting to short sell If the broker runs out of shares to borrow, the position must be closed immediately (called a short-squeeze) The investor who short sells a security must pay the broker the dividends declared on the underlying security A margin account must be maintained by the investor with the broker to cover adverse market movements In the United States short sells can only be executed after an uptick 5.3 Assumptions and Notation The following assumptions are made for some large market participants 1. No transaction costs. Same tax rate is applied to all net trading profits 3. Can borrow and lend at the same risk-free rate 4. Take advantage of arbitrage opportunities JAM July 007 Section 1 Page 7 of 135

5.4 Forward Price for an Investment Asset Assuming the underlying asset S has no income (e.g. a non-dividend-paying stock), the forward price is F rt 0 = Se 0, where r is the risk-free rate This must be the forward price to preclude arbitrage opportunities If the actual forward price is too low, then create an arbitrage portfolio by 1. Taking the long position in a forward contract. Short selling one unit of the underlying security 3. Investing the proceeds from the short-sale a the risk-free rate Arbitrage Profit = Se F rt 0 0 If the underlying asset can not be sold short, the above argument is still valid provided enough investors hold the asset for investment purposes only; they could simply sell their holdings and buy it back later at the forward price If the actual forward price is too high, then create an arbitrage portfolio by 1. Taking the short position in a forward contract. Buying one unit of the underlying security 3. Borrowing at the risk-free rate to purchase the security Arbitrage Profit = F Se 0 0 rt 5.5 Known Income ( ) F S I e rt 0 = 0, where I is the present value of the income over the life of the forward contract If the actual forward price is too low, then create an arbitrage portfolio by 1. Taking the long position in a forward contract. Short selling one unit of the underlying security 3. Investing the proceeds from the short-sale a the risk-free rate A portion must be invested so that it matures when the income is paid because the investor that holds a short position in a security must pay the income generated on the security ( ) Arbitrage Profit = S I e F rt 0 0 JAM July 007 Section 1 Page 8 of 135

If the actual forward price is too high, then create an arbitrage portfolio by 1. Taking the short position in a forward contract. Buying one unit of the underlying security 3. Borrowing at the risk-free rate to purchase the security A portion is only borrowed over a term that coincides with the income payments; the income received from the long position in the underlying security will be used to pay back that portion of the borrowing 0 0 ( ) Arbitrage Profit = F S I e rt 5.6 Known Yield F ( ) r qt 0 = Se 0, where q is the continuous yield on the underlying security over the life of the forward contract 5.7 Valuing Forward Contracts Initially the value of a forward contract is zero; later the value can be positive or negative Key Notation K is the delivery price for a forward contract negotiated in the past F 0 is the forward price today f is the value of a forward contract today For long forward contracts, ( ) For short forward contracts, = ( ) f = F K e = S I Ke = Se Ke rt rt qt rt 0 0 0 f K F e rt 0 JAM July 007 Section 1 Page 9 of 135

5.8 Are Forward Prices and Futures Prices Equal? If the risk-free interest rate is constant and the same for all maturities, the forward and futures prices are equal In reality the risk-free interest rate changes unpredictably and is not the same for all maturities Futures contracts are marked-to-market daily, whereas forward contracts only settle at maturity If the underlying security is negatively correlated with interest rates (e.g. a bond), then the futures price will be less than the forward price Marking-to-market adversely affects the long position If the underlying security decreases in price, the long party in the futures contract must put money in the margin account This must be financed at a high interest rates due to the negative correlation (the underlying security price drops when interest rates rise) If the underlying security increases in price, the long party in the futures contract may take money out of the margin account The funds can only be invested at a low interest rate (the underlying security price increases when interest rates drop) Other factors such as taxes and default risk could cause forward and futures prices to diverge Overall the prices are very close 5.9 Futures Prices of Stock Indices If one assumes the dividend yield is constant on the index, then F 0 0 ( ) = Se The value q should represent the average dividend yield over the life of the futures contract If the above equation does not hold, index arbitrage is possible; this is often implemented through program trading r qt JAM July 007 Section 1 Page 10 of 135

5.10 Forward and Futures Contracts on Currencies Define S 0 and F 0 as dollars per unit of foreign currency Equivalent Investment Strategies Assume an investor starts with 100 units of foreign currency Could invest at the risk-free rate in the foreign country and convert to dollars at the end via a forward contract Alternatively, could convert to dollars immediately and invest at the risk-free rate in the United States These two options must be equivalent, so 100e f F0 = 100Se 0 ( ) This implies r r f T F0 = Se 0, which is equivalent to assuming the income yield (q) on the foreign currency is the risk-free rate in that foreign country (r f ) rt rt If the actual forward price is too low, then create an arbitrage portfolio by 1. Taking the long position in a forward contract. Borrowing in the foreign currency, converting immediately to dollars, and investing at the risk-free rate domestically Arbitrage Profit = Se Fe rt 0 0 rt f This is the arbitrage profit in dollars at maturity per unit of initial foreign currency borrowed If the actual forward price is too high, then create an arbitrage portfolio by 1. Taking the short position in a forward contract. Borrowing in dollars, converting immediately to the foreign currency, and investing at the risk-free rate in the foreign country 1 rt 1 f Arbitrage Profit = e e S F 0 0 rt This is the arbitrage profit in foreign currency at maturity per unit of initial dollars borrowed Alternatively could express it as dollars at maturity per unit of initial foreign currency borrowed rt f 0 0 Fe Se rt JAM July 007 Section 1 Page 11 of 135

Example T = r = 5% r = 7% S = 0.600 F = 0.6600 f 0 0 The forward exchange rate should be ( ) ( 0.07 F e 0.05)( ) forward exchange rate is too high Take advantage of the arbitrage opportunity by 0= 0.600 = 0.6453, so the actual 1. Borrow one dollar for a two-year period at 7% interest (must pay back ( 0.07)( ) e = 1.1503 in two years). Immediately convert the dollar borrowed to foreign currency ( 1 0.6= 1.619) 3. Invest for two years at the foreign risk-free rate ( )( ) 0.05 Maturity value will equal 1.619 e = 1.785 4. Taking short positions in forward contracts Each contract will require selling one unit of foreign currency for 0.66 dollars in two years Therefore, will need to enter into 1.785 short forward contracts 5. After two years, exchange 1.785 units of foreign currency for 1.1765 dollars using the short forward contract 1.1765 = 0.66 1.785 6. Pay back the dollar loan of 1.1503, which leaves a profit of 1.1765 1.1503 = 0.06 dollars at time two The arbitrage profit can also be calculated using the following formula rt f rt ( 0.05)( ) ( 0.07)( ) ( )( Fe 0 Se 0 ) ( )( e e ) 1.619 = 1.619 0.66 0.6 = 0.06 JAM July 007 Section 1 Page 1 of 135

5.11 Futures on Commodities Income and Storage Costs ( ) rt ( r ut ) 0 0 0 F = S + U e = Se +, where U is the present value of all storage costs and u is the storage costs per annum Storage costs are like negative income Consumption Commodities If ( ) F S U e would rt 0 < 0 +, investors holding the commodity solely for investment purposes 1. Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interest rate. Take a long position in a forward contract However, individuals and companies holding the commodity for its consumption value would not be inclined to substitute a long forward contract This means ( + ) F S U e 0 0 rt Convenience Yields Holding some commodities is convenient (e.g. an oil refiner holding crude oil) ( ) Fe S U e yt rt 0 = 0 +, where y is the convenience yield The convenience yield is higher if the market thinks a future shortage is likely 5.1 The Cost of Carry The cost of carry summarizes the relationship between the futures price and the spot price c = r q+ u y F = Se 0 0 ct 5.13 Delivery Options Forward contracts usually specify delivery on a particular day, while futures contracts allow the short party to choose the day within a range If c > y (the futures price rises with time to maturity) the short party is encouraged to deliver at the earliest possible date If c < y (the futures price decreases with time to maturity) the short party is encouraged to deliver at the latest possible date JAM July 007 Section 1 Page 13 of 135

5.14 Futures Prices and Expected Future Spot Prices Keynes and Hicks These economists argued speculators will demand a greater return than hedgers If speculators tend to hold the long positions in futures contracts, the futures price will be lower than the expected spot price (to generate higher returns) If speculators tend to hold the short positions in futures contracts, the futures price will be higher than the expected spot price (for the same reason) Risk and Return Investments with greater systematic risks will generate higher expected returns An investment with negative systematic risk could have an expected return less than riskfree rate; this could apply to short futures positions in index funds The Risk in a Futures Position 0 [ ] ( ) F E S e r kt = T, where k is the investor s required return If the underlying asset is uncorrelated with the market, k = r and F = E[ S ] 0 T If the underlying asset is positively correlated with the market, k > r and F < E[ S ] 0 T If the underlying asset is negatively correlated with the market, k < r and F > E[ S ] Normal backwardation occurs when F < E[ S ] Contango occurs when F > E[ S ] 0 T 0 T 0 T Recommended Questions and Problems You can certainly work all the practice problems, but the following are most beneficial for exam preparation. 3, 4, 9, 10, 11, 1, 14, 7 Solutions to Recommended Questions and Problems 3. 4. ( 0.1)( 0.5) F0 = Se rt 0 = 30e = 31.86 ( r qt ) ( 0.08 0.04)( 0.333) F0 = Se 0 = 350e = 354.70 JAM July 007 Section 1 Page 14 of 135

9. 0 0 ( 0.10)( 1) Initially: F = Se rt = 40e = 44.1 f = 0 10. 0 0 ( 0.10)( 0.5) rt After Six Months: F = Se = 45e = 47.31 ( ) rt ( 0.10)( 0.5) [ ] f = F K e = 47.31 44.1 e =.95 ( r qt ) ( 0.07 0.03)( 0.5) F0 = Se 0 = 150e = 15.88 3 11. ( )( ) ( )( ) 1. 14. q = 0.05 + 0.0 = 0.03 5 5 ( r qt ) ( 0.09 0.03)( 0.417) F0 = Se 0 = 300e = 307.34 0 0 ( r qt ) ( 0.10 0.04)( 0.333) Theoretical Price= Se = 400e = 408.08 This is greater than the actual price of 405, so should Enter into long futures contract on the stock index Sell the stock index short and invest the proceeds at the risk-free rate ( r rf ) T Theoretical Price 0.65 0.08 = Se = e 0.03 0.167 = 0.6554 0 ( ) ( )( ) This is less than the actual futures price of 0.6600, so should Enter into short position in futures contract (deliver Swiss franc and receive USD) Borrow USD, convert to Swiss franc immediately at 0.65 exchange rate, and invest at 3% risk-free rate The following was not asked for in the problem, but it is a useful analysis Assume the investor borrows 100 USD for two months at the risk-free rate of 8% ( )( ) 0.08 0.167 Must pay back 100e = 101.34 Immediately convert the 100 USD to Swiss francs 100 0.65 = 153.85 Invest at the Swiss risk-free rate of 3% to get a maturity value in two months of ( ) ( 0.03)( 0.167) 153.85 e = 154.6 Convert the maturity value back to dollars at an exchange rate of 0.66 by entering into short futures contracts ( 154.6)( 0.66) = 10.05 Pay back the USD loan of 101.34 and keep the arbitrage profit of 0.71 JAM July 007 Section 1 Page 15 of 135

7. 449e < F < 450e 0.55 0.06 0 If F0 450 0.06 > e, then exploit arbitrage by Borrowing $450 at 6% Buying one ounce of gold for $450 Locking in sales price of F 0 If F0 449 0.055 < e, then exploit arbitrage by Selling gold short for $449 Investing the $449 at 5.5% Locking in purchase price of F 0 to cover short position JAM July 007 Section 1 Page 16 of 135

Options, Futures, and Other Derivatives (Sixth Edition) by Hull Chapter 7: Swaps 7.1 Mechanics of Interest Rate Swaps Swaps have grown quickly since they were introduced in the early 1980s A plain vanilla (fixed for floating) is the most common type of swap LIBOR The floating rate in most swaps is the London Interbank Offer Rate (LIBOR) It represents the rate banks will deposit with other banks in the Eurocurrency market Illustration In a plain vanilla swap, one party is the fixed-rate payer while the other party is the floating-rate payer The notional amount, index, settlement frequency, and term (maturity) must be specified in advance The floating payments are made at the end of each period based on the floating rate at the beginning of the period This means the first exchange of payments is known The cash flows are calculated for both the fixed and floating counterparties; the net amount is exchanged The principal (notional) amount is not exchanged at the beginning and end of the swap because it would be the same amount for both parties (a wash transaction) From the fixed-rate-payer perspective, the swap is analogous to a long position in a floating-rate bond and a short position in a fixed-rate bond Using the Swap to Transform a Liability A company could convert a floating-rate loan to a fixed rate loan by entering a swap as the fixed-rate payer Assume a company has borrowed $100 million at a floating rate of LIBOR plus 50 basis points The current swap rate is pay fixed 5% and receive LIBOR The company could convert the floating rate loan by entering into a swap with a $100 million notional amount Net Payment = (LIBOR + 0.5%) + (5% LIBOR) = 5.5% JAM July 007 Section 1 Page 17 of 135

Using the Swap to Transform an Asset A company could convert a fixed rate asset to a floating rate asset by entering into a swap as the fixed-rate payer Assume a company has a $100 million investment that earns 6% The current swap rate is pay fixed 5% and receive LIBOR The company could convert the fixed rate asset by entering into a swap with a $100 million notional amount Net Receipt = (6%) + (LIBOR 5%) = LIBOR + 1% Role of Financial Intermediary A financial institution such as a bank will normally enter into offsetting swap (e.g. be the fixed payer in one swap and the floating payer in another swap) Usually the financial intermediary will earn 3 to 4 basis points in the offsetting transactions This spread will cover default risk and profits Market Makers Many large financial institutions will enter into swaps without directly offsetting transactions They hedge their exposure in aggregate using securities such as bonds, forwards, and interest rate futures The swap rate is the average of the bid and offer fixed rates 7. Day Count Issues LIBOR-based floating-rate cash flows are calculated on an actual/360 basis n ( L)( R)( ) Payment =, where L is the principal (notional amount), R is the relevant 360 LIBOR rate, and n is the number of days since the last payment date The fixed rate is quoted on an actual/365 or 30/360 basis Day count issues are ignored in this chapter 7.3 Confirmations A confirmation is the legal agreement underlying a swap Some standardized master agreements are produced by the International Swaps and Derivatives Association (ISDA) JAM July 007 Section 1 Page 18 of 135

7.4 The Comparative-Advantage Argument If a company has a comparative advantage to borrow at a fixed rate compared to a floating rate but desires a floating rate note, a swap could be used to convert the fixed rate loan to a floating rate loan Illustration Assume the following borrowing rates are available for two companies, CorpA and CorpB Company Fixed Rate Floating Rate CorpA 4.0% LIBOR + 0.3% CorpB 5.4% LIBOR + 0.8% CorpA has a comparative advantage for fixed rate loans and CorpB has a comparative advantage for floating rate loans If CorpA wants to borrow at a floating rate and CorpB wants to borrow at a fixed rate, the two companies could engage in a swap to reduce the cost for both CorpA CorpB Borrowing Rate -4.0% -(LIBOR + 0.8%) Pay on Swap -LIBOR -4.15% Receive on Swap 4.15% LIBOR Net Rate -(LIBOR 0.15%) -4.95% In the table above a negative amount is used to designate and amount paid and a positive amount is used for receipts Notice that each company saved 0.45% (or 45 basis points) by using a swap CorpA borrowed at LIBOR 0.15% instead of LIBOR + 0.3% CorpB borrowed at 4.95% instead of 5.4% The total savings is always a b is the difference between the floating rates, where a is the difference between the fixed rates and b In this example a = 1.4% and b = 0.5%, so the total savings is 0.9% Normally the total savings will be shared equally by the two companies If a financial intermediary is used for the swap, some of the savings (~4 basis points) will go to it Criticism of the Comparative-Advantage Argument The differential could exist because floating rate lenders have an option to review the creditworthiness of the borrower at each reset date; fixed-rate lenders do not have that luxury and thus must charge a greater spread In the example above, the net fixed rate of 4.95% for CorpB is only valid if CorpB can maintain the 0.8% spread over LIBOR on the floating rate loan JAM July 007 Section 1 Page 19 of 135

7.5 The Nature of Swap Rates The swap rate is the average of the fixed rate the market maker will pay (its bid rate) and the fixed rate the market maker will receive (its offer rate) LIBOR rates are close to risk-free rates 7.6 Determining LIBOR/Swap Zero Rates Eurodollar futures are used to produce the LIBOR zero curve out to years Swap rates are used to extend the LIBOR zero curve past years Since the present value of the floating rate bond embedded in a swap must initially be B = B zero, the present value of the embedded fixed rate bond must also be zero ( fix fl ) The bootstrap method is used to calculate the LIBOR/swap zero curve from the swap rates See problem #7.18 for an example 7.7 Valuation of Interest Rate Swaps An interest rate swap is worth zero initially; over its life it can have positive and negative values Valuation in Terms of Bond Prices The floating-rate payer has a long position in a fixed-rate bond and a short position in a floating-rate bond From their perspective, the value of the swap is Vswap = Bfix Bfl The value of the swap is the opposite for the fixed-rate payer The value of the fixed-rate bond is simply the present value of the future payments The floating-rate bond will have a value equal to par (L) right after the next payment (at time t*) of k* This means the value of the floating-rate bond today is ( ) ** LIBOR/swap zero rate for a maturity of t* L+ k* e r t, where r* is the JAM July 007 Section 1 Page 0 of 135

Example A financial institution is paying 6-month LIBOR and receiving 8% fixed (with semiannual compounding) Notional = L = $100 Remaining Life = 1.5 years LIBOR/swap zero rates (continuously compounded) 3-month 9-month 15-month r* 10.0% 10.5% 11.0% The 6-month LIBOR rate on the last payment date was 10.% t* = 0.5 because there are just three months before the next payment date k * = ( 0.5)( 0.10 )( $100 ) = $5.10 fix fl It is multiplied by 0.5 because payments are made twice a year ( ) ( ) ( )( ) ( ) ( 0.5)( 0.10) ( )( ) ( ) ( )( ) B = e + e + e = B 0.5 0.10 0.75 0.105 1.5 0.11 4 4 104 98.4 = 5.10+ 100 e = 10.51 Right before the next coupon date the value of the floating rate bond is the par value plus the coupon payment at that date Vswap = Bfix Bfl = 4.7 JAM July 007 Section 1 Page 1 of 135

Valuation in Terms of FRAs The future floating rate payments in the swap could be calculated by assuming the forward rates will be realized; the net payments could then be discounted at the LIBOR/swap zero rates Example Use the same given information as in the previous example Must first calculate the forward rates from the given zero rates ( 0.5)( 0.10) ( 0.5)( ) ( 0.75)( 0.105) R cont e e = e, where R cont is the continuously compounded 6-month forward LIBOR rate in three months R cont = 10.75% Rsemi ( 0.5) R Convert to a semiannual basis using 1+ = e cont Rsemi = 11.04% ( 0.75)( 0.105) ( 0.5)( ) ( 1.5)( 0.11) R cont e e = e, where R cont is the continuously compounded 6-month forward LIBOR rate in nine months R cont = 11.75% Rsemi ( 0.5) R Convert to a semiannual basis using 1+ = e cont Rsemi = 1.10% The value of the swap is the present value of the net payments (pay floating, receive fixed) ( ) ( ) ( )( ) ( ) ( )( ) ( ) Vswap = 100 4% e + 4% e + 4% e V = 4.6 swap ( )( ) 10.% 0.5.010 11.04% 0.75.0105 1.10% 1.5.011 If the term structure of interest rates is rising, the forward rates will increase over time which for the floating rate receiver will make the early net expected cash flows negative and the later net expected cash flows positive 7.8 Currency Swaps Principal and interest payments in one currency are exchanged for principal and interest payments in another The principal amounts must be exchanged at the beginning and end of the swap The principal amount exchanged at the end is exactly opposite of the initial exchange; it is not based on the exchange rates at that time Currency swaps can be used to transform liabilities or assets to other currencies JAM July 007 Section 1 Page of 135

Comparative Advantage This is very similar to the analogous section on interest rate swaps Assume the following borrowing rates are available for two companies, CorpA and CorpB Company US Dollars (USD) Australian Dollars (AUD) CorpA 5.0% 1.6% CorpB 7.0% 13.0% CorpA has a comparative advantage in USD and CorpB has a comparative advantage in AUD If CorpA wants to borrow in AUD and CorpB wants to borrow in USD, the two companies could engage in a swap (probably using a financial institution) to reduce the cost for both The total savings potential is.0% 0.4% = 1.6% Assuming the financial institution will make 0 basis points, the two companies should each save 70 basis points CorpA Financial Intermediary CorpB Borrowing Rate -5.0% USD -13.0% AUD Pay on Swap -11.9% AUD -5.0% USD Receive on Swap 5.0% USD 11.9% AUD Pay on Swap -13.0% AUD -6.3% USD Receive on Swap 6.3% USD 13.0% AUD Net Rate -11.9% AUD 1.3% USD 1.1% AUD -6.3% USD The financial intermediary makes its 0 basis points spread, albeit with some currency risk; this exposure could be eliminated with forward contracts JAM July 007 Section 1 Page 3 of 135

7.9 Valuation of Currency Swaps Valuation in Terms of Bond Prices Vswap = BD SB 0 F if dollars are received and Vswap = SB 0 F BD if dollars are paid B F is the value of the foreign bond measured in foreign currency and S 0 is the spot exchange rate (in dollars per unit of foreign currency) Assume the term structure of LIBOR/swap interest rates is flat in both Japan (4%) and the United States (9%), continuous compounding A financial institution receives 5% per annum in yen and pays 8% per annum in dollars The principal amounts are $10 million and 100 yen The swap expires in 3 years and the current exchange rate is 110 yen = $1 D F 0 ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) B = e + e + e = 0.09 1 0.09 0.09 3 0.8 0.8 10.8 9.64 ( )( ) B = e + e + e = S 0.05 1 0.05 0.05 3 60 60 160 130.55 $1 = = 0.00909dollars per yen 110 yen V = SB B = 1.55 million swap 0 F D Valuation as Portfolio of Forward Contracts The future net payments in the swap could be converted to one currency by assuming the forward exchange rates will be realized; the net payments could then be discounted at the LIBOR/swap zero rates in that currency Calculate the 1-year, -year, and 3-year forward exchange rates ( ) ( 0.09 )( 1 ) ( 0.04 )( 1 ) ( ) ( ) ( 0.09 )( ) ( 0.04 )( ) ( ) ( ) ( 0.09 )( 3 ) ( 0.04 )( 3 ) ( ) 0.00909 e = e F F = 0.00956 1 1 0.00909 e = e F F = 0.01005 0.00909 e = e F F = 0.01056 3 3 The equations above show it is equivalent to 1. Convert one unit of foreign currency to dollars and invest at the risk-free rate in the United States; and. Invest one unit of foreign currency at the risk-free rate in the foreign country and lock in an exchange to dollars via a forward contract ( ) ( ) ( )( ) ( 0.09)( 3) ( ) V e e 160 0.01056 10.8 e = 1.54 million ( )( ) 0.09 1 0.09 swap = 60 0.00956 0.8 + 60 0.01005 0.8 + For the payer of the high-interest currency (as in this example), the value of the swap will tend to be positive during most of its life JAM July 007 Section 1 Page 4 of 135

7.10 Credit Risk A financial institution has credit risk in a swap only when the value of the swap is positive from the financial institution s perspective For this reason potential losses from swaps are less than potential losses from loans of the same principal 7.11 Other Types of Swaps Variations on the Standard Interest Rate Swap Payment frequencies (tenors) of 1 month, 3 months, and 1 months trade regularly Other floating rates besides LIBOR are used Floating-for-floating interest rate swaps are available Amortizing swaps have declining principal balances while step-up swaps have increasing principal balances Forward swaps have delayed start dates A constant maturity swap (CMS) exchanges LIBOR for a swap rate A constant maturity Treasury swap (CMT) exchanges LIBOR for a particular Treasury rate A compounding swap accumulates the payment to the end A LIBOR-in-arrears swap uses the LIBOR rate at the payment date to determine the floating payment; regular swaps use LIBOR rate at the prior payment date Other Currency Swaps A fixed-for-floating currency swap exchanges a floating rate in one currency for a fixed rate in another currency; this is called a cross-currency interest rate swap A floating-for-floating currency swap is also possible A diff swap (or quanto) applies a rate observed in one currency to a principal amount in another currency An equity swap exchanges the total return on an equity index for a fixed or floating rate Options could be embedded in swaps (e.g. extendable, puttable) Commodity swaps are a series of forward contracts on a commodity Volatility swaps have payments based on historical volatility JAM July 007 Section 1 Page 5 of 135

Recommended Questions and Problems You can certainly work all the practice problems, but the following are most beneficial for exam preparation. 1,, 3, 5, 9, 10, 11, 18, 3 Solutions to Recommended Questions and Problems 1. Total Gain = (13.4% 1.0%) (0.6% 0.1%) = 0.9% Financial Intermediary Margin = 0.1% Company A Savings = Company B Savings = 0.4% Company A Financial Intermediary Company B Borrowing Rate -1.0% -(LIBOR + 0.6%) Pay on Swap -LIBOR -1.3% Receive on Swap 1.3% LIBOR Pay on Swap -LIBOR -1.4% Receive on Swap 1.4% LIBOR NET RATE -(LIBOR 0.3%) 0.1% -13.0%. Total Gain = (6.5% 5.0%) (10.0% 9.6%) = 1.1% Financial Intermediary Margin = 0.5% Company X Savings = Company Y Savings = 0.3% Company X Financial Intermediary Company Y Borrowing Rate -5.0% Yen -10.0% Dollars Pay on Swap -9.3% Dollars -5.0% Yen Receive on Swap 5.0% Yen 9.3% Dollars Pay on Swap -10.0% Dollars -6.% Yen Receive on Swap 6.% Yen 10.0% Dollars NET RATE 9.3% Dollars 1.% Yen 0.7% Dollars -6.% Yen 3. Floating Payer Value = Bfix Bfl 4 10 10 ( 0.1) ( 0.1) ( 0.1) 1% 1 1 1 Bfix = ( $100 )( ) e + e + ( $100 ) e = $103.33 ( )( 9.6% ) 4 ( 0.1) 1 Bfl = $100 + $100 e = $101.36 V = B B = $1.97 swap fix fl JAM July 007 Section 1 Page 6 of 135

5. Coupon ( )( ) Coupon ( )( ) B B D F 0 D = $30 10% = $3 = 0 14% =.8pounds 3 15 1 1 ( )( ) ( )( ) = $3 1.08 + $33 1.08 = $3.9 million 3 15 1 1 ( )( ) ( )( ) =.8 1.11 +.8 1.11 =.74 million pounds S=1.6500 Sterling Payer Value = B SB = $4.60 million Dollar Payer Value = SB B = $4.60 million 9. Total Gain = (8.8% 8.0%) (0) = 0.8% Financial Intermediary Margin = 0.% 0 D F Company X Savings = Company Y Savings = 0.3% 0 D F Company X Financial Intermediary Company Y Borrowing Rate -8.0% -LIBOR Pay on Swap -LIBOR -8.3% Receive on Swap 8.3% LIBOR Pay on Swap -LIBOR -8.5% Receive on Swap 8.5% LIBOR NET RATE -(LIBOR 0.3%) 0.% 8.5% 10% 10. ( )( ) Coupon = $10 = $0.50 B B fix fl fix 4 0.08 t 0.08 4 = ( $0.50) ( 1 + ) + ( $10)( 1 + ) = $10.86 t= 0 = $10 + $10 = $10.45 9% ( )( ) F This is the value of the floating rate bond right before the coupon payment at the end of year 3 Vswap = Bfix Bfl = $0.41million 11. Coupon ( )( ) Coupon ( )( ) B B S D F 0 D = $7 8% = $0.56 million = 10 3% = 0.30million francs 4 t 4 = ( $0.56) ( 1.08 ) + ( $7)( 1.08 ) = $7.56 t= 0 4 t 4 = ( 0.30) ( 1.03) + ( 10)( 1.03) = 10.30 francs t = 0 = 0.80 dollars per franc Dollar Payer Value = SB 0 F BD = $0.68 million, which is the default cost for the financial institution F JAM July 007 Section 1 Page 7 of 135

18. Use the -year,.5-year, and 3-year swap rates to solve for the LIBOR zero rates A bond paying coupons equal to the semiannual swap rates will have a par value = e + e + e + e 0.05 0.5 0.05 1.0 0.05 1.5 100.7.7.7.7 R R.0 0.05 0.5 0.05 1.0 0.05 1.5 0.0534.0 100.75.75.75.75 10.75 R.5 = 5.34% = e + e + e + e + e = 5.44% R.5.5 0.05 0.5 0.05 1.0 0.0 3.0 = + + 51.5 0.0534.8.0.8 0.0544.5 R + e + e + 10.8e 3.0 100.8e.8e.8e R 3.0 = 5.544% 3. Financial Intermediary Margin = 0.5% Financial Intermediary Margin = 0.5% Company X Savings = Company Y Savings Company X Financial Intermediary Company Y Borrowing Rate -1.0% pounds -10.5% Dollars Pay on Swap -10.75% Dollars -1.0% pounds Receive on Swap 1.0% pounds 10.75% Dollars Pay on Swap -10.5% Dollars -1.5% pounds Receive on Swap 1.5% pounds 10.5% Dollars NET RATE -10.75% Dollars 0.5% Dollars + 0.5% pounds -1.5% pounds This maintains the 1.5% difference between Company X and Company Y JAM July 007 Section 1 Page 8 of 135

Options, Futures, and Other Derivatives (Sixth Edition) by Hull Chapter 8: Mechanics of Options Markets (Sections 8.3 to 8.13 only) 8.3 Underlying Assets Stock Options Most trading is on the exchanges (e.g. Chicago Board Options Exchange, AMEX) Each contract is for the right to buy or sell 100 shares Foreign Currency Options Most are traded over-the-counter Both European and American options are offered The size of one contract depends on the currency Index Options Most popular ones are on the S&P 500, Nasdaq, and Dow Jones; all trade on the Chicago Board Options Exchange (CBOE) Most are European (the S&P 100 is American) One contract usually represents the right to buy or sell 100 times the index Futures Options A futures option normally matures just before the delivery period in the futures contract 8.4 Specification of Stock Options Expiration Dates The month of expiration is used to identify the option Options expire at the end of the third Friday of the month LEAPS (long-term equity anticipation securities) have expirations up to 3 years Strike Prices Strike prices are spaced $.50, $5.00, or $10.00 apart, depending on the stock price When new expirations are introduced, the exchange usually picks the two or three closest strike prices Other strike prices are offered as the stock price moves JAM July 007 Section 1 Page 9 of 135

Terminology Options of the same type (calls or puts) on the same asset are referred to as an option class An option series includes all the options in a class with the same expiration date and strike price Options can be in the money, at the money, or out of the money The intrinsic value is the value if the option were exercised immediately FLEX Options Traders on the exchange floor agree to nonstandard terms Competing with the over-the-counter markets Dividends and Stock Splits Early OTC options were adjusted for cash dividends by reducing the strike price; exchange-traded options are not usually adjusted for cash dividends Exchange traded options are adjusted for stock splits For example, if a company makes a 3-for-1 stock split the strike price will be divided by 3 and the number of shares to buy or sell will be multiplied by 3 If a company pays a 0% stock dividend the strike price will be divided by 1. and the number of shares to buy or sell will be multiplied by 1. Position Limits and Exercise Limits The Chicago Board Options Exchange (CBOE) specifies position limits for option contracts This specifies the maximum number of option contracts on one side of the market The exercise limit usually equals the position limit 8.5 Newspaper Quotes Many newspapers (such as the Wall Street Journal) carry option prices Information such as the strike, maturity, volume, and price are readily available JAM July 007 Section 1 Page 30 of 135

8.6 Trading Many exchanges are fully electronic, so traders do not have to be at the same physical location Most exchanges use market makers to quote a bid (buy) and offer (sell) price on the option to promote liquidity The offer price exceeds the bid price, which is how the market maker is compensated The exchange sets upper limits on the bid-offer spread An investor can close out a position by issuing an offsetting order The open interest will increase by one contract when an option is traded if neither investor is closing an existing position The open interest will decrease by one contract when an option is traded if both investors are closing existing positions 8.7 Commissions Discount brokers generally charge less than full-service brokers The commission is usually a fixed amount plus a percentage of the dollar trade The commission upon option exercise is the same as for buying or selling the stock Retail investors usually save commission by selling options rather than exercising them The bid-offer spread is a hidden cost 8.8 Margins In the United States, an investor can typically borrow up to 50% of the share value when purchasing stocks Margin purchases are not allowed for options with less than 9 months to maturity because there is already substantial leverage Investors can borrow up to 5% for options with maturities greater than 9 months JAM July 007 Section 1 Page 31 of 135

Writing Naked Options A naked option is not offset by a position in the underlying stock Initial margins required by the CBOE for written naked call options is the greater of 1. Sale proceeds plus 0% of underlying share price less amount option is out of the money (if any). Sale proceeds plus 10% of underlying share price Initial margins required by the CBOE for written naked put options is the greater of 1. Sale proceeds plus 0% of underlying share price less amount option is out of the money (if any). Sale proceeds plus 10% of exercise share price The 0% above is reduced to 15% if the underlying security is a broad index Other Rules The CBOE has special rules for margin requirements on trading strategies (e.g. covered calls, protective puts) 8.9 The Options Clearing Corporation (OCC) The OCC guarantees option writer performance and keeps a record of all positions Members must maintain a required capital amount The option writer maintains a margin account with the broker, who maintains a margin account with the OCC member Exercising an Option When an investor wants to exercise an option, he notifies the broker who notifies the OCC member The OCC randomly selects a member with a short position, who picks a particular investor The chosen investor must sell if a call is exercised and buy if a put is exercised Many exchanges require in-the-money options to be exercised at maturity 8.10 Regulation The exchange and the OCC govern traders Federal and state regulators are also involved Self-regulation is prominent also, and no major defaults of OCC members have occurred JAM July 007 Section 1 Page 3 of 135

8.11 Taxation Taxation for options is complicated Except for professional traders, the US taxes stock options as capital gains or losses The gain or loss is recognized when the option is sold or expires worthless If an option is exercised the gain is rolled into the position Wash Sale Rule If a security is repurchased within 30 days of the sale, any loss on the sale is not deductible This also applies if an option is purchased to later acquire the stock Constructive Sales Prior to 1997 short sales could be used to offset long positions without incurring a gain Now appreciated property is deemed constructively sold if one of the following occurs 1. Enters into short sale. Enters into future or forward 3. Enters into position that eliminates most of the risk (loss and gain) 8.1 Warrants, Executive Stock Options, and Convertibles The exercise of most options does not affect the company of the underlying stock An exercise of a warrant, executive stock option, or convertible bond leads to a new share being issued 8.13 Over-the-Counter (OTC) Markets Options in the OTC market are very flexible, but the option writer may default Recommended Questions and Problems You can certainly work all the practice problems, but the following are most beneficial for exam preparation. 1,, 11, 1, 17,, 3 JAM July 007 Section 1 Page 33 of 135

Solutions to Recommended Questions and Problems 1. ( ) Profit = 40 3 S + T This will be positive if S T < 37 The option will be exercised if S T < 40, ignoring transaction costs. Profit = 4 ( 50) S T This will be positive if S T < 54 + The option will be exercised if S T > 50, ignoring transaction costs 11. A table can be used to show the two are equivalent ST S Long Forward Long Put K = F 0 Portfolio Long Call K = F 0 S F F S 0 0 < F0 T 0 0 T T > F0 ST F0 0 ST F0 ST F0 The above table shows a Long Call = Long Forward + Long Put Since the forward contract has a zero initial value, the Long Call = Long Put 1. A table can be used to show the trader s profit Long Call K = 45 Long Put K = 40 Portfolio S T < 40-3 ( 40 S T ) 4 33 ST 40< S T < 45-3 -4-7 45 S 45 3-4 S 5 S T > ( T ) T Positive if S < 33 or S > 5 T T 17. a) Divide strike price by 1.1 and multiply number of shares by 1.1 Strike = $36.36 Shares = 550 b) There is no effect for a cash dividend c) Divide strike price by 4 and multiply number of shares by 4 Strike = $10 Shares = 000 JAM July 007 Section 1 Page 34 of 135

. The initial margin is the greater of the following two calculations ( )( ) ( )( ) ( ) ( 5)( 100 ) $3.50+ ( 10% )( $57 ) = $4600 5 100 $3.50+ 0% $57 $60 $57 = $5950 Initial Margin = $5950 3. Illustrate with the following tables 100 Shares 100 Short Calls K = 50 S < 30 ( 100) S T (100)($5) 100 Long Puts K = 30 100 $30 ( )( S T ) T ( 100 )( $7) < < ( 100) S T (100)($5) -(100)($7) ( ) ( 100 )( $5) S T > 50 ( 100) S T ( 100 )( $50) 30 S T 50 S T Portfolio $800 100 S $00 -(100)($7) $4800 T 100 Shares 00 Short Calls K = 50 S < 30 ( 100) S T (00)($5) 00 Long Puts K = 30 00 $30 Portfolio ( )( S T ) T ( ) ( 00 )( $7) < < ( 100) S T (00)($5) -(00)($7) ( ) ST ( 00 )( $5) S T > 50 ( 100) S T -(00)($7) ( ) ( 00 )( $50) 30 S T 50 S T $5600 100 ST 100 $400 $9600 100 ST JAM July 007 Section 1 Page 35 of 135

Options, Futures, and Other Derivatives (Sixth Edition) by Hull Chapter 1: Wiener Processes and Itô s Lemma 1.1 The Markov Property A Markov process is a type of stochastic process in which only the present value of a variable is relevant for predicting the future (i.e. future prices do not depend on past prices) Stock prices are assumed to follow a Markov process This is consistent with the weak form of the Efficient Market Hypothesis 1. Continuous-Time Stochastic Processes Continuous-time stochastic processes can have changes at any time; discrete-time stochastic processes con only change at certain fixed points in time Assume a variable follows a Markov stochastic process with its price change over one year normally distributed with a mean of zero and standard deviation of 1 φ ( 0,1) Over a short time period (Δt), the price change distribution is φ ( 0, t ) Wiener Processes A Wiener process is a Markov stochastic process with a mean change of zero and variance rate of 1.0 per year (just like the example above) It is sometimes referred to as Brownian motion (from physics) There are two key properties of a Wiener process 1. z ε t =, where ε is normally distributed φ ( 0,1). The values of Δz for any two different time intervals are independent This implies it is a Markov process A change over a longer period of time, T, is normally distributed with a mean of 0 and standard deviation of T [ zt ( ) z(0) ] φ( 0, T ) dz has the properties of Δz as Δt approaches zero JAM July 007 Section 1 Page 36 of 135

Generalized Wiener Process The drift rate is the mean change per unit time and the variance rate is the variance per unit time A variance rate of 1.0 means the variance equals T over a time interval length of T In continuous time, dx = adt + bdz In discrete time, x = a t + bε t Δx is normally distributed with a mean of aδt and standard deviation of b The drift rate is a and the variance rate is b t Itô s Process This is a Wiener process in which the parameters a and b are functions of the value of the underlying variable x and time t In continuous time, dx = axtdt (,) + bxtdz (,) In discrete time, x = axt (,) t+ bxt (,) ε t 1.3 The Process for a Stock Price The expected percentage return on a stock price is constant rather than the expected drift rate ds S = µ dt+ σdz where µ is the expected rate of return; and σ is the volatility of the stock price Discrete-Time Model S S (, ) S = µ t+ σε t φ µ t σ t S Monte Carlo Simulation Can simulate results by generating random samples for ε from φ ( 0,1) JAM July 007 Section 1 Page 37 of 135