11/2/2015 Nattawoot Koowattanatianchai 1 Derivatives Analysis - PowerPoint PPT Presentation

Equity forwards: Stock indices  Example  Asset manager’s possible options:  Option 1: Taking a specific portfolio of stocks to a forward contract dealer and obtaining a forward contract on that portfolio.  Option 2: Obtaining a forward contract on the FTSE 100 that has a better price quote because the dealer can more easily hedge the risk with other transactions. 11/2/2015 Nattawoot Koowattanatianchai 27

Equity forwards: Stock indices  Example  Asset manager’s actions:  Assume that the manager decides to protect £15,000,000 of stocks. She chooses Option 2 and obtains a quote price of £6,000 on a short forward contract covering £15,000,000 from the dealer. The index contract is nearly always cash settled. 11/2/2015 Nattawoot Koowattanatianchai 28

Equity forwards: Stock indices  Example  At expiration, the index is at £5,925.  The index declines by 1.25%. Thus, the manager should receive 0.0125 × £15,000,000 = £187,500 from the dealer.  If the portfolio perfectly matches a FTSE 100 index fund, then it could be viewed that the loss of 1.25% of the portfolio initially worth £15,000,000 (loss of £187,500) is covered by the forward contract.  In reality, the portfolio is not an index fund and such a hedge is not perfect. 11/2/2015 Nattawoot Koowattanatianchai 29

Equity forwards: Dividends  Equity forwards typically have payoffs based only on the price of the equity, value of the portfolio, or level of the index. They do not ordinarily pay off any dividends paid by the component stocks.  An exception is some equity forwards on stock indices are based on total return indices, e.g., S&P 500 Total Return Index.  The variablity of prices is much more important than the variability of dividends. 11/2/2015 Nattawoot Koowattanatianchai 30

Forward contracts on bonds  Definition  Bond forward is a contract calling for the purchase of an individual bond, a bond portfolio, or a bond index at a later date.  Similarities between equity forwards and bond forwards  A bond may pay a coupon, which corresponds somewhat to the dividend that a stock may pay. 11/2/2015 Nattawoot Koowattanatianchai 31

Forward contracts on bonds  Differences between equity forwards and bond forwards  A bond matures. Thus, a forward contract on a bond must expire prior to the bond’s maturity date.  Bonds often have many special features such as calls and convertibility.  A bond carries the risk of default. So, a forward contract written on a bond must contain a provision to recognize how default is defined, what it means for the bond to default, and how default would affect the parties to the contract. 11/2/2015 Nattawoot Koowattanatianchai 32

Forward contracts on bonds  Example  Consider a forward contract on a default-free zero-coupon bond (or a Treasury bill or T-bill) in which one party agrees to buy the T-bill at a later date, prior to the bill’s maturity, at a price agreed on today.  Suppose the underlying is a 180-day T-bill, which is selling at a discount of 4%.  Its price per $1 par will be 1- 0.04(180/360) = $0.98.  The bill will therefore sell for $0.98.  If purchased and held to maturity, it will pay off $1. 11/2/2015 Nattawoot Koowattanatianchai 33

Forward contracts on bonds  Example  Consider a forward contract that calls for delivery of a 90-day T-bill in 60 days.  Suppose the contract sells for $0.9895. This implies a discount rate of 4.2%.  $1 – 0.042(90/360) = $0.9895  T-bills are typically sold at a discount from par value, a procedure called “discount interest”, and the price is quoted in terms of the discount rate.  The use of 360 days is the convention in calculating the discount. 11/2/2015 Nattawoot Koowattanatianchai 34

Forward contracts on bonds  Example  Consider a forward contract on a default-free coupon-bearing bond, or a Treasury bond or T- Bond.  The T-bond pays interest, typically in semiannual installments, and can sell for more (less) than par value if the yield is lower (higher) than the coupon rate.  Prices of T-bonds are typically quoted without the interest that has accrued since the last coupon date. But we shall always work with the full price – that is, the price including accrued interest.  Prices are often quoted by stating the yield. 11/2/2015 Nattawoot Koowattanatianchai 35

Interest rate forward contracts  Eurodollar  A dollar deposited outside the US.  Banks borrow dollars from other banks by issuing Eurodollar time deposits, which are essentially short-term unsecured loans.  The rate on such dollar loans is call the London Interbank Rate.  The lending rate, called the London interbank offered rate (Libor), is more commonly used in derivative contracts. 11/2/2015 Nattawoot Koowattanatianchai 36

Interest rate forward contracts  Libor  Libor is the rate at which London banks lend dollars to other London banks.  Libor is considered to be the best representative rate on a dollar borrowed by a private, i.e., non governmental high-quality borrower. 11/2/2015 Nattawoot Koowattanatianchai 37

Interest rate forward contracts  Example: Eurodollar time deposit  Suppose a London bank such as NatWest needs to borrow $10 million for 30 days. It obtains a quote from the Royal Bank of Scotland for a rate of 5.25%.  30-day Libor is 5.25%.  If NatWest takes the deal, it will owe $10,043,750 in 30 days.  $10,000,000 x [1 + 0.0525(30/360)] = $10,043,750 11/2/2015 Nattawoot Koowattanatianchai 38

Interest rate forward contracts  Example: Eurodollar time deposit  Unlike the T-bill market, the interest is not deducted from the principal. Rather, it is added on to the face value, a procedure appropriately called “add - on interest”.  The rates on Eurodollar time deposits are assembled by a central organization and quoted in financial newspapers. 11/2/2015 Nattawoot Koowattanatianchai 39

Interest rate forward contracts  Example: Other time deposit instruments  Eurosterling trades in Tokyo.  Euroyen trades in London.  A euro-denominated loan  One bank borrows euros from another.  There are two rates on such euro deposits.  EuroLibor is complied in London by the British Bankers Association.  Euribor is complied in Frankfurt and published by the European Central Bank. 11/2/2015 Nattawoot Koowattanatianchai 40

Interest rate forward contracts  Forward rate agreement (FRA)  FRA is a contract in which the underlying is neither a bond nor a Eurodollar or Euribor deposit but simply an interest payment made in dollars, Euribor, or any other currency at a rate appropriate for that currency. 11/2/2015 Nattawoot Koowattanatianchai 41

Interest rate forward contracts  Example: FRA  Consider an FRA expiring in 90 days for which the underlying is 180-day Libor. Suppose the dealer quotes this instrument at a rate of 5.5%. Suppose the end user goes long and the dealer goes short. The contract covers a given notional principal of $10 million.  The end user will benefit if rates increase.  In contrast, the dealer will benefit if rates decrease. 11/2/2015 Nattawoot Koowattanatianchai 42

Interest rate forward contracts  Example: FRA  At expiration in 90 days, the rate on 180-day Libor is 6%.  The 6% interest will be paid 180 days later.  The present value of a Eurodollar time deposit at that point in time would be: 11/2/2015 Nattawoot Koowattanatianchai 43

Interest rate forward contracts  Example: FRA  At expiration in 90 days, the rate on 180-day Libor is 6%.  The end user, the party going long the FRA, receives the following payment from the dealer, which is the party going short: 11/2/2015 Nattawoot Koowattanatianchai 44

Interest rate forward contracts  Example: FRA  At expiration in 90 days, the rate on 180-day Libor is 6%.  The numerator indicates that the contract is paying the difference between the actual rate that exists in the market on the contract expiration date and the agreed- upon rate, adjusted for the fact that the rate applies to a 180-day instrument.  The divisor appears because it is necessary to adjust the FRA payoff to reflect the fact that the rate implies a payment that would occur 180 days later on a standard Eurodollar deposit. 11/2/2015 Nattawoot Koowattanatianchai 45

Interest rate forward contracts  FRA payoff formula (from the perspective of the party going long) 11/2/2015 Nattawoot Koowattanatianchai 46

Interest rate forward contracts  FRA descriptive notation and interpretation Notati ation on Contract ract expire res s in Underl rlyin ying rate 1 × 3 1 month 60-day Libor 1 × 4 1 month 90-day Libor 1 × 7 1 month 180-day Libor 3 × 6 3 moths 90-day Libor 3 × 9 3 months 180-day Libor 6 × 12 6 months 180-day Libor 12 × 18 12 months 180-day Libor 11/2/2015 Nattawoot Koowattanatianchai 47

Currency forward contracts  Definition  A currency forward contract enables one party to lock in a pre-agreed upon exchange rate at which it will sell one currency and buy another currency at a later date.  Use  Currency forwards are widely used by banks and corporations to manage foreign exchange rate risk. 11/2/2015 Nattawoot Koowattanatianchai 48

Currency forward contracts  Example  Suppose Microsoft has a European subsidiary that expects to send it € 12 million in three months. When Microsoft receives the euros, it will then convert them to dollars.  Microsoft is essentially long euros since it will have to sell euros, i.e., Microsoft has euro-nominated assets that exceed in value its euro-nominated liabilities.  Microsoft is essentially short dollars because it will have to buy dollars. 11/2/2015 Nattawoot Koowattanatianchai 49

Currency forward contracts  Example  Microsoft’s actions  Obtaining a quote on a currency forward for € 12 million in three months.  JP Morgan Chase quotes a rate of $0.925.  Under this contract, Microsoft would know it could convert its € 12 million to 12,000,000 × $0.925 = $11,100,000 in three months.  To hedge its position, Microsoft has to go short the forward contract, i.e., goes short the euro and long the dollar. 11/2/2015 Nattawoot Koowattanatianchai 50

Currency forward contracts  Example  At expiration, the spot rate for euros is $0.920.  Delivery: Microsoft is pleased that it is locked in a rate of $0.925. It simply delivers the euros and receives $11,100,000 at an exchange rate of $0.925.  Cash settlement: The dealer pays Microsoft 12,000,000 × ($0.925 - $0.920) = $60,000. Microsoft has to convert the euros to dollars at the current spot exchange rate of $0.920, receiving 12,000,000 × $0.920 = $11,040,000. 11/2/2015 Nattawoot Koowattanatianchai 51

Other types of forward contracts  Examples  Commodity forwards  The underlying asset is oil, a precious metal, various sources of energy (electricity, gas, etc.), or some other commodity.  Forward contracts on whether  The underlying is a measure of the temperature or the amount of disaster damage from hurricanes, earthquakes, or tornados. 11/2/2015 Nattawoot Koowattanatianchai 52

Pricing and valuation  In most markets, price of an asset is believed to always equal its value or the price would quickly converge to the value.  Price of an asset is what it will sell for.  Value of an asset is what it is worth. 11/2/2015 Nattawoot Koowattanatianchai 53

Pricing and valuation  With respect to certain derivatives, however, value and price take on slightly different meanings.  Value is what you can sell something for or what you must pay to acquire something.  Accordingly, valuation is the process of determining the value of an asset or service 11/2/2015 Nattawoot Koowattanatianchai 54

Pricing and valuation  Pricing a forward contract  A forward price is the fixed price or rate at which the transaction scheduled to occur at expiration will take place.  Pricing means to determine the forward price or forward rate that both parties agree on the contract initiation date. 11/2/2015 Nattawoot Koowattanatianchai 55

Pricing and valuation  Valuing a forward contract  Determining the amount of money that one would need to pay or would expect to receive to engage in the transaction.  Alternatively, if one already held a position, valuation would mean to determine the amount of money one would either have to pay or expect to receive in order to get out of the position. 11/2/2015 Nattawoot Koowattanatianchai 56

Pricing and valuation  Notation  S 0 = the price of the underlying asset in the spot market at time 0 (S t at time t, and S T at time T).  F(0,T) = the price of a forward contract initiated at time 0 and expiring at time T.  V 0 (0,T) is the value at time 0 of the forward contract initiated at time 0 and expiring at time T (V t (0,T) is the value at time t, and V T (0,T) is the value at expiration) 0 T t today expiration 11/2/2015 Nattawoot Koowattanatianchai 57

Pricing and valuation  Value at expiration  V T (0,T) = S T – F(0,T)  If this equation does not hold, an arbitrage profit can be easily made.  Example: Suppose F(0,T) = $20 and S T = $23.  V T (0,T) must be $3.  If V T (0,T) > $3, the long would be able to sell the contract to someone for more than $3 – someone would be paying the long more than $3 to obtain the obligation of buying a $23 asset for $20. Obviously, no one would do that. 11/2/2015 Nattawoot Koowattanatianchai 58

Pricing and valuation  Value at expiration  Example: Suppose F(0,T) = $20 and S T = $23.  If V T (0,T) < $3, the long would have to be willing to sell for less than $3 the obligation of buying a $23 asset for $20. Obviously, the long would not do that. 11/2/2015 Nattawoot Koowattanatianchai 59

Pricing and valuation  Value today  Example 1: Consider a contract that expires in one year. Suppose S 0 is $100 and that F(0,T) is $108. Assume that the interest rate is 5%.  We borrow $100 to buy the asset and sell the forward contract for $108. We hold the position until expiration.  We lose $5 in interest on the $100 tied up in the asset.  At expiration, we receive $108 for the asset regardless of S T . This would result in an arbitrage profit of $3.  V 0 (0,1) = $100 - $108/1.05 = -$2.8571 (the short pays $2.8571 to the long). Note that $2.8571 × 1.05 = $3. 11/2/2015 Nattawoot Koowattanatianchai 60

Pricing and valuation  Value today  Example 2: Consider the same situation but now F(0,T) is set to $103.  If the asset were a financial asset, we could short sell the asset for $100.  We invest that $100 at the 5% rate and simultaneously buy a forward contract.  At expiration, we would take delivery of the asset paying $103 and then deliver it to the party from whom we borrowed it (for short sale).  Arbitrage profit = $105 - $103 = $2. 11/2/2015 Nattawoot Koowattanatianchai 61

Pricing and valuation  Value today  Example 2: Consider the same situation but now F(0,T) is set to $103.  If short selling is not permitted, too difficult, or too costly, a person who already owns the asset could sell it at the spot market, invest $100 at 5%, and buy a forward contract at the same time.  At expiration, that person would pay $103 to buy the asset back. This means an arbitrage profit of $2.  V 0 (0,1) = $100 - $103/1.05 = $1.9048 (the long pays $1.9048 to the short).  $1.9048 × 1.05 = $2 11/2/2015 Nattawoot Koowattanatianchai 62

Pricing and valuation  Value today formula  V 0 (0,T) = S 0 – F(0,T)/(1+r) T  r represents the interest rate.  V 0 (0,T) must equal 0, otherwise one party is required to make a payment to the other upfront to eliminate an arbitrage opportunity.  Pricing formula  Since V 0 (0,T) = S 0 – F(0,T)/(1+r) T and no money changes hands at the start with a forward contract, V 0 (0,T) = 0 giving F(0,T) = S 0 (1+r) T . 11/2/2015 Nattawoot Koowattanatianchai 63

Pricing and valuation  Value at a point during the life of the contract (from the perspective of the party going long)  The long will have to pay F(0,T) at T  The long will receive the underlying asset worth S T at T  V t (0,T) = S t – F(0,T)/(1+r) (T-t)  S t represents the present value of the asset’s future value.  F(0,T)/(1+r) (T-t) represents the present value of the payment of F(0,T) to be made at T. 11/2/2015 Nattawoot Koowattanatianchai 64

Pricing and valuation  Advanced problem:  An investor holds title to an asset worth $125.72. To raise money for an unrelated purpose, the investor plans to sell the asset in nine months. The investor is concerned about uncertainty in the price of the asset at that time. The investor learns about the advantages of using forward contracts to manage this risk and enters into such a contract to sell the asset in nine months. The risk- free interest rate is 5.625%. 11/2/2015 Nattawoot Koowattanatianchai 65

Pricing and valuation  Advanced problem:  A. Determine the appropriate price the investor could receive in nine months by means of the forward contract.  Solution: $130.99  B. Suppose the counterparty to the forward contract is willing to engage in such a contract at a forward price of $140. Explain what type of transaction the investor could execute to take advantage of the situation.  Solution: Overpriced contract should be sold. 11/2/2015 Nattawoot Koowattanatianchai 66

Pricing and valuation  Advanced problem:  C. Calculate the rate of return (annualized), and explain why the transaction in B is attractive.  Solution: Annualized ROR = 15.43%  D. Suppose the forward contract is entered into at the price in A. Two months later, the price of the asset is $118.875. The investor would like to evaluate her position with respect to any gain or loss accrued on the forward contract.  Solution: V 2/12 (0,9/12) = -$8.0 11/2/2015 Nattawoot Koowattanatianchai 67

Pricing and valuation  Advanced problem:  E. Determine the value of the forward contract at expiration assuming the contract is entered into at the price in A and S T is $123.50.  Solution: V 9/12 (0,9/12) = -$7.49  F. Explain how the investor did on the overall position of both the asset and the forward contract in terms of the rate of return at expiration.  Solution: The ROR in 9 months is 4.19%. This means that the annualized ROR is 5.625% 11/2/2015 Nattawoot Koowattanatianchai 68

Pricing equity forwards  The effects of dividends must be incorporated into the pricing process.  Given a series of these dividends of D 1 , D 2 , … D n , whose values are known, that occur at times, t 1 , t 2 , … t n , the present value will be computed as: 11/2/2015 Nattawoot Koowattanatianchai 69

Pricing equity forwards  The effects of dividends must be incorporated into the pricing process.  The future value will be computed as: 11/2/2015 Nattawoot Koowattanatianchai 70

Pricing equity forwards  Recall that the forward price is the spot price compounded at the risk-free interest rate.  In the presence of dividends, the general pricing formula is adjusted to:  Holders of long positions in forward contracts do not benefit from dividends in comparison to holders of long positions in the underlying stock.  Alternatively, 11/2/2015 Nattawoot Koowattanatianchai 71

Pricing equity forwards  Example  The risk-free rate is 4%. The forward contract expires in 300 days and is on a stock currently priced at $5, which pays quarterly dividends according to the following schedule: Days to Ex-di dividend dend date Dividend dend ($) 10 0.45 102 0.45 193 0.45 283 0.45 11/2/2015 Nattawoot Koowattanatianchai 72

Pricing equity forwards  Example 11/2/2015 Nattawoot Koowattanatianchai 73

Pricing equity forwards  Continuous compounding  Assume that the stock, portfolio, or index pays dividends continuously at a rate of δ c and the continuously compounded equivalent of the discrete risk-free rate r is r c = ln(1+r). 11/2/2015 Nattawoot Koowattanatianchai 74

Pricing equity forwards  Continuous compounding  Example: Consider a forward contract on France’s CAC 40 Index. The index is at 5475, δ c is 1.5%, and r c is 4.625%. The contract life is 2 years. 11/2/2015 Nattawoot Koowattanatianchai 75

Valuing equity forwards  Recall that the value of a forward contract is the asset price minus the forward price discounted back from the expiration date.  For discrete compounding:  For continuous compounding: 11/2/2015 Nattawoot Koowattanatianchai 76

Valuing equity forwards  Recall that the value of a forward contract is the asset price minus the forward price discounted back from the expiration date.  V 0 (0,T) is set to zero because no cash exchanges hands at the contract initial date.  At expiration, no dividends remain, so the valuation formula reduces to V T (0,T) = S T – F(0,T) 11/2/2015 Nattawoot Koowattanatianchai 77

Valuing equity forwards  Advanced example:  An asset manager anticipates the receipt of funds in 200 days, which he will use to purchase a particular stock. The stock he has in mind is currently selling for $62.50 and will pay a $0.75 dividend in 50 days and another $0.75 dividend in 140 days. Assume that r is 4.2%. The manager decides to commit to a future purchase of the stock by going long a forward contract on the stock. 11/2/2015 Nattawoot Koowattanatianchai 78

Valuing equity forwards  Advanced example:  A. At what price would the manager commit to purchase the stock in 200 days through a forward contract.  Solution: PV(D,0,T) = $1.48 and F(0,T) = $62.41  B. Suppose the manager enters into the contract at price in A. Now 75 days later, the stock price is $55.75. Determine V t (0,T) at this point.  Solution: The second dividend only remains and will be paid in 65 days. PV(D,t,T) = $0.74 and V t (0,T) = -$6.53. 11/2/2015 Nattawoot Koowattanatianchai 79

Pricing and valuing bond forwards  Notation  T = the expiration date of the forward contract  Y = the remaining maturity of the bond on the forward contract expiration  B c = a coupon bond  B t c (T+Y) = the bond price at time t  CI = the coupon interest over a specified period of time 11/2/2015 Nattawoot Koowattanatianchai 80

Pricing and valuing bond forwards  Pricing and valuation formulae 11/2/2015 Nattawoot Koowattanatianchai 81

Pricing and valuing bond forwards  Example  Consider a bond with semiannual coupons. The bond has a current maturity of 583 days and pays the next four semiannual coupons in 37 days, 219 days, 401 days, and 583 days, at which time the principal is repaid. Suppose that the bond price, which includes accrued interest, is $984.45 for a $1,000 par, 4% coupon bond. Assume that:  r = 5.75%  T = 310 days  T+Y = 583 days, implying Y = 273 days 11/2/2015 Nattawoot Koowattanatianchai 82

Pricing and valuing bond forwards  Example  Only the first two coupons occur during the life of the forward contract.  PV(CI,0,T) = $20/1.0575 37/365 + $20/1.0575 219/365 = $39.23  F(0,T) = ($984.45 - $39.23)(1.0575) 310/365 = $991.18  15 days later, the new bond price is $973.14. Let the risk free rate now be 6.75%.  PV(CI,t,T) = $20/1.0675 22/365 + $20/1.0675 204/365 = $39.20  V t (0,T) = ($973.14 - $39.20) - $991.18/(1.0675) 295/365 = -$6.28 11/2/2015 Nattawoot Koowattanatianchai 83

Pricing and valuing FRAs  Notation  h = the day on which the FRA expires  g = an arbitrary day prior to the expiration  h + m = days from today until the maturity date of the Eurodollar instrument on which the FRA rate is based. h  0 m g h today maturity expiration 11/2/2015 Nattawoot Koowattanatianchai 84

Pricing and valuing FRAs  Notation  L i (j) = the rate on a j-day Libor deposit on an arbitrary day i  The bank that borrows $1 on day i for j days will pay back the following amount in j days.  L h (m) is the rate for m-day Libor on day h  FRA(0,h,m) = the rate on an FRA established on day 0, expiring on day h, and based on m-day Libor. 11/2/2015 Nattawoot Koowattanatianchai 85

Pricing and valuing FRAs  For a $1 notional principal, the FRA payoff at expiration is:  This formula is the same as the previous one: 11/2/2015 Nattawoot Koowattanatianchai 86

Pricing and valuing FRAs  FRA rate is given by the following formula:  The numerator is the future value of a Eurodollar deposit of h+m days.  The denominator is the future value of a shorter- term Eurodollar deposit of h days.  Multiplying by (360/m) annualizes the rate. 11/2/2015 Nattawoot Koowattanatianchai 87

Pricing and valuing FRAs  The value of an FRA during its life  V g (0,h,m) = the value of an FRA on day g, which was established on day 0, expires on day h, and is based on m-day Libor  The first term is the present value of $1 received at day h.  The second term is the present value of 1 plus the FRA rate to be received on day h+m. 11/2/2015 Nattawoot Koowattanatianchai 88

Pricing and valuing FRAs  Example  Consider a 3 × 9 FRA. This instrument expires in 90 days and is based on 180-day Libor. Thus, the Eurodollar deposit on which the underlying rate is based begins in 90 days and matures in 270 days.  h = 90  m = 180  h+m = 270 11/2/2015 Nattawoot Koowattanatianchai 89

Pricing and valuing FRAs  Example  Let the current rate be:  L 0 (h) = 90-day Libor = 5.6%  L 0 (h+m) = 270-day Libor = 6%  Determine the rate that will be quoted on the 3 × 9 FRA. 11/2/2015 Nattawoot Koowattanatianchai 90

Pricing and valuing FRAs  Example  Assume that we go long the FRA, and it is 25 days later (g = 25). Interest rates have moved significantly upward to the following:  L g (h-g) = L 25 (65) = 65-day Libor = 5.9%  L g (h+m-g) = L 25 (245) = 245-day Libor = 5.9%  Determine the market value oft the FRA for a $1 notional principal. 11/2/2015 Nattawoot Koowattanatianchai 91

Pricing currency forwards  Exchange rates  We shall always quote exchange rates in terms of units of the domestic currency per unit of the foreign currency.  Example  From the perspective of a US investor, a stock that sells for $50 is quoted in units of the domestic currency per unit (share) of stock.  Likewise, if the euro exchange rate is quoted as $0.90, then the euro sells for $0.90 per unit, which is one euro. 11/2/2015 Nattawoot Koowattanatianchai 92

Pricing currency forwards  Notation  S 0 = current exchange rate (units of the domestic currency per one unit of the foreign currency)  r f = the foreign interest rate  r = the domestic interest rate  F(0,T) = exchange rate in a T- year forward contract 11/2/2015 Nattawoot Koowattanatianchai 93

Pricing currency forwards  Consider the following transactions executed today (time 0), assuming a contract expiration date of T:  Take S 0 /(1+ r f ) T units of the domestic currency and convert it to 1/(1+ r f ) T units of the foreign currency.  Sell a forward contract to deliver one unit of the foreign currency at the rate F(0,T). 11/2/2015 Nattawoot Koowattanatianchai 94

Pricing currency forwards  At expiration  The 1/(1+ r f ) T units of foreign currency will accrue interest at the rate r f and grow to one unit of the foreign currency at T.  Deliver one unit of the foreign currency to the holder of the long forward contract, who pays F(0,T) units of the domestic currency.  This amount was known at the start of the transaction. 11/2/2015 Nattawoot Koowattanatianchai 95

Pricing currency forwards  Pricing formula  Since the payment of F(0,T) is riskless. The present value of F(0,T) must equal the initial outlay of S 0 /(1+ r f ) T . Otherwise, an arbitrage opportunity exists. 11/2/2015 Nattawoot Koowattanatianchai 96

Pricing currency forwards  Arbitrage  If F(0,T) is too high  Sell the forward contract at the market rate.  Borrow S 0 /(1+ r f ) T from a domestic bank at the rate r to buy 1/(1+ r f ) T units of the foreign currency.  Hold the position, earning interest on the foreign currency.  At maturity of the forward contract, deliver one unit of the foreign currency and be paid the forward rate.  Return S 0 /(1+ r f ) T (1+ r) T to the local bank. 11/2/2015 Nattawoot Koowattanatianchai 97

Pricing currency forwards  Arbitrage  If F(0,T) is too low  Buy the forward contract at the market rate.  Sell 1/(1+ r f ) T units of the foreign currency, receive S 0 /(1+ r f ) T units of domestic currency and deposit this amount in a domestic bank at the rate r .  Hold the position, earning interest on the domestic currency.  At maturity, buy back 1/(1+ r f ) T units of the foreign currency and pay the forward rate.  Withdraw S 0 /(1+ r f ) T (1+ r) T from the local bank. 11/2/2015 Nattawoot Koowattanatianchai 98

Pricing currency forwards  Continuous compounding  Notation  r fc = ln(1+r f ) = the continuously compounded foreign interest rate  r c = ln(1+r) = the continuously compounded domestic interest rate  Pricing formula 11/2/2015 Nattawoot Koowattanatianchai 99

Pricing currency forwards  Example  Suppose a domestic currency is the US dollar and the foreign currency is the Swiss franc. Let the spot exchange rate be $0.5987, the US interest rate be 5.5%, and the Swiss interest rate be 4.75%. We assume that these interest rates are fixed and will not change over the life of the forward contract. We also assume that these rates are based on annual compounding and are not quoted as Libor-type rates. Thus, we compound using formulas like (1+r) T . 11/2/2015 Nattawoot Koowattanatianchai 100