Lecture 2. Agenda: Basic descriptions for derivatives. 1. Standard derivatives Forward Futures Options

Transcription

1 Lecture 2 Basic descriptions for derivatives Agenda: 1. Standard derivatives Forward Futures Options 2. Nonstandard derivatives ICON Range forward contract

2 1. Standard derivatives ~ Forward contracts: An agreement to buy or sell an asset at a certain future time for a certain price. Payoff Payoff pattern from Forward contracts: S T : forward price (delivery price) S T : the spot price of the asset at maturity 1

3 Example 1: Arbitrage 01/05/11 07/05/11 Gold: $435/ounce Forward rate: $450 6-month T-bill rate: 2.75% Borrow: $435 at 2.75% for 6 months. Delivery gold at $450 Buy Gold at spot market at $435 Repay $435(1+2.75%) 0.5 Sell Gold forward contracts at $450 Cash flow: $0 Cash inflow: $450 Cash outflow:$ Net profit: $ Gold: $435/ounce Forward rate: $430 6-month T-bill rate: 2.75% Sell Gold at spot market at $435 Buy gold at $430 Lend $435 at 2.75% for 6 months. Receive $435(1+2.75%) 0.5 Buy Gold forward contracts at $430 Cash flow: $0 Cash inflow: $ Cash outflow: $430 Net profit: $ F = S 0 (1+r ) T Forward rate: $435(1+2.75%) 0.5 =$

4 Example 2: Hedging Pay BP 1 mil. 01/05/11 07/05/11 6-month forward: BP/USD: Quotation in FX markets: Direct quotation: Based on $ units. British relevant countries (i.e., the British pound, Australian dollar, New Zealand dollar, Irish pound) and Euro. Futures markets: Example: Indirect quotation: Based on foreign currency units. For FX spot rate: For forward contracts: Example: 3

5 Example 3: Forward rate calculation in practice. Given the following spot FX spot and money market rates, what should be the theoretical 90 day forward FX rate? Spot rate = USD/CAD Canadian 90-day Libor = 4.50% US 90-day Libor = 3.80% Interest Rate Parity 0 T Borrow $1 at 3.8% Repay: $1*(1+3.8%*90/360) Convert into CAD 1.35 and deposit at 4.5% Receive: CAD 1.35*(1+4.5%*90/365) Buy USD/CAD forward at F Convert back to USD at F Cash flow=0 Cash flow = 0 [CAD1.35*(1+4.5%*90/365) ]/F = $1*(1+3.8%*90/360) F = [1.35*(1+4.5%*90/365)] / (1+3.8%*90/360) = T Borrow CAD 1.35 at 4.5% Repay: CAD1.35*(1+4.5%*90/365) Convert into USD 1 and deposit at 3.8% Receive: USD 1*(1+3.8%*90/360) Sell USD/CAD forward at F Convert back to CAD at F Cash flow=0 Cash flow = 0 [CAD1.35*(1+4.5%*90/365) ]= $1*(1+3.8%*90/360) * F F = [1.35*(1+4.5%*90/365)] / (1+3.8%*90/360) = F = S 0 (1+ r d T) / (1+ r f T) (Because the FX is expressed in terms of CAD, r d is the Canadian risk free rate, r f is the US risk free rate) The fair forward FX rate is $ USD/CAD. The fair basis is $0.0021, i.e., 21 basis points. In forward markets, the 21 basis points are said to be the swap points. Suppose the forward rate was USD/CAD: USD is too expensive: Borrow CAD 1.35 at 4.5% for 90 days Convert into $1 and deposit USD at 3.8% Sell USD (buy CAD) forward at 1.36 Suppose the forward rate was USD/CAD: USD is too cheap: Borrow $1 at 3.8% for 90 days Convert into CAD 1.35 and deposit CAD at 4.5% Buy USD (sell CAD) forward at

6 Quoting Conventions The previous example was somewhat oversimplified. There was no bid-ask spread in the spot FX rate and the loan and investment interest rates were assumed equal. In reality, to calculate valid forward prices one needs to have the valid bid-ask prices of the spot rates, and separate loan and investment interest rates. Forward rates can be quoted in two ways, as an "outright" quote, or as forward points (also called a swap rate). The outright quote is simply a bid-ask price same as the spot market quotes. The forward points are the amount that needs to be added to or subtracted from the spot rates. The following illustrates the two quoting methods: Bid Ask Spot rate USD/CAD: Swap rate Outright forward rate Premiums and Discounts As previously discussed, the forward rates are closely related to the spot rates and interest rates of the two countries. A result of the IRP theory is that for the country with the higher interest rate, its currency is weaker in the forward market than in the spot market. As shown in the previous example, the Canadian interest rate was higher than the US interest rate, and the resulting theoretical forward rate was USD/CAD, compared with the spot rate of USD/CAD. The terms premium and discount refer to whether the forward rates are higher or lower than the spot rates. A premium means the forward price is higher than the spot price and a discount means lower. In this case the forward rate is then at a premium of 21 points. If F > Spot: the SWAP points are premiums (if F > Spot: it means r d > r f ) If F < Spot: the SWAP points are discounts (if F > Spot: it means r d < r f ) 5

7 Spot spread and interest rate spread: Given the following information, what should be the quotes for USD/CAD forward rates? Bid Ask Spot rate USD/CAD: Canadian 90-day Libor = 4.25% 4.75% US 90-day Libor = 3.60% 4.00% Bid side: if you want to sell USD forward, you sell at bid side It means that you short (borrow) USD at 4.00% and long (lend) CAD at 4.25%. F= (1+4.25%*90/365) / (1+4.00%*90/360) Ask side: If you want to buy USD forward, you buy at ask side It means that you long (lend) USD at 3.60% and short (borrow) CAD at 4.75%. F= (1+4.75%*90/365) / (1+3.60%*90/360) Given the following information, what should be the quotes for USD/CAD forward rates? Bid Ask Spot rate USD/CAD: Canadian 90-day Libor = 2.25% 2.75% US 90-day Libor = 4.00% 4.40% (You do it.) 6

8 ~ Futures contracts: An agreement between tow parties to buy or sell an asset at a certain time in the future for a certain price. Difference between Futures and Forward contracts: Futures Exchange Standard contracts Range of delivery dates Settled daily (mark to market) Usually squared prior to maturity No credit risk Forward OTC Non-standard contracts 1 specified delivery date Settled on maturity date Delivery or cash settlement Credit risk 7

9 ~ Options: Options are financial contracts that give the buyer the right to purchase or sell a security at a pre-specified price (strike) within a specified period of time. Call (Put) options give buyers the right to buy (sell) a security at a strike price. American options: can be exercised any time before expiration. European options: can be exercised only on expiration date. Asian options: strike price depends on the average of the underlying stock price over certain time interval. ~ Concept of Option Pricing: The expected price of a security equals the weighted average present value of all the potential future payoffs. Example: i = 5% 0 1 year Price? $0 $10 $20 $30 $40 40% 10% 15% 20% 15% Profit Profit S T S T Prob. Prob. E(C) = e -r t S T (S T ) L (S T )ds T E(P) = e -r t 0 ( S T ) L (S T )ds T S T 8

10 Example: Put-Call Parity Assumptions: 1) no frictions, 2) no early exercise, 3) borrow at risk free rate. C P = S t ( 1 r ) Where C and P have the same t, S, and. Actual Theoretical S (stock price) $50.00 (strike price) $50.00 T (time to maturity) 0.5 r (annual interest rate) 6% C (call premium) $4.75 $4.75 P (put premium) $3.00 $3.31 (It means Put premium is too low.) (It means Call premium is too high.) Arbitrage Process: Sell Call $4.75 Buy Put $3 Buy S $50 Borrow $48.56 at 6% for six months Profit (Loss) Stock Price at Option Expiration (ST) $0.00 $50.00 $ Call $4.75 $4.75 ($45.25) Put $47.00 ($3.00) ($3.00) Loan ($1.44) ($1.44) ($1.44) Stock ($50.00) $0.00 $50.00 Total $0.31 $0.31 $0.31 C P = S ( 1 r ) t, C P S + t ( 1 r ) = 0 ( >0, if C is too high or P is too low) Sell Call (+C) Buy stock (-S) Buy Put (-P) Borrow t ( 1 r ) Does Put-Call Parity hold in the real world? Different payoff patterns between options and futures: Options: insurance Futures: neutralize risk 9

11 2. Nonstandard derivatives (Exotic options) The possibilities for designing new interesting nonstandard derivatives are limitless. To satisfy the needs of clients, financial institutions can design various derivatives by combining different derivatives and underlying assets. Example: A bond s interest payment is linked with oil prices. At the bond maturity, the bond issuer pays a par value of $1,000 plus the product of 170 and the excess of the price of a barrel of oil at maturity over $25. The maximum additional amount paid was $2550 (which corresponds to a price of $40 per barrel). Payoff 2 1 S T 10

12 Example: ICON (index currency option notes) The bonds in which the amount received by the holder at maturity varies with a foreign exchange rate: Two exchange rates are specified, 1 and 2, where 1 > 2. If the exchange rate, S T, at the bond s maturity is above 1, the bondholder receives the full face value. If 1 > S T > 2, a portion of the full face value is received. If S T < 2, the bondholder receives nothing. More specifically, the payoff pattern is: S T > 1 $1,000 1 > S T > 2 $1,000-max[0, 1000*( 1 /S T -1 )] S T < 2 $0 Payoff 2 1 S T

13 Example: Range forward contract (flexible forwards) A currency range forward contract has the chosen band between 1.92 and If the spot rate at the maturity is less than 1.92, the buyer pays 1.92; if it is between 1.92 and 1.96, the buyer pays the spot rate; if it is greater than 1.96, the buyer pays S T <Lower bound $S T $1.92 (Loss) Higher bound > S T >lower bound $0 S T >Higher bound $S T $1.96 (Gain) Payoff S T 12