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Unformatted text preview: 1 INTRODUCTION Options, Futures, and Other Derivatives, 7th Ed., John C. Hull (2008) : Chapter 1‐2,5 Size of OTC and Exchange‐Traded Markets 2
550
500
450
400 Size of
Market
($ trillion) OTC
Exchange 350
300
250
200
150
100
50
0
Jun98 Jun99 Jun00 Jun01 Jun02 Jun03 Jun04 Jun05 Jun06 Jun07 Source: Bank for International Settlements. Chart shows total principal
amounts for OTC market and value of underlying assets for exchange
market Forward Contracts A forward contract is an agreement to buy or sell an asset at a certain date in the future for a certain price. A spot contract is an agreement to buy or sell an asset today. When you assume a long position, you agree to buy the underlying asset You assume a short position when you agree to sell the asset at the same date for the same price. Forward contracts are traded over‐the‐counter Foreign Exchange Quotes for USD/GBP, July 20, 2007 4 Spot Bid
2.0558 Offer
2.0562 1month forward 2.0547 2.0552 3month forward 2.0526 2.0531 6month forward 2.0483 2.0489 Forward Price 5 The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero) The forward price may be different for contracts of different maturities Example 6 On July 20, 2007 the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 2.0489 This obligates the corporation to pay $2,048,900 for £1 million on January 20, 2008 What are the possible outcomes? 7 Profit from a Long Forward Position Profit K Price of Underlying
at Maturity, ST 8 Profit from a Short Forward Position Profit K Price of Underlying
at Maturity, ST Futures Contracts 9 Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract Whereas a forward contract is traded OTC, a futures contract is traded on an exchange Gold: An Arbitrage Opportunity? 10 Suppose that: The spot price of gold is US$900 The 1‐year forward price of gold is US$1,020 The 1‐year US$ interest rate is 5% per annum Is there an arbitrage opportunity? Gold: Another Arbitrage Opportunity? 11 Suppose that: ‐ The spot price of gold is US$900 ‐ The 1‐year forward price of gold is US$900 ‐ The 1‐year US$ interest rate is 5% per annum Is there an arbitrage opportunity? The Forward Price of Gold 12 If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1‐year (domestic currency) risk‐
free rate of interest. In our examples, S = 900, T = 1, and r =0.05 so that F = 900(1+0.05) = 945 Oil: An Arbitrage Opportunity? 13 Suppose that: ‐ The spot price of oil is US$95 ‐ The quoted 1‐year futures price of oil is US$125 ‐ The 1‐year US$ interest rate is 5% per annum ‐ The storage costs of oil are 2% per annum Is there an arbitrage opportunity? Futures Contracts 14 Available on a wide range of assets Exchange traded Specifications need to be defined: What can be delivered, Where it can be delivered, & When it can be delivered Delivery 15 If a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices and Eurodollars) are settled in cash Convergence of Futures to Spot 16 Futures
Price Spot Price
Futures
Price Spot Price Time (a) Time (b) Futures Prices for Gold on Jan 8, 2007: Prices Increase with Maturity 17 Futures Price ($ per oz) 650 640 630 620 610 Contract Maturity Month
600
Jan07 Apr07 Jul07 Oct07 Jan08 Futures Prices for Orange Juice on January 8, 2007: Prices Decrease with Maturity 18 210 Futures Price (cents per lb) 205
200
195
190
185
180
175 Contract Maturity Month
170
Jan07 Mar07 May07 Jul07 Sep07 Nov07 19 Forward Contracts vs. Futures Contracts FORWARDS FUTURES Private contract between 2 parties Exchange traded Nonstandard contract Standard contract Usually 1 specified delivery date
Settled at end of contract
Delivery or final cash
settlement usually occurs
Some credit risk Range of delivery dates
Settled daily
Contract usually closed out
prior to maturity
Virtually no credit risk Basis Risk 20 Basis is the difference between the spot and futures price Basis = spot price of asset to be hedged – futures price of the contract used Basis risk arises because of the uncertainty about the basis when the hedge is closed out Convergence of Futures to Spot (Hedge initiated at time t1 and closed out at time t2) 21 Futures
Price Spot
Price
Time
t1 t2 Short Selling 22 Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends and other benefits the owner of the securities receives Short Selling 23 Consider the position of an investor who shorts 500 IBM shares in April when the price per share is $120 and closes out the position by buying them back in July when the price per share is $ 100. Suppose that a dividend of $1 per share is paid in May. The investor: receives 500 x $120 = $60,000 in April when the short position is initiated. The dividend leads to a payment by the investor of 500 x $1 = $500 in May. The investor also pays 500 x $100 = $50,000 when the position is closed out in July. The net gain is therefore $60,000 ‐ $500 ‐ $50,000 = $9,500 Assumptions 24 1. 2. 3. 4. The market participants are subject to no transactions costs when they trade. The market participants are subject to the same tax rate on all net trading profits. The market participants can borrow money at the same risk‐free rate of interest as they can lend money. The market participants take advantage of arbitrage opportunities as they occur. 25 Notation for Valuing Futures and Forward Contracts S0: Spot price today F0: Futures or forward price today T: Time until delivery date (in years) r: Risk‐free, per annum, interest rate for maturity T, expressed with continuous compounding An Arbitrage Opportunity? 26 Suppose that: The spot price of a non‐dividend paying stock is $40 The 3‐month forward price is $43 The 3‐month US$ interest rate is 5% per annum Is there an arbitrage opportunity? T=0 Borrow for three months Buy one unit of the asset T=1 $ 40 Repay loan (40e0.05x3/12) $ 40.5 ‐40 Short one 3 month forward contract Sell one unit of asset using the forward contract $ 0 43 $ 2.50 Another Arbitrage Opportunity? 27 Suppose that: The spot price of non dividend‐paying stock is $40 The 3‐month forward price is US$39 The 1‐year US$ interest rate is 5% per annum Is there an arbitrage opportunity? T=0 Sell one unit of the asset Lend for 3 months T=1 $ 40 ‐40 Receive repayment of loan Long one 3 month forward contract Buy one unit of the asset using forward contract $ 0 $ 40.5 ‐39 $ 1.50 The Forward Price 28 If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then F0 = S0erT where r is the 1‐year risk‐free rate of interest. In our examples, S0 =40, T=0.25, and r=0.05 so that F0 = 40e0.05×0.25 = 40.50 29 When an Investment Asset Provides a Known Dollar Income Consider a long forward contract to purchase a coupon‐bearing bond whose current price is $900. We will suppose that the forward contract matures in 9 months, that coupon payments of $40 are expected after 4 month. We assume the 4‐month and 9‐month risk‐free interest rates (continuously compounded) are 3% per annum and 4% per annum, respectively. Suppose first that the forward price is relatively high at $910. Is there an arbitrage opportunity? 30 When an Investment Asset Provides a Known Dollar Income T =0 T= 4 months Borrow for 4 months $ Receive coupon 39.60 payment Borrow for 9 months 860.4 Pay back 3 0 month loan Buy one unit of the asset T= 9 months $ 40 Sell asset $ 910 under forward ‐40 Pay back 9 ‐ 886.60 month loan ‐900 Short a 9 month forwards contract 0 0 $ 23. 40 31 When an Investment Asset Provides a Known Dollar Income F = (S – I )erT 0 0 where I is the present value of the income during life of forward contract So in our example, S0 = 900, I = 40e‐0.03x4/12, r = 0.04 and T = 0.75 F0 = ( 900 – 39.60)e0.04x0.75= $ 886.60 When an Investment Asset Provides a Known Yield 32 F0 = S0 e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding) Consider a six‐month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a six‐month period. The risk‐free rate of interest (with continuous compounding) is 10% per annum. The asset price is $25. In this case S0 = 25, r = 0.10, and T = 0.5. The yield is 4.04% per annum with semiannual compounding. This is 3.96% per annum with continuous compounding. It follows that q = 0.0396, so that the forward price F0 is given by: Fo = 25e(10‐0.0396)x0.5 = $25.77 Valuing a Forward Contract 33 Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short forward contract is (K – F0 )e–rT Valuing a Forward Contract 34 Example: A long forward contract on a non‐
dividendpaying stock was entered into some time ago. It currently has six months to maturity. The risk‐free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24. In this case So = 25, r = 0.10, T = 0.5, and K = 24. The six‐month forward price, Fo, is given by: F0 = 25e0.1x0.5 = $26.28 f= (26.28 ‐ 24)e‐0.1x0.5 = $2.17 Valuing a Forward Contract 35 Recall, that F0 = S0erT Then the value of a long forward on a non income asset is: f= S0 – Ke‐rT Similarly, the value of a short forward contract on a non‐ income asset is: f= Ke‐rT – S0 36 Valuing Forwards for a Dividend Paying Stock Similarly, for a dividend or income paying stock, recall that F0 = (S0 – I)erT. Therefore the value of a long forward on an income paying contract is: f= S0 – I –Ke‐rT or S0e‐qT – Ke‐rT for a known yield q And the value of a short forward on an income paying contract is: f= Ke‐rT – S0 + I or Ke‐rT ‐ S0e‐qT for a known yield q Stock Index Futures 37 Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F0 = S0 e(r–q )T where q is the average dividend yield on the portfolio represented by the index during life of contract For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio Index Arbitrage 38 What if F0 > S0e(rq)T When F0 > S0e(rq)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(rq)T an arbitrageur buys futures and shorts or sells the stocks underlying the index Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally simultaneous trades are not possible and the theoretical no‐arbitrage relationship between F0 and S0 does not hold Futures and Forwards on Currencies 39 A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk‐free interest rate It follows that if rf is the foreign risk‐free interest rate ( r rf ) T
F0 S 0 e This is the well known interest‐rate parity relationship Why the Relation Must Be True 40 1000
1000 units of
foreign currency
at time zero 1000 e rf T units of foreign
currency at time T 1000 F0e rf T dollars at time T 1000S0 dollars
at time zero 1000 S0e rT
dollars at time T The Cost of Carry 41 The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F0 = S0ecT For a non dividend paying stock, c=r For a dividend yielding stock, c= r – q For a FX, c= r – rf For a commodity that provides income at rate q but requires cost (e.g. storage) at rate u, c = r + u – q 42 Futures Prices & Expected Future Spot Prices Suppose k is the expected return required by investors on an asset We can invest F0e–r T at the risk‐free rate and enter into a long futures contract so that there is a cash inflow of ST at maturity This shows that F e rT e kT E ( S )
0 T F0 E ( ST )e ( r k )T What happens when asset return is positively correlated with the market? 43 Futures Prices & Future Spot Prices (continued) If the asset has no systematic risk, then k = r and F0 is an unbiased estimate of ST positive systematic risk, then k > r and F0 < E (ST ) negative systematic risk, then k < r and F0 > E (ST ) ...
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This note was uploaded on 04/09/2012 for the course FINN 321 taught by Professor Farahsaid during the Spring '12 term at Alvin CC.
 Spring '12
 FarahSaid

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