金融学概论讲义(北大光华管理学院)lecture06

Principles of FinanceLecture 06Forward and Futures ContractsThe Nature of Derivatives∙ A derivative is an instrument whose value depends on the values of other more basic underlying variables∙Examples of derivatives-F orward-F utures-O ptions-S waps……Derivative Markets ∙Exchange Traded-Standard products-Trading floor or computer trading-Virtually no credit risk∙Over-the-Counter (OTC)-Non-standard products-Telephone market-Credit riskForward Contracts∙ A forward contract is an agreement made today to buy or sell an asset at a certain time in the future for a certain price (referred to as the forward price or the delivery price)∙The delivery price is usually chosen so that the initial value of the contract is zero; No money changes hands when contract is first negotiated and it is settled at maturity∙An OTC agreement between two parties and both parties are subject to credit risk∙Both parties have the obligation to honor the contractSettlement of Forward Contracts∙Physical: requires delivery of actual assets∙ Cash settled: requires only the exchange of the difference between the delivery price and the prevailing spot price at maturity∙Suppose that:Long 3-month Gold forwardDelivery price $300Spot price at t = 3 months: $320P/L from a Long Forward PositionS, TP/L from a Short Forward PositionSTFutures Contract∙Futures are standardized forward contracts.∙Whereas a forward contract is traded OTC a futures contract is traded on an exchange∙Specifications need to be defined:-The underlying asset-Delivery location-Maturity date and delivery time-Method of settlement∙Most contracts are closed out before maturityFeatures Promoting Liquidity ∙Standardized Contract-Maturity dates-Contract size-Price tick size, i.e. minimum price movement-The underlying asset (especially commodities) ∙Organized exchangesFeatures Reducing Credit Risk∙Daily settlement: Futures contracts are marked to market and settled at the end of every business day∙Margin account: To buy or sell a futures contract, the investor is required to post a specified margin to guarantee contract performance∙Clearinghouse: The clearinghouse does not take a position in any trade but interpose itself between two parties in every transactionMargin Accounts∙ A margin is cash or marketable securities deposited by an investor with his or her broker∙The balance in the margin account is adjusted to reflect daily settlement (profit or loss)∙Initial margin: The amount a trader must deposit into his/her trading account (i.e. margin account) when establishing a futures position∙When the balance in the margin account falls to, or below, a maintenance margin level, the trader receives a margin call and is requested to top up the account to the initial level. The extra funds deposited are known as a variation margin∙If the balance in the margin account exceeds the initial margin level, the trader is entitled to withdraw the excess funds in the accountExample of the Margin Account ∙An investor takes long position in $/£ futures∙Contract size: £62,500∙Initial margin: 1,485$∙Maintenance margin: 1,100$Date SettlementPrice OpeningBalance($)DailyP/L ($)ClosingBalance($)Margincall ($)Cumulative P/L ($)1.6500 1,48501/11 1.6508 1,485 50 1,535 50 02/11 1.6412 1,535 -600 935 550 -550 03/11 1.6384 1,485 -175 1,310 -725 04/11 1.6456 1,310 450 1,760 -275 05/11 1.6492 1,760 225 1,985 -50The Economic Function of Futures MarketsThe futures markets facilitate the re-allocation of exposure to commodity price risk among market participants.By providing a means to hedge the price risk associated with storing a commodity, futures contracts make it possible to separate the decision of whether to physically store a commodity from the decision to have financial exposure to price changes.The Economic Function of Futures MarketsThe existence of the futures market for wheat conveys information to all producers, distributors, and consumers; and this eliminates the necessity for market participants to gather and process information in order to forecast the future spot priceSuppose the commodity is wheat, and next year’s crop is expected to be much higher than average, then futures prices may be lower than the spot, (the spread may be negative,) nobody will store wheat.The Law of One Price and Arbitrage∙In a competitive market, if two assets are equivalent they will tend to have the same price∙The law of one price is enforced by a process called arbitrage∙Arbitrage is the purchasing of a set of assets, and immediate sale of another set of assets, in such a way as to earn a sure profit from price differences∙Arbitrage process brings two equivalent assets to the same price, this is known as market clearing.An Arbitrage Opportunity?∙Shares of General Motors (GM) are listed on both NYSE and LSE ∙The quoted price is £100 in London and $148 in New York∙The current exchange rate is $1.4500/£∙An arbitrage opportunity?Another Arbitrage Opportunity?∙There are two investment portfolios: portfolio A and portfolio B∙The payoffs at maturity are as follows:State 1 State 2Portfolio A $70 $100Portfolio B $70 $100∙The current quoted price of portfolio A is $80 and the current quoted price for portfolio B is $82∙An arbitrage opportunity?Framework for Forward/Future Pricing ∙Future price: price of the future∙Spot price: price of the underlying asset at present∙Future spot price: price of the underlying asset in the futureFramework for Forward/Future Pricing Suppose you have some spare cash, and you want to invest it in gold in a year’s time. There are 2 ways to do it:A.Buy gold at the spot price with your money, store it for a year(which means you incur some storage costs), sell it at the future spot price.B.Enter into a forward/future contract of gold, put your money in abank for a year, buy A at the forward/future price in the end of the year, sell it at the future spot price.Since the two strategies are equivalent, they must provide the same return so that there are no arbitrage opportunities.Framework for Forward/Future PricingDenote S as the spot price of gold, F as the forward/future price of gold, FS as the future spot price of gold, s as the storage cost of gold as a fraction of spot price, r as the risk-free interest rate:Return of A: FS S s S -- Return of B: FS FrS -+FS S FS Fs rS S---=+Therefore: (1)F r s S =++Framework for Forward/Future Pricing∙Forward price must be arbitrage-free∙Suppose that-The spot price of gold is US$300-The 1-year US$ interest rate is 8% per annum with annual compounding-Storage costs 2% of gold.-The forward-spot-price-parity relation implies that the one-year forward price is:+=++F⨯rs=S+08)30033002.0)1(.01(=∙Forward prices above $330 permit arbitrage-Suppose the forward price is $340-At time t = 0- Sell gold forward at $340- Borrow $300 at 8% pa- Purchase gold in the spot market at $300, store for a year (storage costs $6)-At time t = 1 year- Deliver gold and receive $340-Pay back loan with interest ($324)-Pay storage cost: ($6)- Make a profit of $10: 340-324-6=10∙Forward prices below $330 permit arbitrage-Suppose the forward price is $320-At time t = 0-Buy gold forward at $320-Sell short gold in the spot market at $300 (borrow gold and sell it immediately)-Deposit $300 at 8% pa-At time t = 1 year-Accept delivery of gold for $320: ($320)-Return the gold and receive storage cost: $6-Receive deposit with interest of $324-Make a profit of $10: 324+6-320=10Financial FuturesThe underlying asset of a financial future is a financial instrument, e.g. stock, bond, foreign currency, etc.Example: Share A has a spot price of $100 (S=100), the risk-free interest rate is 8% (r=0.08) with annual compounding, what’s the forward price?Forward-spot-price-parity for a share with no dividend with maturity of T years:T1(+=SrF)Financial FuturesThe investor has two equivalent investment strategies:1.buy one share A, hold it for a year, and sell it at the future spotprice of 1S. Cash flow in a year’s time: 1S2.buy a forward/future contract of share A at the price of F, make adeposit in a risk-free asset with future value of F, take the money out after a year, and buy the share at F, sell it in the market at the future spot price of 1S. Cash flow in a year’s time: 1SThe law of one price says that they should have the same price today since they produce the same amount of cash flow in a year’s time!Financial FuturesPrice of strategy 1: the spot price of share A: SPrice of strategy 2: the amount of cash invested into the risk-free asset so as to generate F in a year’s time: F/(1+r)Therefore: S=F/(1+r)Rearranging, we have:F+=S)1(rIf the futures contract matures in T years, it becomes:T=1(+rSF)Financial FuturesWhat if share A pays dividend of D in a year’s time?Again the investor has two equivalent strategies:Strategy 1: buy share A at the spot price S, hold it for one year, receive dividend of D, and sell the share at the future spot price of S. Cash flow in a year’s time: 1S+ D1Strategy 2: buy a forward/future contract at the price of F, make an investment in a risk-free asset with future value of F+D, take the money out after a year, buy the share at F, sell it in the market at the future spot price of 1S. Cash flow in a year’s time: 1S+DFinancial FuturesPrice of strategy 1: the spot price of share A: SPrice of strategy 2: the amount of cash invested into the risk-free assetso as to generate F+D in a year’s time (F+D)/(1+r)Applying the law of one price, we have: S=(F+D)/(1+r) Rearranging, we have the forward-spot-price-parity of a share with dividend payment:F=S(1+r)-DThe Forward Price is not a Forecast of Future Spot Price The forward price is obtained without risk from the current spot and risk free investmentThe spot value at a future date is obtained by investing in the security and accepting (market) risk, and this risk must be rewardedFX Forward RateDefine HC r and FC r as the effective interest rates at home andabroadFCHCr r S F ++=11where F and S are defined as the number of units of HC per unit of FC. For example, suppose £1=$1.6, then F =1.6 and $r r HC =, £r r FC = if you are buying a pound future; F =0.625 and £r r HC =, $r r FC = if you are buying a dollar future.The FX Forward RateSuppose an investor wants to buy a futures contract of pound sterling at F , i.e. he can buy pound at the price of £1=$F in a year’s time. The spot price of pound is S , i.e. £1=$S . So here pound is the foreign currency, dollar is the home currency. The risk-free interest rates are: $r r HC =, £r r FC =. What is the proper price of the future contract?The investor has two equivalent strategies: Stragegy 1:At t=0: Enter into a futures contract of pound with futures price of F . Cash flow: 0At t=1: Buy pound at F . Cash flow: FThe FX Forward RateStrategy 2:At t=0: Borrow )1/(£r S + of US dollars, change into £1/(1)r + of pounds, put it in a bank at the pound interest rate. Cash flow: 0At t=1, take the money out (in pound), pay back the dollar loan. Cashflow: $£11r Sr ++The FX Forward RateSince the two strategies both will give you one pound in a year’s time, the law of one price says that they have the same price, i.e. the amount of investment of these two strategies must be the same:$£11r F Sr +=+More generally:FCHCr r SF ++=11Pricing FX Forward Contract∙ Suppose that:Spot $/£: 1.4222One-year $ interest rate: %00.5 per annum with annual compounding One-year £ interest rate: %00.6 per annum with annual compounding∙ The six-month £ forward rate:415.10296.010247.014222.11111£$=++⨯=++=++=r r S r r S F FC HCAs a rule of thumb, if the foreign currency offers a higher interest rate, the future price of the foreign currency will be lower than the spot price.Corporate Applications: Hedging∙Receive FC payment at a future date ⇒ sells FC forward short∙Boeing has just sold 10 Boeing-747s to British Airways with total price of £200m payable in one year’s time∙Boeing can hedge this cash flow in £ by selling £ forward short∙If the one-year forward rate is $1.60/£, so Boeing will receive $320m no matter how exchange rate $/£ movesCorporate Applications: Hedging∙Make FC payment at a future date ⇒ buys FC forward (long)∙An US company imported some goods from Switzerland and is due to pay SFr100m in six months’ time∙The company can hedge its exposure to SFr by buying SFr forward. The six-month forward rate is SFr1.54/$, so the company is required to pay $64.94mThe Role of Expectations in Determining Exchange Rates Consider a world in which there are two countries, Domestic & Foreign, and conditions are such in each country that the yield curves are flat, with yields of 5% and 10% respectively.Further assume that the exchange rate is 1 todayThe 1-year forward is 1*1.05/1.10=0.9545The Role of Expectations in Determining Exchange Rates If the interest rate in Foreign is higher than in Domestic, one explanation may be that the rate of inflation is higher.Assume no taxes, and the interest rate difference is the result inflation being 5% and 10% respectively.Then the price dynamics of both countries will result in an exchange rate of 0.9545 next year, which is also the forward rate.The Role of Expectations in Determining Exchange Rates In real life, things are not so simple, but several mechanisms may be postulated that support the expectations hypothesis.International investor confidence, and their forecasts of inflation, place price pressure on both spot and forward exchange rates through the international bond market。