期权、期货及其他衍生产品课件3金融工程学.ppt

5.1,Determination of Forward and Futures Prices,Chapter 5,5.2,Consumption vs Investment Assets,Investment assets are assets held by significant numbers of people purely for investment purposes(Examples:stock,bond,gold,silver)Consumption assets are assets held primarily for consumption(Examples:copper,oil),5.3,Short Selling(Page 99-101),Short selling involves selling securities you do not ownYour broker borrows the securities from another client and sells them in the market in the usual wayAt some stage you must buy the securities back so they can be replaced in the account of the client,5.4,Short Selling(continued),If at any time while the contract is open the broker is not able to borrow shares,the investor is forced to close out the position,even if not ready to do so,called short-squeezed(挤空，挟仓).You must pay dividends and other benefits the owner of the securities receives,5.5,Example,The investor is required to maintain a margin account with the broker.,5.6,Assumptions,The market participants are subject to no transaction costs when they trade.The market participants are subjects 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.,5.7,Notation for Valuing Futures and Forward Contracts,5.8,Forward price for an investment asset,5.9,When Interest Rates are Measured with Continuous Compounding,F0=S0erT 远期价格大于即期价格 if F0 S0erT,arbitrageurs can buy the asset and short forward contracts on the asset.if F0 S0erT,they can short the asset and enter into long forward contracts on it.This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs.,5.10,When an Investment Asset Provides a Known Dollar Income,5.11,套利者净收益：910.00-886.60=23.40,5.12,When an Investment Asset Provides a Known Dollar Income(page 105,equation 5.2),F0=(S0 I)erT where I is the present value of the income during life of forward contract.In our example,So that,5.13,When an Investment Asset Provides a Known Yield(Page 107,equation 5.3),F0=S0 e(rq)T where q is the average yield during the life of the contract(expressed with continuous compounding),5.14,Example Consider a 6-month forward contract on an asset that is expected to provide income equal to 2%of the asset price once during a 6-month period.The risk free rate of interest is 10%per annum.The asset price is$25.The yield is 4%per annum,it follows that q=0.0396(=2ln(1+4%/2).,5.15,Valuing a Forward ContractPage 108,Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the contract todayThe value of a long forward contract,is=(F0 K)erT Similarly,the value of a short forward contract is(K F0)erT,5.16,5.17,Example,A long forward contract on a not-dividend paying stock was entered into some time ago.It currently has 6 months to maturity.The risk-free rate of interest is 10%per annum,the stock price is$25,and the delivery price is$24.,5.18,The value of a long forward contract,No income,income,Known yield,5.19,Forward vs Futures Prices,Forward and futures prices are usually assumed to be the same.When interest rates are uncertain they are,in theory,slightly differentA strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward priceA strong negative correlation implies the reverse,5.20,Stock Index(Page 110-112),Can be viewed as an investment asset paying a dividend yieldThe futures price and spot price relationship is therefore F0=S0 e(rq)T where q is the average dividend yield on the portfolio represented by the index during life of contract,5.21,Stock Index(continued),For the formula to be true it is important that the index represent an investment assetIn other words,changes in the index must correspond to changes in the value of a tradable portfolioThe Nikkei index viewed as a dollar number does not represent an investment asset(See Business Snapshot 5.3,page 111),5.22,Index Arbitrage,When F0 S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futuresWhen F0 S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index,5.23,Index Arbitrage(continued),Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally(e.g.,on Black Monday)simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold,5.24,A foreign currency is analogous to a security providing a dividend yieldThe continuous dividend yield is the foreign risk-free interest rateIt follows that if rf is the foreign risk-free interest rate,Futures and Forwards on Currencies(Page 112-115),5.25,Why the Relation Must Be True Figure 5.1,page 113,5.26,Example,Suppose that the 2 year interest rates in Australia and the United States are 5%and 7%,respectively,and the spot exchange rate is 0.6200 USD per AUD.The 2 year forward exchange rate should be,5.27,Suppose that the 2 year forward exchange rate is less than this,say 0.6300.An arbitrageur can:1.Borrow 1000 AUD at 5%per annum for 2 years,convert to 620 USD and invest the USD at 7%.2.Enter into a forward contract to buy 1105.17 AUD for 1105.17*0.63=696.26 USD,5.28,Suppose that the 2 year forward rate is 0.6600.An arbitrageur can:1.Borrow 1000 USD at 7%per annum for 2 years,convert to 1000/0.6200=1616.90 AUD,and invest the AUD at 5%.2.Enter into a forward contract to sell 1782.53 AUD for 1782.53*0.66=1176.47 USD.,5.29,A foreign currency can be regarded as an investment asset paying a known yield.The yield is the risk free rate of interest in the foreign currency.,5.30,Futures on Consumption Assets(Page 117-118),F0 S0 e(r+u)T where u is the storage cost per unit time as a percent of the asset value net of any yield earned on the asset.Alternatively,F0(S0+U)erT where U is the present value of the storage costs,net of income.Storage costs can be treated as negative income.,5.31,Suppose F0(S0+U)erT,a arbitrageur can:1.Borrow an amount S0+U at the risk free rate and use it to purchase one unit of the commodity and to pay storage cost.2.Short a forward contract on one unit of the commodity.Equation F0(S0+U)erT cannot hold.Suppose F0(S0+U)erT,1.Sell the commodity,save the storage costs,and invest the proceeds at the risk free interest rate.2.Take a long position in a forward contract.,5.32,Individuals and companies who own a consumption commodity usually plan to use it in some way.They are reluctant to sell the commodity in the spot market and buy forward for futures contracts,because forward and futures contracts cannot be consumed.we can assert for a consumption commodity is:F0(S0+U)erT F0 S0 e(r+u)T,5.33,Convenience Yields y,5.34,The Cost of Carry(Page 118-119),The cost of carry,c,is the storage cost plus the interest costs less the income earned c=storage cost+interest cost-income earnFor an investment asset F0=S0ecT For a consumption asset F0 S0ecTThe convenience yield on the consumption asset,y,is defined so that F0=S0 e(cy)T,5.35,Futures Prices&Expected Future Spot Prices(Page 119-121),Suppose k is the expected return required by investors on an assetWe can invest F0er T at the risk-free rate and enter into a long futures contract so that there is a cash inflow of ST at maturity,5.36,Value this investment:The discount rate we should use for the expected cash flow at time T equals an investors required return on the investment k.The present value of this investment is:We can assume that all investments in securities markets are priced so that they have zero net present value.This shows that,5.37,Futures Prices&Future Spot Prices(continued),If the asset has no systematic risk,then k=r and F0 is an unbiased estimate of STpositive systematic risk,then k r and F0 E(ST)(contango，期货溢价),5.38,5.28 一家公司和一家银行签订了远期合约，规定公司将在 时刻以 卖出外汇。时刻的汇率为。公司询问银行是否可以将合约向前延展到 时刻。银行同意了，并给出了新的交割价格。请解释新的交割价格应如何计算。,