Chapter 3Forward and Futures PricesDate1This Chapter Covers:Relationships between forward/futures prices and the price of the underlying.Forward price and futures price are very close to each other.Distinguish between investment assets and consumption assets.No arbitrage pricing methods Date2Compounding FrequencyThe compounding frequency used for an interest rate is the unit of measurementThe difference between quarterly and annual compounding is analogous to the difference between miles and kilometersDate3Terminal ValuesCompounded once per annum:A(1+R)nCompounded m times per annum:A(1+R/m)mnCompounded continuously:Date4Continuous CompoundingCompounded continuously is almost equal to compounded per day.$100 grows to $100eRT when invested at a continuously compounded rate R for time T$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is RDate5Conversion FormulasDefineRc : continuously compounded rateRm: same rate with compounding m times per yearDate6Conversion FormulasWhen m=1Rc=ln(St/St-1)because: Rm=(St-St-1)/St-1=St/St-1-11+Rm=St/St-1Date7Short Selling 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 wayDate8Short Selling (continued)At some stage you must buy the securities back so they can be replaced in the account of the clientYou must pay dividends & other benefits the owner of the securities receivesDate9AssumptionsNo transaction costsSame tax rateBorrow or lend at the same risk-free rate of interestNo arbitrage opportunitiesDate10Gold ExampleFor the gold example in chapter 1,F0 = S0(1 + r )T (assuming no storage costs)If r is compounded continuously instead of annually F0 = S0erTDate11Extension of the Gold ExampleFor any investment asset that provides no income and has no storage costs F0 = S0erT Date12When an Investment Asset Provides a Known Dollar Income F0 = (S0 I )erT where I is the present value of the incomeDate13When an Investment Asset Provides a Known Dividend Yield F0 = S0 e(rq )T where q is the average dividend yield during the life of the contractDate14Valuing a Forward ContractSuppose that K is delivery price in a forward contract & 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 )erTDate15Forward vs Futures PricesForward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:A 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 Date16Stock Index Can be viewed as an investment asset paying a continuous dividend yieldThe futures price & spot price relationship is therefore F0 = S0 e(rq )T where q is the dividend yield on the portfolio represented by the indexDate17Stock 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 assetDate18Index ArbitrageWhen F0S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futuresWhen F0 r and F0 E (ST )Date27Relationship between F and SF,SFT=E(ST)F0S0, (S0-I),or Se-qTTime premiumTimeRisk premiumDate28