华东师范大学：《金融工程》英文版 Chapter 3 Forward and Futures prices

3.1 Forward and Futures prices Chapter 3 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.1 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Forward and Futures Prices Chapter 3

3.2 Two Kinds of Underlying Assets Investment assets: held for investment purposes by a significant numbers of investors. EXamples: stockS, bonds, gold Three different situations 1. The asset provides no income 2. The asset provides a known dollar income 3. The asset provides a known dividend yield Consumption assets: held primarily for consumption. EXamples: commodities such as copper, oil and live hogs Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.2 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Two Kinds of Underlying Assets • Investment assets: held for investment purposes by a significant numbers of investors. Examples: stocks, bonds, gold. Three different situations: 1. The asset provides no income 2. The asset provides a known dollar income 3. The asset provides a known dividend yield • Consumption assets: held primarily for consumption. Examples: commodities such as copper, oil and live hogs

3.3 Arbitrage Arguments is workable for the determination of the forward and futures prices of investment assets from spot and other observable variables is not possible to determine the forward and futures prices of consumption The forward price and futures price are very close to each other when the maturities of the two contracts are the same Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.3 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Arbitrage Arguments • is workable for the determination of the forward and futures prices of investment assets from spot and other observable variables. • is not possible to determine the forward and futures prices of consumption. • The forward price and futures price are very close to each other when the maturities of the two contracts are the same

3.4 Compounding frequency The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.4 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Compounding Frequency • The compounding frequency used for an interest rate is the unit of measurement • The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

3.5 Continuous Compounding(Page 51) In the limit as we compound more and more frequently we obtain continuously compounded interest rates R lmm→ A(1+-) Ae $100 grows to $100eR/when invested at a continuously compounded rate R for time T' $100 received at time t discounts to $100e-R7 at time zero when the continuously compounded discount rate is r Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.5 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Continuous Compounding(Page 51) • In the limit as we compound more and more frequently we obtain continuously compounded interest rates • $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 R mn Rn m Ae m R lim → A(1+ ) =

3.6 Conversion Formulas (Pages 52, 53) Define Rc: continuously compounded rate Rm: same rate with compounding m times per year R R=mIn 1+-m 77 AeRen=A1+m Rm= mle r/m EXamples:3.1, 3.2(page 53) Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.6 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Conversion Formulas (Pages 52,53) • Define Rc : continuously compounded rate Rm: same rate with compounding m times per year Examples: 3.1, 3.2(page 53)

3.7 Short selling (Page 53) 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 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.7 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Short Selling (Page 53) • 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

3.8 Short selling (continued At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends other benefits the owner of the securities receives Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.8 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Short Selling (continued) • At some stage you must buy the securities back so they can be replaced in the account of the client • You must pay dividends & other benefits the owner of the securities receives

3.9 Assumptions and notations The market participants are subject to no transaction costs when they trade are subject to the same tax rate on all net trading profits can borrow money at the same risk-free rate of interest as they lend money take advantage of arbitrage opportunities as they occor Notations T: time when the forward contract matures ( years) So: price of asset underlying the forward contract today Fo: forward price today r: risk-free rate of interest per annual with continuous compounding, for an investment maturing at T Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.9 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Assumptions and Notations The market participants • are subject to no transaction costs when they trade • are subject to the same tax rate on all net trading profits • can borrow money at the same risk-free rate of interest as they lend money • take advantage of arbitrage opportunities as they occor Notations: T: time when the forward contract matures (years) S0 : price of asset underlying the forward contract today F0 : forward price today r: risk-free rate of interest per annual, with continuous compounding, for an investment maturing at T

3.10 Gold Example For the gold example in chapter 1 F0=S0(1+r)7 (assuming no storage costs) If r is compounded continuously instead of annually Fo= Soer 0 0 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

3.10 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Gold Example • For the gold example in chapter 1, F0 = S0 (1 + r ) T (assuming no storage costs) • If r is compounded continuously instead of annually F0 = S0 e rT