3.1

The Determination of Forward & Futures Prices

3.2

Compounding Frequency• The compounding frequency

used for an interest rate is the unit of measurement

• The difference between quarterly & annual compounding is analogous to the difference between miles & kilometers

3.3

Continuous Compounding

• 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

3.4

Conversion Formulas

• Define

Rc: continuously compounded rate

Rm: same rate with compounding m times per year

Rc=m ln(1+Rm/m)

Rm=m[exp(Rc/m)-1]

3.5

Short Selling• Short selling involves selling

securities you do not own

• Your broker borrows the securities from another client & sells them in the market in the usual way

3.6

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

3.7Repo Rate

• The repo rate is the relevant rate of interest for many arbitrageurs

• A repo (repurchase agreement) is an agreement where one financial institution sells securities to another financial institution & agrees to buy them back later at a slightly higher price

• The difference between the selling price & the buying price is the interest earned

3.8

Gold Example

• For gold F = S (1 + r )T whereF : forward price, S : spot price, & r : T -year rate of interest (This assumes no storage costs )

• If r is compounded continuously instead of annually

F = S er T

3.9

Extension of the Gold Example

• For any investment asset that provides no income & has no storage costs

F = S er T (Equation 3.5, p.53)

3.10When an Investment Asset Provides an Income Which is

Known in Dollar Terms

F = (S – I )er T (Equation 3.6, p.56)

where I is the present value of the income

3.11

When an Investment Asset Provides a Known Dividend

Yield

F = S e(r–q )T (Equation 3.7, p.57)

The asset is assumed to provide a return during time t equal to qS t where q is the dividend yield and S is the asset price

3.12

Valuing a Forward Contract• Suppose that

• K : delivery price in a forward contract &

• F : forward price that would apply to the contract today

• The value of a long forward contract, ƒ, is ƒ = (F – K )e–r T (Equation 3.8, p.58)

• Similarly, the value of a short forward contract is

(K – F )e–r T

3.13

Forward vs Futures Prices

• Forward & futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:

- strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price

- strong negative correlation implies the reverse

3.14

Stock Index• Can be viewed as an investment

asset paying a continuous dividend yield

• The futures price & spot price

relationship is therefore F = S e(r–q )T (Equation 3.12, p. 62)

where q is the dividend yield on the portfolio represented by the index

3.15

Stock Index(continued)

• 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

• The Nikkei index viewed as a dollar number does not represent an investment asset

3.16

Index Arbitrage

• When F>Se(r-q)T an arbitrageur buys the stocks underlying the index and sells futures

• When F<Se(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

3.17

Index Arbitrage(continued)

• Index arbitrage involves simultaneous trades in futures & many different stocks

• Very often a computer is used to generate the trades

• Occasionally (eg “Black Monday”) simultaneous trades are not possible & the theoretical no-arbitrage relationship between F & S may not hold

3.18

• A foreign currency is analogous to a security providing a continuous 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

Futures on Currencies

F S r r Tf e ( )

(Equation 3.13, p.64)

3.19

Futures on Consumption Assets

F S e(r+u )T (Equation 3.20, p.68)

where u is the storage cost per unit time as a percent of the asset value.

Alternatively,

F (S+U )er T (Equation 3.19, p.68)

where U is the present value of the storage costs.

3.20

The Cost of Carry• The cost of carry, c , is the storage cost plus

the interest costs less the income earned

• For an investment asset F = SecT

• For a consumption asset F S ec T

• The convenience yield on the consumption asset, y , is defined so that F = S e(c–y )T (Equation 3.23, p.70)

3.21

Futures Prices & Expected Future Spot Prices

• Suppose k is the expected return required by investors on an asset

• We can invest F e –r T now to get ST back at maturity of the futures contract

• This shows that

F = E (ST )e(r–k )T(Equation 3.24, p.71)

3.22Futures Prices & Future Spot Prices (continued)

• If the asset has

- no systematic risk, then k = r , & F is an unbiased estimate of ST

- positive systematic risk, then

k > r , & F < E (ST )

- negative systematic risk, then

k < r , & F > E (ST )