Chapter 3

Assigned problems

3.1-3.7, 3.10-3.17

Investment assets and consumption assets

Mechanics of short selling

Forward price of an investment asset

Future values and present values with continuous compounding

Logarithmic scale

Forward price with known income (bonds)

A stock trades at $50 and is expected to pay a $2 dividend after two months.

What is the forward price for the stock three months ahead?

Interest rate is 6% per annum (continuously compounded)

S₀ = 50 I = 2 e^{-(0.06)(2/12)}= 1.9809

[First compute rt rt = 0.01 Then compute e^-rt]

F₀ = (50 – 1.9809) e^(0.06)(3/12) = 48.745

Forward price with known yield (foreign currency)

Continuous yield of q

F₀ = S₀ e^(r-q)t

Buy stock

Borrow S₀

Reinvest the dividends in the same stock

Number of share at the end of t = e^qt

Sell e^qt contracts

e^qtF₀ = S₀e^rt

Assumptions for pricing under ideal conditions

a) No transactions costs

b) All participants have the same tax rates

c) Borrowing and lending at the same interest rate

d) Arbitrage opportunities are immediately exploited

Value of a forward contract

f = (F₀ – K) e ^–rt

K is the F₀ at the time of entering into the contract. (zero value contract)

Is forward price the same as futures price?

Is S is positively correlated with interest rates

futures contract will be priced higher than forward .

Why?

Imagine you take a long position. Price goes up. Gains are invested at a high r

If price goes down, can borrow at a low r to pay the losses

If S is negatively correlated with interest rates, futures price should be

lower than forward prices

Stock index futures

cash settlement

dividend not included

S&P 500 Index has a continuous dividend yield of 1%. The risk free rate is 1.5%

If the current index value is 800 what is the price of a 6 month futures contract?

Currency and continuous yield

The interest rate in the US is 1% and in the UK it is 4%. If the current exchange rate is $1.62 per pound what is the 3 month forward rate?

Storage costs

F₀ = (S₀ + U) e ^rt

U = present value of storage costs (negative dividend)

For consumption assets

F₀ £ (S₀ + U) e ^rt

F₀ e ^yt = (S₀ + U) e ^rt

y = convenience yield

if storage cost is proportional to spot price (u)

F₀ e ^yt = S₀ e ^(r+u)t

F₀ = S₀ e ^(r+u-y)t

r+u can be viewed as cost of carry