BFF3751 Derivatives 1
Week 3
Chapter 5, Hull
Determination of Forward and Future Prices
Investment vs. Consumption Assets
 Investment assets: assets held primarily for
investment purposes.
 E.g.: stock, bonds, gold, silver.
 Do not have to be held exclusively for investment.
 Silver has a number of industrial uses.
 Consumption assets: assets held primarily for
consumption purposes.
 E.g.: copper, oil, pork bellies.
 Use arbitrage arguments to determine the forward
and futures prices of an investment asset but not for
consumption assets.
Short Selling
 Short selling involves selling assets you do not own.
 Possible for some but not all investment assets.
 Your broker borrows the securities from another
client and sells them in the market in the usual way.
 Later you must buy the securities back, so they can
be replaced in the account of client.
 The investor takes a profit if the stock price has
declined and a loss if it has risen.
Short Selling
 You must pay dividends and other benefits the owner
of the securities should receive on the shares.
 Margin account is kept with the broker to guarantee
that you do not walk away from your obligations.
Short Selling Example
 Investor sells short 100 shares of Commonwealth
Bank at $100 each in March 2016.
 Total value: 100 * 100 = $10,000 (receive).
 In April, stock pays a dividend of $2 per share.
 Total dividend cost: 2*100 = $200 (pay).
 In May, investor closes the position by buying back
the shares on the market at $110 each.
 Total buy back cost: 110 * 100 = $11,000 (pay).
 Profit/loss = $10,000 - $200 - $11,000 = - $1,200.
Assumption and Notation
 Assumption market participants:
 subject to no transaction costs when they trade.
 subject to same tax rate on all net trading profits.
 can borrow money at the same risk-free rate of interest
as they can lend money.
 take advantage of arbitrage opportunities as they
occur.
S0: Spot price today.
F0: Futures or forward price today.
T: Time to maturity.
r: Risk-free interest rate.
Arbitrage Opportunity?
 Consider a long forward contract to purchase a non-
dividend-paying stock in 3 months.
 Current stock price is $40.
 3-months risk-free rate is 5% per annum.
 Is there an arbitrage opportunity?
Arbitrage Opportunity?
 Suppose forward price is relative high at $43.
 Today:
 Borrow $40 @ 5%.
 Buy one share.
 Short one forward to sell one share at $43 in 3 months.
 In 3 months:
 Sell the stock at $43.
 Repay the loan: 40*e(0.05*3/12) = $40.50.
 Profit = 43 - 40.50 = $2.5.
 This arbitrage works when F > $40.50.
Arbitrage Opportunity?
 Suppose forward price is relative low at $39.
 Today:
 Short one share to realize $40.
 Invest $40 @ 5% for 3 months.
 Long one forward to buy one share at $39 in 3 months.
 In 3 months:
 Buy the stock at $39, and close the short position.
 Receive from investment: 40*e(0.05*3/12) = $40.50.
 Profit = 40.5 - 39 = $1.5.
 This arbitrage works when F < $40.50.
 There is no arbitrage only when F = $40.50.
Forward Pricing
 If the spot price is S0 and the forward price for a
contract deliverable in T years is F0, r is the risk-free
interest rate.
 To avoid arbitrage, F0 should be equal to:
F0 = S0erT
 This equation relates the forward price and the spot
price for any investment asset that provides no
income and has no storage costs.
 E.g., non-dividend-paying stocks, zero-coupon bond.
 In our examples, S0 = 40, T = 3/12, and r = 0.05.
F0 =S0erT = 40*e(0.05*3/12) = $40.5.
Forward Pricing: Known Income
 Consider a forward contract on an investment asset
that will provide a perfectly predictable cash income to
the holder.
 E.g., stock paying known dividends and coupon-
bearing bonds.
F0 = (S0 – I)erT
 where I is the present value of the income during life
of forward contract.
Forward Pricing: Known Income
• Consider a long forward contract to purchase a coupon-
bearing bond whose current price is $900.
• The forward contract matures in 9 months.
• A coupon payment of $40 (i.e., income) is expected after
4 months.
• 4- and 9-month rates (continuously compounded) are 3%
and 4% per annum.
Forward Pricing: Known Income
• F=$910: use the income to pay part of the borrowing in 4
month. Take away the PV(I) from the $900 borrowing today.
• F=$870: use part of the investment to pay the income of the
shorted asset in 4 month. Take away the PV(I) from the $900
investment today.
Forward Pricing: Known Yield
 Consider the situation where the asset underlying a
forward contract provides a known yield rather than a
known cash income.
F0 = S0 e(r–q)T
 where q is the average yield during the life of the
contract (expressed with continuous compounding).
Forward Pricing: Known Yield
 Stock price is $25 today.
 6-month forward contact.
 Dividend yield q=3.96% per annum with continuous
compounding.
 Risk-free interest rate is 10% per annum.
 What is the 6-months forward price of this stock?
F0 = S0 e(r–q)T = 25 e(10%–3.96%)*6/12 = $25.77
Valuing Forward Contracts
 When a forward contract is first entered into, its value
is zero.
 Suppose that K is delivery price in a forward contract
and F0 is forward price that would apply to the
contract today.
 The value of a long forward contract is
ƒ = (F0 – K)e–rT
 Similarly, the value of a short forward contract is
ƒ = (K – F0)e–rT
Valuing Forward Contracts
 One month ago, you entered a long forward contract
on gold with delivery price of $1200 per ounce.
 The contract has two months remaining.
 If the spot price of gold is $1400, and continuously
compounded interest rate is 6%, what is the value of
this contract now?
 First, find the current forward price of gold:
F0 = S0erT = 1400e0.06*2/12 = $1414.07
 Now we can value the contract:
f = (F0 – K)e–rT = (1414.07 - 1200)e-0.06*2/12 = $211.94
Forward vs. Futures Pricing
 Forward and futures prices are the same when short term
interest rates are constant and same maturities.
 When interest rates are unpredictable they are slightly
different.
 If a strong positive correlation between IR and asset price.
Futures price is slightly higher than forward price.
 Consider a long future:
 asset price increases (IR increases)  immediate gain
(daily settlement)  the gain is reinvested at a higher
rate.
 asset price decreases (IR decreases)  immediate
loss (daily settlement)  the loss is refinanced at a
lower rate.
 Underlying asset: one unit of the foreign currency.
 Holder of the currency can earn risk-free IR prevailing
in foreign country.
 Foreign currency  a security providing a dividend
yield (i.e., foreign risk-free IR).
Forward and Futures on Currencies
 S0: spot price in local currency in one unit of foreign
currency.
 F0: forward/futures price in local currency in one unit
of foreign currency.
 rf: foreign risk-free IR.
 r: domestic risk-free IR.
Forward and Futures on Currencies
Trr feSF )(00

Why the Relation Must Be True
1000 units of
foreign currency
at time zero
units of foreign
currency at time T
Trfe1000
dollars at time T
TrfeF01000
1000S0 dollars
at time zero
dollars at time T
rTeS01000=
• Starts with 1,000
units of the foreign
currency.
• Two ways it can be
converted to dollars
at time T.
1. Investing for T years
at rf. Entering a forward
contract to sell the
proceeds for dollars at
time T.
2. Exchanging foreign
currency for dollars in
the spot market today
and investing for T
years at rate r.
Arbitrage Example 5.6
 2-years IR are 5% in Australia and 7% in US
(domestic).
 Spot exchange rate = 0.62 USD/AUD.
 2-years forward exchange rate must
= 0.62e(0.07 – 0.05)*2 = 0.6453.
 If forward rate = 0.63 USD/AUD  possibility of
arbitrage.
 If forward rate = 0.66 USD/AUD  possibility of
arbitrage.
If forward rate = 0.63 USD/AUD
 Borrow 1,000 AUD @ 5% (need to repay
1,105.17AUD = 1,000e0.05*2).
 Convert into 620 USD and invest @ 7% for 2 years.
 Enter a 2-years forward contract to buy 1,105.17
AUD for 1,105.17*0.63 = 696.26 USD.
 In 2 years, receive investment 620e0.07*2 = 713.17
USD.
 Pay 696.26 USD to buy 1,105.17 AUD and repay the
loan.
 Profit = 713.17 – 696.26 = 16.91USD.
If forward rate = 0.66 USD/AUD
 Borrow 1,000 USD @ 7%.
 Convert into 1,000/0.62=1612.90 AUD and invest @
5% for 2 years.
 Enter a 2-years forward contract to sell AUD:
1,782.53AUD*0.66 =1,176.47 USD.
 In 2 years, receive investment 1,612.90e0.05*2=
1,782.53 AUD. Then, exchange to 1,176.47 USD.
 Pay US borrowing 1,000e0.07*2=1,150.27.
 Profit = 1,176.47 – 1,150.27 = 26.20USD.
Futures on Investment Commodities
 Gold and silver: can provide income, but have
storage costs.
 U: present value of all storage costs, net of income,
during the life of forward contract.
F0 = (S0+U)erT
 If storage costs u, net of income, are proportional to
commodity price:
F0 = S0 e(r + u)T
 Storage costs can be treated as negative yield.
Futures on Consumption Assets
 Consumption assets: high storage costs (U) and no
income, but consumption value.
 Suppose we have F0 > (S0+U)erT.
 An arbitrageur can borrow (S0+U), buy and store the
commodity, and short a futures contract.
 At expiration, the riskless profit = F0 - (S0+U)erT > 0
 There is a tendency for S0 to increase and F0 to
decrease, thus the relation can not hold for long.
Futures on Consumption Assets
 Suppose we have F0 < (S0+U)erT.
 An arbitrageur can sell the commodity, save the
storage costs, invest in risk-free rate, and long
futures contact.
 At expiration, the riskless profit = (S0+U)erT - F0 > 0
 There is a tendency for S0 to decrease and F0 to
increase, thus the relation can not hold for long.
 Wait a minute, is that true?
 The commodity is used for consumption, the holders
are reluctant to sell. The strategy does not work.
 There is nothing to stop the relation from holding.
Futures on Consumption Assets
 In case of consumption assets we have inequality:
F0  (S0+U)erT
 Alternatively,
F0  S0 e(r+u)T
 where u is the annual storage cost as a percent of the
asset value.
Convenience Yield
 For consumption commodities users, the ownership
of physical commodities are not obtained by future
contract.
 Ownership of physical asset enables a manufacturer to
keep a production process running and perhaps profit
from temporary local shortages.
 E.g., the crude oil in inventory can be an input to the
refining process, but not for futures contact.
Convenience Yield
 Convenience yield: benefit derived from holding the
physical asset.
 Only for consumption assets.
 Reflects the market’s expectations concerning the
future availability of the commodity.
 If inventories are low, shortages are more likely and the
convenience yield is usually higher.
 Convenience yield y balances out the previous
inequality:
F0 = S0 e(r+u-y)T
The Cost of Carry
 The relationship between futures prices and spot prices
can be summarized in terms of the cost of carry.
 Cost of carry (c) = storage cost + interest costs -
income earned
 Non-dividend-paying stock: c = r.
 Dividend-paying stock: c = r – q.
 Currency: c = r - rf .
 Commodity: c = r - q + u.
 For an investment asset: F0 = S0ecT
 For a consumption asset: F0 = S0e(c–y)T