PRICING FUTURES

1
PRICING FUTURES
Factors that determine futures prices 1.
Supply and demand like any current commodity
price - elasticities of supply and demand -
less elastic more price volatility 2. Cost of
carry the opportunity costs of buying a
commodity now and holding (carrying) it for
future use. - foregone interest - storage
costs 3. Convenience yield benefit from
holding a commodity - available inventory,
dividend, coupon
2
General Futures Pricing Equation
Ft,T MinEt(ST), St(1 Rf,T-t CCT-t -
CYT-t) t a point in time, say, now t0 T
future date beyond time t Ft,T futures
price covering time t to time T St spot
price at time t Et(ST) expected spot price in
the future at time T Rf,T-t risk-free
interest rate from time t to T CCT-t cost to
carry from time t to T such as storage CYT-t
convenience yield from time t to T This
equation says the futures price should be set
equal to either the expected future spot price
or the cost of carry price whichever is less.
3
Explaining Some of the Costs
Examples of each alternative. Perishables -
Strawberries, milk - Ft,T E(ST) Durables -
Gold, Oil Ft,T St(1 Rf,T-t CCT-t -
CYT-t) Seasonal crops like corn a mixture of
both alternatives Interest carry costs Suppose
interest rates are positive, then interest
payments make holding a commodity (which pays no
interest) less attractive. It makes futures
contracts more attractive. This is like options
where the option price depends upon interest
rates because the value of competing positions
depend on interest rates.
4
Other carrying costs Suppose it costs something
to store the commodity such as grain silos,
insurance, etc. This adds to the cost of buying
now and holding for future use. Convenience
yield (CY) Suppose that you are Kelloggs and
make corn flakes. You can buy a fixed number of
futures for corn delivery each month, but if you
get an unusually large order for corn flakes one
month, you may not be able to fill it. Therefore,
you may hold an inventory of corn for future
use. Look at Quotes for seasonality, oil -
futuresource.com
5
QUESTION How should T-bond futures be priced?
(Assumes zero coupon paid). ANSWER Ft,T
St(1 Rf,T-t) - no carry costs except for
interest forgone - Expectations are already
reflected in the spot price (Pure Expectations
Theory) QUESTION How should this be adjusted
for SP futures or bonds with coupons? -
must consider dividends - D Ft,T St1
Rf,T-t - E(DT-t)/St
6
NOTE The futures price does not include the
expected return of the SP over the contract
period. This is because, if you buy a contract
you buy the systematic risk of the SP and should
be rewarded, that is Ft,T lt Et(ST). -
the seller is selling the risk and the buyer buys
it, so the buyer must be compensated by paying a
low Ft,T now and expecting to receive a higher
Et(ST) later.
7
- the futures price is not set at Et(ST)
because St(1 Rt,T -Dt,T/St) lt
Et(ST) ACCORDING TO THE FUTURES PRICING
MECHANISM, WE GO WITH THE LOWER PRICE. If the
market expects an extra return in the SP beyond
its normal expected return required for its risk
then both St and Ft,T will move up to discount
the extra no-risk return and thus the
relationship Ft,T St(1 Rt,T - Dt,T/St)
still holds.
8
EXAMPLE TESTING SP FUTURES FOR ARBITRAGE
OPPORTUNITIES
Spot SP Nov. 6, 2008 1118.86 Futures
Dec. 2008 1121.00 Mar.
2009 1122.40 SP dividend yield 1
annualized 2 Month Tbill yield 1.7
annualized 4 Month Tbill yield 1.8
annualized Prompt date is on Thurs. before third
Friday of month Theoretical Price
Actual - Theoretical FN,D
1118.861.017(44/365)-.01(44/365)
1119.80 1121 - 1119.80
1.20 FN,M 1118.861.018(128/365)-.01(128/365)
1122
1122.4 - 1122 0.40 Profit only if
differences between actual and theoretical
futures prices exceed trading commissions.
9
EXPLAINING ARBITRAGE
TRADING GOLD - SPOT MARKET N.Y. London gold
price / oz 350 360 5 shipping
costs Arbitrage opportunity - buy in NY / sell
in London Buy gold in NY -350 Agree to sell
in London 360 Shipping -5 Guaranteed
profit / oz. 5
10
TRADING CORN FUTURES - ASSUME R 10 AND
IGNORE OTHER CARRY COSTS
Corn price - spot 3.00/bu Corn price - one
year futures 3.50 Theoretical futures
price 3.30 3(1 .10) Arbitrage
opportunity - buy spot / sell futures Buy spot
corn now -3.00 Sell futures now /deliver corn
later - receive 3.50 Pay interest carry
costs -.30 Guaranteed profit/bu.
.20 No matter what happens to price of corn
after you buy corn and sell futures, you still
make .20.
11
TRADING SP FUTURES
SP Nov. 6, 2008 spot price 1118.86 SP Dec.
futures price 1121.00 Theoretical December
futures price FN,D 1118.861.017(44/365)-.01(4
4/365) 1119.80 Assuming no transaction
costs Arbitrage opportunity - sell futures / buy
spot Sell futures now and make delivery in
Dec. 1121.00 Buy spot now
-1118.86 Forgo interest on funds
1118.86.017(44/365) -2.24 Receive dividends
1118.86.01(44/365) 1.30 Guaranteed profit
no matter what SP does 1.20
12
SWAPS - NEW, LARGE, AND EXPANDING MARKET
Definition Contract to swap cash flows or
principal value of one asset for those of another
of equal value at origination date. The assets
are not actually traded, payments are made on
designated dates consisting of the differences in
their values or cash flows over time EXAMPLE
Interest rate swap payments (Long rate -
Short rate) x (notional principal)
13
WHY ARE SWAPS POPULAR?

• Avoid time and expense of issuing new securities
  and retiring old ones
• Avoid regulation
• Change risk level but avoid explicitly changing
  balance sheet
• Swaps are actually a series of forward contracts.
• Caps, floors, collars, and swaptions combine
  swaps and options

14
HEDGING WITH FUTURES - ADVANCED
If we have futures contracts in the commodity we
wish to hedge, we simply use that contract and
match the amount of the commodity we wish to
hedge with the amount of futures. 1 million
bushels of wheat -gt 1 million bushels in
futures
15
CROSSHEDGING
If we do not have futures in the identical
commodity we may "crosshedge" with a similar
commodity. For an effective hedge we want the
change in the value of the spot commodity to be
equal to minus the change in the value of the
futures. The amount of futures needed per unit of
spot is h hedge ratio -DSpot price /
DFutures price - this means if the spot and
futures price move together (opposite), sell
(buy) futures to hedge.
16
For example, suppose you hold IBM bonds but only
Treasury bond futures are available. You can
hedge your IBM position by knowing the change in
the price of your bond when the Tbond price
changes. Your bonds price change is (1) Py
Price of your bond Dury Duration of your
bond Yoy old yield of your bond Yny new
yield of your bond
17
The Treasury bonds price change
is (2) PT price of treasury
bond DurT Duration of Treasury bond YoT Old
yield on Treasury bond YnT New yield on
Treasury bond
18
To get the hedge ratio divide (1) by
(2) Where h the units (dollars)
of futures to be sold per unit (dollar) of spot.
This is reliable in most cases less reliable if
durations change much with rate changes. It
assumes a parallel shift in yield curve. NOTE
This hedges only interest rate risk - default
risk is ignored.
19
CAN USE THE SAME IDEA FOR STOCKS
You need the betas of the spot portfolio and the
futures portfolio. h -Beta(Spot) /
Beta(Futures) This hedges only systematic risk
- stock-specific risk is ignored.
20
EXAMPLE COMPLEX HEDGING - SOUTHEAST CORP CASE

• On Jan 6, 2008 Southeast authorized 60 million
  of 25 year bonds to fund a building project which
  would be needed in August 2008.
• Bonds are Aa rated and have Yield of 12.88 if
  issued today.
• The bonds have a duration of 7.8
• A regression of Aa yield changes on Tbond yield
  changes has a slope of 1.123

21

• The Tbond futures of September 2008 had a price
  of 69 - 08 or 69.25
• The futures contract price is 69,250 on a
  100,000 face value 8 contract
• The cheapest to deliver bond for the September
  contract has an 11.80 Yield.
• It also has a duration of 8.5 years

22
FINDING THE NUMBER OF FUTURES CONTRACTS NEEDED TO
HEDGE

• FIND THE HEDGE RATIO
• h 7.8/8.5 x 1.123/1 x 1.118/1.1288 1.021
• FIND DOLLAR AMOUNT OF FUTURES NEEDED TO HEDGE
• F 60,000,000 1.021 61,260,000
• FIND NUMBER OF FUTURES CONTRACTS NEED
• NF 61,260,000/69,250 885

23
FOREIGN EXCHANGE FUTURES
SIMPLE DEFINITION Buying one currency with
another INTEREST RATE PARITY - implies that all
countries have the same interest rate after one
adjusts up or down for the change in the
country's currency value. INTEREST RATES ARE
THE PRICE OF MONEY - Thus the futures price of,
say, the dollar in terms of the Euro, will depend
on their relative prices, i.e., the respective
countries interest rates.
24
INTEREST RATE PARITY DEFINES THE RELATIONSHIP
BETWEEN FUTURES AND SPOT
Ft,T the future price of one unit of foreign
currency in terms of the domestic currency e.g,
2/1. St the spot price of one unit of
foreign currency in terms of the domestic
currency. RD,T-t the domestic interest rate
covering the contract period. RF,T-t the
foreign interest rate covering the contract
period. Ft,T St(1 RD,T-t)/(1 RF,T-t)
25
This exactly adjusts for differences in
countries' interest rates an approximation
is Ft,T St(1 RD,T-t - RF,T-t)
Like SP futures adjusted for dividends, here
we adjust for the rate earned on the foreign
currency. - if RD RF gt Ft St - if RD gt
RF gt Ft gt St - if RD lt RF gt Ft lt St This
relationship is called interest rate parity.
26
You can always exchange dollars for Euros and get
Euro interest rates so arbitrage forces interest
rates between countries to be the same adjusted
for expected currency depreciation or
appreciation. or, RD,T-t (Ft,T/St)(1
RF,T-t) - 1 Suppose RD increases and RF stays
constant this implies that Ft,T/St increases so
St decreases or Ft,T increases or both.
27
WHAT OFTEN HAPPENS IS St INCREASES BUT Ft,T
INCREASES EVEN MORE. QUESTION Why? ANSWER
Because the spot rate now will be a function of
the expected future spot rate - pure expectation
hypothesis - investors hold currencies that they
expect to appreciate which increases the demand
for them now.
28
EXAMPLE OF INTEREST RATE PARITY AND EXCHANGE
RATES
ASSUME Spot rate of British pound is 1.70 per
pound The annual pound interest rate is RL
.11 The annual dollar interest rate is
R .13 QUESTION What should be the Futures
price of pounds to be delivered in one
year? F 1.70 x (1 .13)/(1 .11) 1.7306
dollars per pound