- Commission Brokers = execute
for others and earn comissions
- Locals = trade for their own
account
- Market order = get me the
contract at whatever its trading at
- Limit order = get me the contract
at my price or better
- Close out = Enter into opposite
trade and closeout. Close out by delivery of underlying
asset is rare.
Future
contract features
The Asset
Possible to have different grades
of assets. Prices may be adjusted depending on the
grade of asset delivered
True for commodities but financial
assets generally don't follow this trend with the
exception of future contracts on Treasuries
Contract size
If its too small execution of
large hedges would be difficult because of the per
contract transaction cost
If its too large, you would exclude
small investors or investors who are looking for small
hedges
Delivery Terms
Location and time. Delivery month
is specified but exact time is not. Maybe the complete
or a shorter period.
Price Quotes
Decimals or 32nd of a dollar
Daily Price & Position
Limits
Used to discourage speculative
activity. Price cannot move beyond the daily limit
during a single day trading. Speculators cannot go
beyond position limits
Margins
An amount on deposit with broker.
All gains or losses to your account are credited to
your margin account.
Mark to market
Effectively a daily settlement
of the contract. All gains and losses to your account
are charged to your margin account.
Initial margin
The initial deposit required
by the broker on the orgination of the contract
Maintenance margin
Less than the initial margin.
If amount in the margin account falls below the maintenance
margin, a margin call is generated
Margin Call
Request to bring up the margin
account to its initial level. Non compliance leads
to closing of the position and the account by your
broker
Variation margin
The extra amount deposited on
a margin call
Overall result
Reduce the risk of default by
reducing the settlement period and the size of the
settlement by using daily mark to market and margins.
Day trade
Close out position on the same
day
Spread Transaction
Two positions, one long, one
short, on different dates on the same underlying asset
There is no difference between
taking a long position or a short position on a future
exchange. This is not always true for cash/basic securities
markets
Clearing Margins
Margin maintained by the broker
at the clearinghouse or with one of its members. Same
as the ordinary margin account except that there are
no maintenance margins.
Convergence
of Spot and Future Prices
Case A: Future price is higher
than Spot during delivery period
1. Sell (Short) a future contract
2. Buy the asset
3. Make delivery
Case B: Future price is lower
than Spot during delivery period
1. Buy (long) a futures contract
2. Take delivery
3. Sell asset
In both cases you make a riskless
profit
Underlying asset refers to assets
underlying the futures contract
Regulation
CFTC (Commodity Future Trading
Commission). Regulates future markets
NFA (National Futures Association).
Industry Body
SEC, Federal Reserves Board and
US Treasury Department
Hedging
Why hedging using futures is not
perfect in practice
1. Underlying asset is different
from hedged asset
2. Mismatch between the settlement
dates of the two assets (the hedged asset and the
underlying asset)
Basis Risk
Basis = Spot price of hedged asset
- Future Price (FP) of contract
When hedged asset and underlying
asset are the same basis is zero at contract orgination
When spot increases more than
FP basis increases = strengthening of the basis
When FP increases more than
Spot, basis decreases = weakening of the basis
SOne = Spot
at time one
STwo = Spot
at time two
FOne = FP at
time one
FTwo = FP at
time two
BOne = Basis
at time one
BTwo = Basis
is time two
Effective purchase price
when purchase was hedged with futures = STwo + FOne
- FTwo = FOne + BTwo
Basis risk is the risk associated
with BTwo
For investment assets (currencies,
stock indices, precious metals) basis risk tends to
be low except under certain conditions.
For commodities basis risk is
higher
When underlying asset and hedged
asset are different
STwoStar = Spot of underlying
asset
Effective purchase price
= FOne + (STwoStar - FTwo) + (STwo - STwoStar)
The first term in brackets is
the traditional term, the second term results from
the difference between assets
Basis risk is affected by
Choice of underlying asset
Choice of delivery month
Optimal
Hedge Ratio
Hedge Ratio = Ratio of position
taken in futures contract to the size of the exposure
DeltaS = change
in spot price
DeltaF = change in
futures price
SigmaS = standard deviation
of DeltaS
SigmaF = standard deviation
of DeltaF
Row
= correlation coefficient for DeltaS and DeltaF
h
= hedge ratio
optimal hedge ratio = Row
x SigmaS / SigmaF
Rolling
Hedges
If the hedging period is longer
than the maturity of any available futures contract
then you may try the following
1. Hedge for the maximum maturity
available
2. When the contract matures
roll it forward for another term
3. Keep on repeating (2) till
you hit the end of the hedging period
The problem is that you now
introduce basis risk for each of the roll overs. If
there are N roll overs, there are now (N-1) sources
of basis risk. There is also the rollover basis which
refers to the price difference between the contract
that just matured and an equivalent new contract
Accounting
FASB Statement 52, Foreign Currency
Translation covers foreing currency futures
FASB Statement 80, Accounting for
Futures Contracts cover all other contracts within US
Change in market value is recognized
when its occurs unless the contract qualifies as a
hedge
If contract qualifies as an hedge,
gain or loss is recognized in the same period as the
gain and loss for the hedge is recognized
Tax
- Nature of gain or loss
- Timining of the recognition
of gain or loss
- Capital Gains are taxed at
the same rate as income
Introduction
& Notation
Continous compounding
Short Selling
Involves selling securities that
you don't own and then buying them back later.
Do this by borrowing shares from owners who are willing
to lend them to you. Short Squeezed is when you run
out of shares to borrow. Can only short shares on
an uptick (price moved up). Dividends on borrowed
shares have be to be paid back to the original owner.
Assumptions
- No transaction costs
- Same tax rate
- freely borrow and lend money
at the risk free rate
- arbitrage opportunities
can be exploited
Assumptions don't need to hold
for the whole market. As long as it holds for a small
section we are happy.
Repo Rate
Sell a security now and buy it
at a slightly higher price a little later. Difference
between the two prices is the interest. Effectively
a short term secured loan where the securities serve
as collateral. Generally low risk. Slightly higher
than treasury rate
Notation
T = maturity
t = current time
S = price of underlying
asset at time t
ST = price of underlying asset
at time T
K = Delivery price/exercise price
f = value of long forward contract
F = forward price at time t
r = annual risk free rate of
interest
T-t is measured in years
The forward price is F is different
from the value of the forward contract f.
F is the delivery price that
would make the contract have a zero value.
At orgination K=F and f = 0
Forward
on security with no income
F = Spot * exp [r
* (T-t)]
if F > Spot * exp[(r
* (T-t)] then
1. Borrow S
2. Buy the Asset
3. Short the forward contract
4. At Maturity settle the forward
contract with the asset and use the proceeds to repay
the loan.
5. Profit = F - Spot * exp[r
* (T-t)]
If F < Spot * exp[r * (T-t)]
then
1. Short the Asset
2. Take a long position in the
forward contract
3. Invest the proceeds (Spot)
from the short sale at rate r
4. At Maturity use the forward
contract to close the short asset position
5. Profit = Spot * exp [r * (T-t)]
- F
Forward
on security with known cash income
Stocks paying known dividends
or coupon bearing bonds
F = (Spot - Income) * exp[
r * (T-t)]
Same logic as above if the relationship
does not hold.
Forward
on security with known dividend yield
Currencies and stock
indices
F = Spot * exp[ (r-dividend)
* (T-t)]
General
Result
Value of Forward Contract
= f = (F - K) * exp[ -r * (T-t)]
When the risk free rate is constant
for all maturities the forward price is equal to the
future price, everything else being the same. When
rates fluctuate a lot the relationship does not hold
anymore.
If you are holding a financial
asset whose value is positively correlated with interest
rates. A forward contract would be unaffected by any
short term fluctuations in interest rates since there
is no daily settlement. However a long future contract
would move with the interest rates (rates go up, asset
goes up, future contract goes up and vice versa).
When Spot is positively correlated
with interest rates, future prices are higher than
forward prices. When Spot is negatively correlated
with interest rates, future prices are lower than
forward prices
Future
Price of Stock Indices
F = Spot * exp [ (r-dividend)
* (T-t)]
If the above equation does not
hold you can do index arbitrage. If F is less than
the right hand side, Index arb is done by a
pension fund (index funds); If F is greater than the
right hand side, index arb is done by a corporation
(short term money market account)
Growth rate of index future
prices = excess return over the risk free rate on
the index
Hedging
Using Index Futures
Pi
= value of the portfolio
F = mF
Optimal number of contracts
= Beta * Pi / F
Adjusting
Betas
(Beta - BetaStar) * Pi
/ F
When BetaStar > Beta then
you short
Absolute [(Beta - BetaStar)
* Pi / F] number of contracts
Not applicable on any foreign
index (local currency) that needs to be converted
to the base currency (Dollars)
Forward
and Futures Contract on Currencies
F = Spot * exp[ (r
- risk free rate in foreign currency) * (T - t)]
Futures
on Commodities
Commodities
held for investment
F = Spot * exp[
r * (T - t)]
F = (Spot + Storage Cost)
* exp [ r * (T - t)]
F = Spot * exp [ (r + Storage
Cost) * (T - t) ]
Commodities
held for consumption
You can't use the arbitrage
pricing argument in this case since owners of commodities
held for consumption will not buy or sell them for
investment reasons.
F = Spot * exp [ ( r +
Storage Costs - Convenience Yield) * (T - t) ]
The convenience yield measures
expectations regarding future availability
Cost of carry = c = storage
cost + interest - income earned
Investment Asset
F = Spot * exp
[ c * (T -t)]
Consumption Asset
F = Spot * exp
[ (c - y) * (T - t)]
Delivery Choices
If prices are going
up deliver early
If prices are going down deliver
late
Future
Price and Future spot
If hedgers hold short
positions and speculators hold long positions, future
prices will be below expected future spot
If positions are reversed then
future prices will be above expected future spot
Normal backwardation = future
price is below future spot
Contango = future price is above
future spot
Risk and Rewards
F = Expected Spot * exp
[ (r - k) * (T - t)]
three cases
k = r implies F
= Expected Spot
k > r implies F <
Expected Spot
k < r implies
F > Expected Spot
Interest
Rate Futures
Contract on an asset
whose value is dependent on interest rates
Hedging Interest Rate risk is
more complicated than hedging commodity risk. An additional
variable/complication is the term structure of interest
rates (rates and maturities) that needs to be defined
before the risk can be hedged where as in commodities,
the price of the commodity is sufficient
N Year Spot = SpotN = rate for
investing now for n years = N year zero coupon yield
R = Spot for T years
RStar = Spot for TStar > T
RTilda = Forward rate between
T and TStar =
(RStar * TStar -
RT) / (TStar - T)
Zero
Coupon Yield Curve
Curve showing relationship
between Spot rates and maturity. Different from the
Coupon Bearing Yield Curve
If you have an upward sloping
yield curve
Forward Rate Curve > Zero Coupon
Yield Curve > Coupon Bearing Yield Curve
If you have a downward sloping
yield curve
Coupon Bearing Yield Curve >
Zero Coupon Yield Curve > Forward Rate
Boot
Strap Method
1. use discount bonds
to calculate annualized Spot rates for short maturities
(upto one year)
2. use rates in (1) and coupon
bearing bonds to calculate spot rates for longer maturities
Day
Count Conventions
X / Y
X = number of days between two
dates
Y = number of days in the reference
period
1. Actual / Actual
Treasury Bonds
2. 30 / 360
Corporate and muncipal bonds
3. Actual / 360
Treasury bills and money
market instruments
Term
Structure Theories
Expectations Theories
Long term interest rates reflect
expected future short term interest rates. Forward
rate = expected future spot
Market Segmentation Theory
No relationship betwen short,
medium and long term interest rates. Different institutions
prefer different maturities, invest in their preferred
maturities and do not switch maturities
Liquidity Preference Theory
Forward rates would always be
higher than expected future spot. Investors prefer
to invest for short periods, borrow prefer to borrow
at fixed rates for long maturities. To match investors
with borrowers and avoid interest rate risk, financial
intermediaries raise long term rates relative to expected
future short term interest rates
Forward Rate
Agreements
Agree now that a rate
will apply in the future on pre agreed principal and
term
Generally settled at the begining
of the pre agreed period.
RK = Pre agreed rate between
T and TStar
RK = (RStart * TStar - RT) /
( TStar - T)
Value of FRA is zero whene RK
equals the period between T and TStar.
Value of FRA =
CashFlow * Accumulated at
FRA rate * Discounted at Forward Rate;( for TStar
- T) - CashFlow * Discounted at Spot Rate; (
for T - t)
Treasury
Bond and Treasury Note Futures
T Bond
Any government bond with more than
15 years to go (maturity or callable)
T Notes
Any government bond with maturity
with 6.5 and 10 years can be delivered.
Quotes are in dollars and 32nds
of a dollar
Quoted price is different from
the cash price. Quoted is the clean price & the
cash price is the dirty price
Cash price = quoted price
+ accrued interest since last coupon date
Conversion
Factor
Cash received by party
with short position = quoted price * conversion factor
+ accrued interest rate
Conversion factor = Value of
the bond assuming flat rate of 8% (semi annual compounding)
for all maturities
exact number of half years -
first coupon paid in six month
if we don't have an exact number
of six months - first coupon is paid after three
months
Cheapest
to deliver bonds
For the shrot party the cheapest
to deliver bond is the bond for which
Cost of purchase - Proceeds
from sale is the minimum
Quoted Price + accrued interest
- [(quoted future price * conversion
factor) + accrued interest] =
Quoted Price - (quoted future
price * conversion factor)
When yields > 8% the
system favours low coupon long maturity
bonds
When yields < 8% the system
favours high coupon, short maturity bonds
Yield curve is upward sloping,
favours long maturity
Yield curve is downward sloping
favours short maturtity
Certain bonds sell for more
than their theoretical value. Unlikely to
be cheapest to deliver under all circumstances
The
wild card play
Trading at CBOT ceases
at 2 pm (Chicago time). TBonds trade till 4 pm. Short
position holder has till 8 pm to issue intention to
deliver
If we know the cheapest to deliver
bond and the delivery date then
Future Price = F= (Spot -
Income) * exp [ r * (T - t)] = equation 4.4
F = cash future price S = cheapest
to deliver bond
Step One =
cash price of cheapest to derliver bond
Step Two =
cash future price from cash bond price from equaiton
4.4
Step three = quoted future
price from cash future price
Step four =
Divide quoted future price by conversion factor
Treasury
Bill Futures
Underlying asset is a
90 day treasury bill.
Deliver day is the issue day
of a 13 week treasury bill and a one year TBill has
13 weeks to expiry. TBill may have 89, 90 or 91 days
to expiry
Future price = F = Spot *
exp [ -r * (TStar - T)]
Arbitrage when forward interest
rates implied by Treasury bills is different from
the rate implied by the TBill futures contract
Arbitrage
Buy the asset that is
priced cheaply = Borrow/ Long at the forward rate which
is lower
Sell the asset that is priced expensively
= Short/Lend the forward rate which is higher
Type One = Short the future
and borrow using the treasury contracts (borrow at
first interval, invest at second interval)
Type Two = Long the future contract,
borrow money using the treasury contracts (borrow
at second interval, lend at first interval)
Involve borrowing at
or close to the Treasury bill rate. Do that using the
repo transaction
Calculate the implied repo rate
= rate of interest on a short term treasury bill implied
by the future price for a contract maturing at the
same time as the short term treasury bill and the
price of a treasury bill maturing 90 days later than
the short term treasury bill.
If the implied repo rate is
greater than the actual short term TBill rate, a Type
One arb is in priniciple possible. If the implied
repo rate is less than the short term TBill rate,
a Type Two arb is possible
Quotes
for TBills
TBill are quoted as
360/n [100 - Y] where Y is the
cash price
Discount rate, it is the annualized
dollar return provided by the treasury bill in 360
days expressed as a percentage of the face value.
This is not the same as the yield on the Bill.
Treasury Bill Future Price quote
= 100 - Treasury Bill price quote
Euro
Dollar Futures
Difference between a Euro
dollar future quote and a treasury bill quote. For
a TBill price converges to 90 day face value TBill.
An ED is settled in cash
The quoted ED rate is the
actual 90 day rate on ED deposits with quaterly compounding.
It is not a discount rate
The ED future contract is
a contract on an interest rate while the TB future
contract is a future contract on the price of a TBill
or a discount rate
Duration
Percentage in Bond Value
= - Duration of Bond * change in yield curve
Duration = Summation [ time
* cashflow * discount factor] / Bond Value
Modified duration = Duration
/ ( 1+ Y/m)
Duration based hedge ratio =NStar
=
[Value of Asset being Hedged
* Duration of Asset being Hedged] /
[ Contract price for irate
future contract * Duration of asset underlying future
contract]