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# Futures Valuation

advertisement ```Determination of Forward
and Futures Prices
Chapter 5
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.1
Consumption vs Investment Assets

Investment assets are assets held by significant
numbers of people purely for investment purposes
(Examples: stock, bonds, gold, silver)

Consumption assets are assets held primarily for
consumption (Examples: copper, crude oil, corn,
etc.)

Use arbitrage argument to determine forward
and futures prices of investment asset.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.2
Short Selling


Short selling involves selling
securities you do not own
Your broker borrows the
securities from another client and
sells them in the market in the
usual way
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.3
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 and
other benefits the owner of
the securities receives
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.4
Example




Consider the position of an investor who shorts 500
IBM shares in April, when the price is \$120 per
share
He closes out the position by buying them back in
July when the price per share is \$100
A dividend of \$1 per share is paid in May
The net gain of the investor is?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.5
Example


The net gain of the investor is 500*\$120-500*\$1500*\$100=\$9,500
The investor is required to maintain a margin
account with the broker.
Note: Net gain of the trader with long position:
500*100-500*120+500 = (\$9500)

Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.6
Assumptions
1 The market participants are subject to no
transaction costs when they trade
2 The market participants are subject to the same
tax rate on all net trading profits (for simplicity we
assume tax rate to be zero)
3 The market participants can borrow and lend at
the same risk-free rate of interest
4 The market participants take advantage of
arbitrage opportunities as they occur . No
restrictions on short-selling.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.7
Notation for Valuing Futures and
Forward Contracts
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for
maturity T (assume continuous
compounding)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.8
Valuation of forward/futures
contracts: Basic Principles

Forwards and futures have net zero
value at the time of inception.

Both long position and short position
should have ZERO value at the time
contract is struck.

Forwards and futures are zero-sum
game i.e. gain on long contract =
loss on short contract and vice versa
5.9
Valuation of forward/futures
contracts: Basic Principles

Imp: There should be no arbitrage
opportunity during the life of the
contract
Explain Arbitrage

Hence, forwards and futures are
valued under the assumption of
NO-ARBITRAGE
5.10
Valuation of forward/futures
contracts: Basic Principles contd..

At the time contract is struck the future cash
flows are known with certainty!
Example:
A trader is considering purchasing a stock and
hedge it using a forward contract.

5.11
Valuation of forward/futures
contracts: Basic Principles contd..
The trader borrows the spot price and buys 1
share of ABC Ltd and shorts 1 forward contract
(lot size is 1 share).
Other Information:
Spot price = Rs 100/share; risk-free interest
rate = 4.88%; expiry of forward/futures contract
= 1 year
5.12
Valuation of forward/futures
contracts: Basic Principles contd..
Trader’s Strategy: To hedge the risk of this
portfolio (i.e. long 1 share) the trader shorts 1
futures contract at price F0.
F0 is the price at which shares will be delivered
at time T. Assume a non-dividend paying stock.
Question: What should be the no-arbitrage
price of the forward contract?
5.13
Valuation of forward/futures
contracts: Basic Principles contd..
What are the cash flows at time T?
From hedging =>
cash inflow = F0
From borrowing =>
cash outflow = 100*e(4.88%*1)
5.14
Valuation of forward/futures
contracts: Basic Principles contd..
Under no-arbitrage argument these two cash
flows should be equal
i.e. F0 = 100*e(4.88%*1)
F0 = Rs 105
What if F0=Rs 95?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.15
Valuation of forward/futures
contracts: Basic Principles contd..
What if F0=Rs 95?
Trader will:
 long forward contract to buy 1 share at Rs 95 on
expiry.
 will short 1 share for Rs 100
 will lend Rs 100 for 1 year at 4.88%
5.16
Valuation of forward/futures
contracts: Basic Principles contd..
Pay-off on expiry:
Cash inflow = 100*e(4.88%*1) = Rs 105
Cash outflow = Buy share at the pre-determined
forward price of Rs 95 and cover the short position.
Risk-less gain = Rs 10
5.17
Valuation of forward/futures
contracts: Basic Principles contd..
To summarize:
Under no-arbitrage the present value of the future
cash flows discounted at the risk-free rate should be
the Spot Price of the asset.
i.e. S0=F0e-(r*t)
And, F0=S0e(r*t)
Since, future cash flows are known with certainty the
future cash flows are risk-less and hence must be
discounted using the risk-free rate.
5.18
1. Gold: An Arbitrage
Opportunity?


Suppose that:
 The spot price of gold is US\$390
 The quoted 1-year forward price of
gold is US\$425
 The 1-year US\$ interest rate is 5% per
annum
 No income or storage costs for gold
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.19
2. Gold: Another Arbitrage
Opportunity?


Suppose that:
 The spot price of gold is US\$390
 The quoted 1-year forward price of
gold is US\$390
 The 1-year US\$ interest rate is 5%
per annum
 No income or storage costs for gold
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.20
The Forward Price of Gold
If the spot price of gold is S and the futures price
for a contract deliverable in T years is F, then
F = S0erT
where r is the 1-year risk-free rate of interest.
In our examples, S=390, T=1, and r=0.05 so that
F = \$410
This is the theoretical forward price of gold.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.21
Another Example
Consider a stock that is currently selling for
\$30
 The quoted 2-years forward price of the
stock is \$35
 The 2-years US\$ interest rate is 5% per
annum
 This means that the stock can be bought
or sold for 35 with delivery in two years
Is there an arbitrage opportunity?


Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.22
Arbitrage Positions:

1.
2.
3.



Arbitrageurs can adopt the following strategy:
Borrow \$30 at 5% for two years
Buy 1 share of the stock
Enter into a short forward contract to sell 1 share for
\$35 in two years
In two-years \$30e0.05*2=33.16 is required to repay
the loan.
The arbitrageur can sell the shares for \$35. The
result is a riskless profit of \$1.84.
It is easy to see that any forward price above \$33.16
gives rise to this type of riskless arbitrage profit
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.23
Continued..
Consider a stock that is currently selling for
\$30
 The quoted 2-years forward price of the
stock is \$31
 The 2-years US\$ interest rate is 5% per
annum
 This means that the stock can be bought
or sold for \$31 with delivery in two years
Is there an arbitrage opportunity?


Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.24
Arbitrage Positions:

1.
2.
3.



Arbitrageurs can adopt the following strategy:
Short 100 shares of the stock
Invest the proceeds (3000) at 5% for two-years
Enter into a long forward contract to repurchase 100
shares for 3100 in two-years
In two-years the 3000 investment grows to
3000e0.05*2=3316.
100 shares are purchased for 3100 and the short
position is closed out. The result is a riskless profit of
216.
This riskless arbitrage opportunity is available
whenever the forward price is below 33.16
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.25
Arbitrage

Riskless arbitrage opportunities, if they are
observed at all, rarely last for long

As traders take advantage of these arbitrage
opportunities, the forward price will be driven
up or down and the arbitrage opportunity will
disappear.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.26
Generalization:



The relationship between F0 and S0 is:
F0 = S0erT
When F0 > S0erT an arbitrageur can buy the
asset and short forward (futures) contacts
on the asset
When F0 < S0erT an arbitrageur can short
the asset and buy forward (futures)
contracts on it
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.27
Example

Consider a four-month forward contract to buy a
zero-coupon bond that will mature one year from
today

The current price of the bond is \$930

The 4-months interest rate is 6% per annum

Estimate the theoretical forward price?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.28
Example

The forward price is obtained by
F0= P0er*T = 930e0.06*4/12 =\$948.79
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.29
When an Investment Asset
Provides a Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income
during life of forward contract
Examples are stocks paying dividends
and coupon bearing bonds.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.30
Example





Consider a one-year forward contract to purchase
a coupon-bearing bond whose current price is
900, that will mature in five years
Coupon payments of 40 are expected after 6 and
12 months
Six-month and one-year riskless rates are 9%
and 10% respectively
So, S0=900, I=40e-0.09*0.5 + 40e-0.1*1 =74.433, and
F0 = (S0-I)er*T=(900-74.433)e0.1*1 =912.39
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.31
Arbitrage

If F0>(S0-I)er*T, an arbitrageur can lock in a
profit by buying the asset and shorting a
forward contract on the asset.

If F0<(S0-I)er*T, an arbitrageur can lock in a
profit by shorting the asset and taking a
long position in a forward contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.32
Example





Consider a 10-month forward contract on a stock
with price of 50
The riskless rate is 8% per annum for all maturities
Dividends of 0.75 per share are expected after
three, six and nine months
The present value of the dividends, I, is given by
I=0.75e-0.08*3/12+0.75e-0.08*6/12+0.75e-0.08*9/12 =2.162
The forward price is given by
F0 = (S0-I)er*T=(50-2.162)e0.08*10/12 =51.14
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.33
When an Investment Asset
Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the
life of the contract (expressed with
continuous compounding)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.34
Example





Consider a 6-month forward contract on a
stock with price of 25
The riskless rate is 10% per annum
The stock is expected to provide income
equal to 2% of the stock price in 6 months.
The yield is 3.96% per annum with continuous
compounding.
The forward price is given by
F0 = S0e(r-q)*T=25e(0.10-0.0396)*0.5 =25.77
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.35
Valuing a Forward Contract



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, ƒ, is
ƒ = (F0 – K )e–rT
Similarly, the value of a short forward contract
is
(K – F0 )e–rT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.36
Example





Consider a six-month long forward contract on a
non-dividend paying stock, that was entered into
some time ago
The stock price is 25, and the delivery price is 24
The risk-free rate of interest is 10% per annum
Then, the forward price, F0, is given by
F0 = S0er*T= 25e0.1*0.5=26.28
The value of the forward contract is
f = (F0-K)e-r*T = S0-Ke-r*T=25-24e-0.1*0.5=2.17
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.37
Example

Consider A Forward Contract That Has A Term Of 2
Years. The Price Of The Asset Underlying The
Contract Is Currently \$200 And The Risk-Free Rate
Is 9%. Given The Forward Price Of \$220, The Value
Of The Forward Contract At Initiation Is Closest To ?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.38
Example

Consider A Forward Contract That Has A Term Of 2
Years. The Price Of The Asset Underlying The
Contract Is Currently \$200 And The Risk-Free Rate
Is 9%. Given The Forward Price Of \$220, The Value
Of The Forward Contract At Initiation Is Closest To ?

200-220*exp(-9%*2) = 16.24
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.39
Stock Index


Can be viewed as an investment asset
paying a dividend yield
The futures price and spot price
relationship is therefore
F0 = S0 e(r–q )T
where q is the average dividend yield on
the portfolio represented by the index
during life of contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.40
Example





Consider a 3-month futures contract on the
S&P 500
The riskless rate is 6% per annum
The stocks underlying the index provide a
dividend yield of 1% per annum.
The current value of the index is 800
The futures price is given by
F0 = S0e(r-q)*T=800e(0.06-0.01)*0.25 =810.06
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.41
Index Arbitrage


When F0 > S0e(r-q)T an arbitrageur buys the
stocks underlying the index and sells
futures
When F0 < S0e(r-q)T an arbitrageur buys
futures and shorts or sells the stocks
underlying the index
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.42
Futures and Forwards on
Currencies



A foreign currency is analogous to a security
providing a 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
F0  S0e
( r rf ) T
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.43
Why the Relation Must Be True
Figure 5.1, page 113
1000 units of
foreign currency
at time zero
1000 e
rf T
units of foreign
currency at time T
1000 F0 e
rf T
dollars at time T
1000S0 dollars
at time zero
1000 S 0 e rT
dollars at time T
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.44
Example



Suppose the 2-year interest rate in Australia
and the US are 5% and 7% respectively
The spot exchange rate is 0.62 USD for 1
AUD
The 2-year forward exchange rate should be
F0 = S0e(r-rf)*T= 0.62e(0.07-0.05)*2 =0.6453
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.45
Arbitrage Position 1:

1.
2.




Suppose the 2-year forward exchange rate is less than this,
say 0.63. An arbitrageur can:
Borrow 1000 AUD at 5% per annum for 2-years, convert to
620 USD and invest the USD at 7%
Enter into a forward contract to buy 1,105.17 AUD for
1,105,17*0.63=696.26 USD
The 620 USD grow to 620e0.07*2=713.17 USD.
696.26 USD are used to purchase 1,105,17 AUD under the
terms of the forward
Repay the principal and interest on the 1000 AUD that are
borrowed (1000e0.05*2=1,105.17)
This riskless profit is 713.17-696.26=16.91 USD
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.46
Arbitrage Position 2:

1.
2.




Suppose the 2-year forward exchange rate is greater than
this, say 0.6600. An arbitrageur can:
Borrow 1000 USD at 7% per annum for 2-years, convert to
1000/0.62=1,612.90 AUD and invest the AUD at 5%
Enter into a forward contract to sell 1,782.53 AUD for
1,782.53*0.66=1,176.47 USD
The 1,612.90 AUD grow to 1,612.90e0.05*2=1,782.53 AUD.
The forward contract has the effect of converting this to
1,176.47 USD
Repay the principal and interest on the 1000 USD that are
borrowed (1000e0.07*2=1,150.27)
This riskless profit is 1,176.47-1,150.27=26.20 USD
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
5.47
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