Transcript Pricing of Forward and Futures Contracts

Pricing of Forward and
Futures Contracts
Finance (Derivative Securities) 312
Tuesday, 15 August 2006
Readings: Chapter 5
Consumption v Investment
Investment assets are assets held by
significant numbers of people purely for
investment purposes (eg gold, silver)
Consumption assets are assets held
primarily for consumption (eg copper, oil)
Short Selling
Selling securities you do not own
Broker borrows securities from another
client and sells on your behalf
Obligation to reverse the transaction at a
later date
Must pay dividends and other benefits to
owner of securities
Arbitrage
Suppose that:
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•
•
•
Spot price of gold is US$390
1-year forward price of gold is US$425
1-year $US interest rate is 5% p.a.
No income or storage costs for gold
Is there an arbitrage opportunity?
What if the forward price is US$390?
Forward Pricing
If spot price of gold is S & futures price for
a contract deliverable in T years is F, then
F = S (1 + r)T
where r is the 1-year (domestic currency)
risk-free rate of interest
From previous example, S = 390, T = 1,
and r = 0.05 so that
F = 390(1 + 0.05) = $409.50
Forward Pricing
Arbitrage strategy:
Now
Borrow US$390
Buy Gold
Short Forward
After 1 year
Sell Gold under Forward
Receive US$425
Repay US$409.50
Initial cost of portfolio = 0
CF after one year = US$15.50 (riskless profit)
Forward Pricing
If forward price is US$390, reverse
strategy:
Now
Sell Gold
Invest US$390
Long Forward
After 1 year
Investment pays $409.50
Buy Gold under Forward
Pay US$390
Initial cost of portfolio = 0
CF after one year = US$19.50 (riskless profit)
Forward Pricing
If using continuous compounding, then:
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
Forward Pricing
If asset provides a known income:
F0 = (S0 – I)erT
where I is the present value of the income
If asset provides a known yield:
F0 = S0e(r–q)T
where q is the average yield during the life of the
contract (continuous compounding)
Pricing with Known Income
Suppose that:
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Coupon bearing bond is selling for $900
Coupon of $40 expected in four months
9-month forward price of same bond is $910
4- and 9-month rates are 3% and 4%
What is the arbitrage strategy?
Pricing with Known Income
Arbitrage strategy:
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Borrow cash, buy bond, short forward
PV of coupon = 40e–0.03(4/12) = $39.60
Borrow $39.60 at 3% for four months
Borrow remaining $860.40 at 4% for nine
months
• Repay 860.40e0.04(9/12) = $886.60, receive
$910 under forward contract, profit = $23.40
What if forward price was $870?
Valuing a Forward Contract
Suppose that:
• K is delivery price in a forward contract
• F0 is forward price that would apply to the
contract today
Value of a long forward contract, ƒ, is
ƒ = (F0 – K)e–rT
Value of a short forward contract is
ƒ = (K – F0)e–rT
Forward v Futures Prices
Forward and futures prices usually
assumed to be the same
When interest rates are uncertain then:
• strong positive correlation between interest
rates and asset price implies futures price is
slightly higher than forward price
• strong negative correlation implies the reverse
Stock Indices
Can be viewed as an investment asset
paying a dividend yield
The futures price and spot price
relationship is therefore
F0 = S0e(r–q)T
where q is the dividend yield on the
portfolio represented by the index
Stock Indices
For the formula to be true it is important
that the index represent an investment
asset
Changes in the index must correspond to
changes in value of a tradable portfolio
Nikkei Index viewed as a dollar number
does not represent an investment asset
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 the stocks underlying
the index
Currency Arbitrage
Foreign currency is analogous to a
security providing a dividend yield
Continuous dividend yield is the foreign
risk-free interest rate
If rf is the foreign risk-free interest rate:
F0  S0e
( r rf ) T
Currency Arbitrage
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
Currency Arbitrage
Suppose that:
• 2-year rates in Australia and the US are 5%
and 7% respectively
• Spot exchange rate is AUD/USD = 0.6200
• 2-year forward rate: 0.62e(0.07–0.05)2 = 0.6453
What if the forward rate was 0.6300?
Currency Arbitrage
Arbitrage strategy:
• Borrow AUD1,000 at 5% for two years,
convert to USD620, and invest at 7%
• Long forward contract to buy AUD1,105.17 for
1,105.17 x 0.63 = USD696.26
• USD620 will grow to 620e0.07x2 = 713.17
• Pay USD696.26 under contract
• Profit = 713.17 – 696.26 = USD16.91
Consumption Assets
F0  S0e(r+u)T
where u is the storage cost per unit time as a
percent of the asset value
Alternatively,
F0  (S0+U)erT
where U is the present value of the storage
costs
Cost of Carry
Cost of carry, c, is storage cost plus
interest costs less income earned
For an investment asset F0 = S0ecT
For a consumption asset F0  S0ecT
Convenience yield on a consumption
asset, y, is defined so that
F0 = S0e(c–y)T
Risk in a Futures Position
Suppose that:
• A speculator takes a long futures position
• Invests the present value of the futures price,
F0e–rT
• At delivery, speculator buys asset under
contract, sells in market for (expected) higher
price
Value of the investment?
Risk in a Futures Position
PV of investment = –F0 e–rT + E(ST )e–kT
If securities are priced based on zero
NPVs, then the PV should equate to zero
F0 = E(ST)e(r–k)T
Expected Future Spot Prices
If asset has
• no systematic risk, then k = r and F0 is
an unbiased estimate of ST
• positive systematic risk, then k > r and
F0 < E(ST)
• negative systematic risk, then k < r and
F0 > E(ST)