pricing of forward and futures contracts finance (derivative securities) 312 tuesday, 15 august 2006...
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Pricing of Forward Pricing of Forward and Futures and Futures ContractsContracts
Finance (Derivative Securities) 312
Tuesday, 15 August 2006
Readings: Chapter 5
Consumption v Consumption v InvestmentInvestment
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 SellingShort Selling
Selling securities you do not ownBroker borrows securities from another
client and sells on your behalfObligation to reverse the transaction at a
later dateMust pay dividends and other benefits to
owner of securities
ArbitrageArbitrage
Suppose that:• 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 PricingForward 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 PricingForward Pricing
Arbitrage strategy:
Now After 1 year
Borrow US$390 Sell Gold under Forward
Buy Gold Receive US$425
Short Forward Repay US$409.50
Initial cost of portfolio = 0
CF after one year = US$15.50 (riskless profit)
Forward PricingForward Pricing
If forward price is US$390, reverse strategy:
Now After 1 year
Sell Gold Investment pays $409.50
Invest US$390 Buy Gold under Forward
Long Forward Pay US$390
Initial cost of portfolio = 0
CF after one year = US$19.50 (riskless profit)
Forward PricingForward 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 PricingForward 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 Pricing with Known IncomeIncome
Suppose that:• 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 Pricing with Known IncomeIncome
Arbitrage strategy:• 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 = $23.40What if forward price was $870?
Valuing a Forward Valuing a Forward ContractContract
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 PricesForward 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 IndicesStock 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 IndicesStock 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 ArbitrageIndex 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 ArbitrageCurrency 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:
F S e r r Tf
0 0 ( )
Currency ArbitrageCurrency Arbitrage
1000 units of
foreign currency at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
1000 units of foreign currency
at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
Currency ArbitrageCurrency 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 ArbitrageCurrency 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 AssetsConsumption 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 CarryCost 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 Risk in a Futures PositionPosition
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 Risk in a Futures PositionPosition
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
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