The Pricing of Forward and Futures
Read Hull Chapter 5
Outline
 Short selling
 Arbitrage
 Asset earns income
 Index futures
 Currency forwards and futures
 Storage costs
 The cost of carry
 Arbitrage examples
Short Sellling
 Consumption and Investment Assets
 People hold investment assets purely for investment purposes such as gold and silver
 People hold consumption assets primarily to consume such as copper and oil
 Short Selling
 Investor sells securities he or she does not own
 Investor or broker borrows the securities from another client and sells them in the market in the usual way
 At maturity, investor must buy the securities so he or she replaces the sold asset in the client's account
 He or she must pay dividends and other benefits to the owner of the securities
 Investor doesn't own the securities
 Investor may pay a small fee for borrowing the securities
 Example
 You short 200 shares when the price is $75 and close out the short position six months later when the price is $65
 During the six months, the stock earns a $4 dividend per share
 What is your profit?
 Short  You sell a 200 shares for $75 and receive $15,000
 Dividend  Tricky
 You sold the stock, so the buyer receives the dividend
 You deduct the $4 dividend, or $800
 Buy 100 shares for $90, or $9,000
 Profit = $15,000  $800 – $13,000 = $1,200
 What would be your loss if you had bought 100 shares?
 Loss if purchased long
 You hold the stock and earn the dividends
 Profit = $13,000 + 800  $15,000 =$1,200
Arbitrage
 Notation to value futures and forward contracts
S_{0} 
Spot price toda while subscript 0 means time = 0 
F_{0} 
Futures or forward price today 
T 
Time until delivery date at time T 
r 
Riskfree interest rate until maturity T 
 An arbitrage example
 The spot price of petroleum is $70 per barrel
 The 4month forward price is $75 per barrel
 The 4month U.S. interest rate is 7% per annum
 Does an arbitrage opportunity exist?
 Theoretical price, F_{0}=S_{0} e^{rt}=$70 e^{0.07 (4/12)}=$71.6525
 Arbitrage opportunity if F_{theoretical} does not equal F_{actual}
 Since F_{actual} $75 > theoretical F_{0} $71.6525
 Buy low and sell high
 You short the forward contract for $75
 Borrow money at 7% from the bank, and you will pay 0.6525 in interest
 At inception, the net cash flows equal zero
 Buy the petroleum at $70
 After four months,
 You receive $75 by selling the petroleum via the forward
 repay bank loan with interest of $71.6525
 profit equals $75  $71.6525 = $3.3475
 Another example
 The spot price of nondividendpaying Apple stock is $200
 The 6month forward price is $199
 The 6month interest rate is 5% per annum
 Does an arbitrage opportunity exist?
 Theoretical price, F_{0}=S_{0} e^{rt}=$200 e^{0.05 (6/12)}=$205.0630
 Arbitrage opportunity if F_{actual} does not equal F_{0}
 Since F_{actual} $199 < theoretical F_{0} 205.0630
 Buy low and sell high
 You short Apple stock for $200
 Go long on forward contract to buy Apple stock at $199
 You deposit the $200 at the bank and earn 5% for six months
 At maturity
 You take out 205.0630 from the bank
 Buy stock at $199 and earn $6.0630 in profit
 Then you return the borrowed shares and close short
 Noncontinuous compounding interest
 The spot price of an investment asset equals S_{0}
 The futures price for a contract deliverable in T years is F_{0}
 The riskfree interest rate, r, that is noncontinuous
 Then F_{0} = S_{0}(1+r)^{T}
 In the previous example
 S_{0} = 200, T = 0.5, and r = 0.05 so that F_{0} = 200(1+0.05/2)^{1} = $205.00
 If Short Sales Are Not Possible
 Formula still works for an investment asset
 Investors who hold the asset will sell it and buy forward contracts when the forward price is too low
Asset Earns Income
 Assets earns a known dollar income
 You do not own the asset in the forward contract
 You do not receive that income, so deduct income from equation
 After paying income, it lowers the value of the asset
 Dividend lowers the spot price of stock after a company pays it.
 F_{0} = ( S_{0} – I ) e ^{rT}
 where I is the present value of the income during life of forward contract
 Remember, S_{0} is the spot price today so income must also be valued today
 An investment asset earns a known yield
 The average yield during the life of the contract equals q
 q is continuous compounding
 q could be a stock earning a percent dividend
 Remove impact of income since holder does not earn it
 F_{0} = S_{0} e^{( r–q )T}
 Valuing a Forward Contract
 A forward contract is worth zero except for bidoffer spread effects when it is first negotiated
 Delivery price, K, price specified in the forward contract
 F_{0} – market value of forward
 Must equal each other at Time 0, otherwise, one party pays the other party money.
 Later, forward contract could have a positive or negative value
 Valuing a Forward Contract
 The market value of the forward contract can change over time
 The delivery price equals K at maturity
 The value of a long forward contract
 PV = (F_{0} − K)e^{−rT}
 You can buy at the delivery price, K, and sell at F_{0}
 e^{rT} is to discount a future cash flow to today's value
 The value of a short forward contract
 PV = (K – F_{0} )e^{–rT}
 You can buy at F_{0} and sell at the delivery price, K
 Forward vs Futures Prices
 When the maturity and asset price are the same, forward and futures prices are usually assumed to be equal.
 Eurodollar futures are an exception because the Eurodollar futures uses an equation to calculate the value of a futures contract
 Forward vs Futures Prices
 When interest rates are uncertain they are, in theory, slightly different:
 A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price
 If asset price increases, then interest rates increase too.
 Thus, the spot rate increases.
 The value of the contract increases because holder buys at the K price and sells to the spot rate
 Balance in margin account increases, so holder can withdraw money and earn the higher interest
 Future have higher value than forward
 A strong negative correlation implies the reverse using similar logic
Stock Index
 Treat as an investment asset paying a dividend yield
 The relationship between futures price and spot price
 F_{0} = S_{0} e^{(r–q )T}
 where q is the dividend yield on the portfolio represented by the index during life of contract
 For the formula to be true, the index should represent the investment asset
 Changes in the index must correspond to changes in the value of a tradable portfolio
 Example
 The Nikkei index comprised of 225 Japanese stock is denominated in Japanese yen
 The futures for Nikkei index pays out in U.S. dollars
 Mismatch in price between the spot index and the futures
 Index Arbitrage
 When F_{actual} > S_{0}e^{(rq)T} an arbitrageur:
 Borrows from the bank
 Buys the stocks underlying the index
 Sells futures
 At maturity
 The arbitrageur gives the holder the stocks
 Takes the money,
 Repays the bank loan
 When F_{actual} < S_{0}e^{(rq)T} an arbitrageur:
 Shorts (or sells) the stocks
 Buys futures
 Invest funds in a bank to earn interest.
 On maturity
 The arbitrageur cashes out investment
 Takes the stocks
 Gives stocks to the source where he/she received them, and earns a profit.
 Index arbitrage involves simultaneous trades in futures and many different stocks
 Very often a computer is used to generate the trades
 Occasionally simultaneous trades are not possible and the theoretical noarbitrage relationship between F_{0} and S_{0} does not hold
Currency Futures and Forwards
 A foreign currency is analogous to a security providing a yield
 Exchange rate = home currency / foreign currency
 The yield is the foreign riskfree interest rate
 It follows that if r_{f} is the foreign riskfree interest rate, then r is the interest rate for the home currency
 Discrete from international finance
 The diagram below shows a box diagram.
 Investor starts with $1.
 He or she can invest in the U.S. to earn interest r
 At the end of the investment, investor has e^{rT}
 Investor can invest in a foreign country
 Investor converts $1 to foreign currency by dividing by the spot rate, S_{0}
 Investor earns foreign interest e^{rfT} / S_{0}
 Investor uses currency forward to return money home, which equals F_{0} e^{rft} / S_{0}
 Since investor can invest in domestic or foreign country, arbitrage causes both investment paths to equal
 The pricing of a currency forward is shown below
 Assume the investor believes he or she has the same risk in the foreign country
Storage Costs
 Consumption Assets
 Storage is negative income, the opposite of income
 F_{0} ≤ S_{0} e^{( r+u )T}
 where u is the storage cost per unit time as a percent of the asset value.
 Alternatively, F_{0} ≤ (S_{0}+U )e^{rT} where U is the present value of the storage costs.
The Cost of Carry
 Cost of carry  includes insurance, storage and interest on the invested funds
 For futures markets, it equals the difference between the yield on a cash instrument and the cost of the funds necessary to buy the instrument
 Nondividend paying stock
 Cost of carry is only r
 No income and no storage costs
 Dividend paying stock
 Cost of carry is r – q
 Earns income but no storage costs
 Consumption asset commodity provides income at q
 Includes storage costs, u
 Must pay interest to borrow from a bank, r
 Asset earns q income as a percent
 The cost of carry, c = r + u – q
 Contango – forward price is greater than spot price
 Cost of carry, c, must be positive
 For an investment asset F_{0} = S_{0}e^{cT}
 Consumption assets can have a lower futures price than spot price
 Backwardation
 For a consumption asset F_{0} ≤ S_{0}e^{cT}
 The convenience yield on the consumption asset, y, is defined
 F_{0} = S_{0} e^{(c–y )T}
 Same equation F_{0} ≤ S_{0}e^{cT}
 Implies the riskfree interest rate is negative
 Add a fudge factor, so riskfree interest rate is positive
 Convenience yield – an implied return to the holder because he or she holds inventory
 Asset has frequent shortages, price spikes, or seasonal effects like a dry spell
 Petroleum and agricultural commodities
 Holder does not use the asset for income purposes
 Expected Future Spot Prices
 Suppose k is the expected return required by investors in an asset
 We can invest F_{0}e^{–r T} at the riskfree rate and enter into a long futures contract to create a cash inflow of S_{T} at maturity
 This shows that F_{0}e^{rT} e^{kT} = E(S_{T})
 or F_{0} = E(S_{T})e^{(rk)T}
 Future price depends on a expected spot price at Time T based on the riskfree interest minus any income earned
No Systematic Risk 
k = r 
F_{0} = E(S_{T}) 
Positive Systematic Risk 
k > r 
F_{0} < E(S_{T}) 
Negative Systematic Risk 
k < r 
F_{0} > E(S_{T}) 
 Positive systematic risk: stock indices
 Negative systematic risk: gold (at least for some periods)
Arbitage Examples
 A gold arbitrage opportunity
 The spot price of gold is US $1,700 per ounce
 The quoted 1year futures price of gold is US$1,800
 The 1year US$ interest rate is 5% per annum
 No income or storage costs for gold
 Does an arbitrage opportunity exists?
 Use the formula
 F = S (1+r)^{T} = 1700(1+0.05)^{1} = 1,785
 Strategy  treat this as a riskfree investment
 If I invest $1,700 today, it grows into $1,785 in one year
 Theoretical (calculated) value of futures contract equals $1,785 but actual value is $1,800.
 Buy low and sell high
 Borrow $1,700 today and buy gold
 Sell futures contract In one year
 On maturity
 Sell the gold in the contract
 Repay loan for $1,785
 Profit = 1,800 – 1,785 = $15 per ounce
 Second gold: arbitrage opportunity
 The spot price of gold is US$1,700
 The quoted 1year futures price of gold is US$1,680
 The 1year US$ interest rate is 5% per annum
 No income or storage costs for gold
 Does an arbitrage opportunity exists?
 Actual futures price < theoretical futures price
 Short gold today at spot price $1,700 i.e. sell gold in the future
 You do not have gold now
 Invest money at 5%
 Buy futures contract and agree to buy $1,680
 On maturity
 You replace the gold In one year
 Buy gold at $1,680
 Close short position
 Receive $1,785 in interest
 Profit = $1,785  $1,680 = $105 per ounce
 An oil arbitrage opportunity
 The spot price of oil is US$80
 The quoted 1year futures price of oil is US$90
 The 1year US$ interest rate is 5% per annum
 The storage costs of oil are 2% per annum
 Does an arbitrage opportunity exists?
 Caculate theoretical price
 F_{0} = S_{0} e^{(r+u)T} = $80 e^{(0.05+0.02)(1)} = $85.80
 Actual $90 > theoretical $85.80
 Buy low and sell high
 Sell the futures for oil
 Borrow funds to buy the oil
 On maturity
 Sell the oil via the futures
 Repay the bank
 Pay the storage costs
 UC has never asked this type of problem
 An oil arbitrage opportunity
 The spot price of oil is US$80
 The quoted 1year futures price of oil is US$75
 The 1year US$ interest rate is 5% per annum
 The storage costs of oil are 2% per annum
 Does an arbitrage opportunity exists?
 Caculate theoretical price F_{0} = S_{0} e^{(r+u)T} = $80 e^{(0.05+0.02)(1)} = $85.80
 Actual $75 < theoretical $85.80
 Buy low and sell high
 Buy the futures for oil
 Short the oil
 Invest the funds to earn interest
 On maturity
 Buy the oil via the futures
 Repay the bank
 Collect the storage costs and close the short
 UC has never asked this type of problem

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