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 Selling

  • 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 200 shares for $65, or -$13,000
        • Profit = $15,000 - $800 – $13,000 = $1,200
      • What would be your loss if you had bought 200 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 Risk-free interest rate until maturity T
  • An arbitrage example
    • The spot price of petroleum is $70 per barrel
    • The 4-month forward price is $75 per barrel
    • The 4-month 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 $70 at 7% from the bank, and you will pay 1.6525 in interest
        • Buy the petroleum at $70
        • At inception, the net cash flows equal zero
      • 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 nondividend-paying Apple stock is $200
    • The 6-month forward price is $199
    • The 6-month 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 inception, cash flows are zero
      • 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
  • Non-continuous 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 risk-free interest rate, r, that is non-continuous
    • 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
      • FV = PV x 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 bid-offer 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, in theory, differ:
    • 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 is comprised of 225 Japanese stock and 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 0e (r-q)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 0e (r-q)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 no-arbitrage 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 risk-free interest rate
  • It follows that if r f is the foreign risk-free interest rate, then r is the interest rate for the home currency

pricing currency forward

  • Continuous

pricing currency forward

  • Discrete from international finance

pricing currency forward

  • 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

international arbitrage

  • Since investor can invest in domestic or foreign country, arbitrage causes both investment paths to converge and become 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

pricing currency forward

Storage Costs

  • Consumption Assets
  • Seller would need to pass storage onto buyer to make profit
  • 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 equals 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 0e cT
  • Consumption assets can have a lower futures price than spot price
    • Backwardation
    • For a consumption asset F 0 ≤ S 0e cT
  • The convenience yield on the consumption asset, y, is defined
    • F 0 = S 0 e (c – y )T
    • Same equation F 0 ≤ S 0e cT
    • Implies the risk-free interest rate is negative
    • Add a fudge factor, y, so risk-free interest rate remains 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 0e –r T at the risk-free rate and enter into a long futures contract to create a cash inflow of S T at maturity
    • This shows that F 0e -rT e kT = E(S T)
    • or F 0 = E(S T)e (r-k)T
    • Future price depends on a expected spot price at Time T based on the risk-free 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)

Value of a Futures/Forward Contract

  • Price in futures/forward is for Time T, at maturity
  • We can place a value on a futures/forward between 0 and T
  • Example
    • We enter a six-month long forward contract on a non-dividend-paying stock
    • Stock price equals $50
    • The risk-free rate of interest is 10% per annum with continuous compounding
    • What is the price of the forward?
      • F 0=S 0e r T
      • F 0=$50 e 0.1x0.5 = $52.5636
    • What is the value of the forward in three months when the stock price is $55 and interest rate remains at 10%
      • Value t = S t - F 0 e -r ( T - t )
      • Remember - the forward is for a price at Time T
      • We discount backwards to time t
      • The value for the long position
      • Value t = $55 - $52.5636 e -0.1 (0.5 - 0.25)
      • Value t = $3.7342
    • What is the value of the forward at time = 0
      • Value t = S t - F 0 e -r ( T - t )
      • Substitute the value of the forward at time 0
      • Value t = S t - S 0 e r T e -r ( T - t )
      • Set t = 0
      • Value 0 = S 0 - S 0 e r T e -r ( T - 0 )
      • Value 0 = S 0 - S 0 e 0 = S 0 - S 0 = 0

Arbitage Examples

  • A gold arbitrage opportunity
    • The spot price of gold is US $1,700 per ounce
    • The quoted 1-year futures price of gold is US$1,800
    • The 1-year US$ 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 risk-free 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
        • On maturity in one year
          • Sell the gold via the futures 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 1-year futures price of gold is US$1,680
    • The 1-year US$ 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 the $1,700 at 5% at a bank
      • Buy futures contract and agree to buy $1,680
      • On maturity in one year
        • Buy gold at $1,680
        • Close short position to replace the gold you sold
        • 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 1-year futures price of oil is US$90
    • The 1-year US$ 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
        • Instructor does not ask for detailed cash flows
  • An oil arbitrage opportunity
    • The spot price of oil is US$80
    • The quoted 1-year futures price of oil is US$75
    • The 1-year US$ 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
        • I do not ask for detailed cash flows