Determination of Forward and Futures Prices Week 4

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Determination of Forward and Futures PricesWeek
4
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Measuring Interest Rates

• A Amount invested
• n Investment period in years
• Rm Interest rate per annum
• m Compounding frequency
• ex. m 2 means that the interest rate is
  paid twice a year (every 6 months)

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Terminal Value of an Investment

• For any n and m, the terminal value of an
  investment A at rate Rm is
• A(1Rm / m)m?n
• limm 8 A(1Rm / m)m?n A eRc ? n
• where RC is the continuously compounded interest
  rate per annum and e 2.71882

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Continuous Compounding

• In the limit as we compound more and more
  frequently we obtain continuously compounded
  interest rates (RC)
• 100 grows to 100eRcT when invested at a
  continuously compounded rate R for time T
• 100 received at time T discounts to 100e-RcT
  at time zero when the continuously compounded
  discount rate is RC

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Conversion Formulas

• Rc continuously compounded rate
• Rm same rate with compounding m times
• per year

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Example Interest Rates

• An interest rate is quoted is quoted as 5 per
  annum with semiannual compounding. What is the
  equivalent rate with
• (a) annual compounding
• (b) monthly compounding
• (c) continuous compounding.

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Example Interest Rates

• (a) annual compounding
• 1.0252-10.050625 or 5.0625
• (b) monthly compounding
• 12 x (1.0251/6-1)0.04949 or 4.949
• (c) continuous compounding
• 2 x ln(1.025)0.04939 or 4.939

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Notation
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Assumptions

• No riskless arbitrage opportunities
• Perfect markets
• Underlying asset not held for consumption or
  production.

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1. Gold An Arbitrage Opportunity?

• Suppose that
• The spot price of gold is US390
• The quoted 1-year forward price of gold is US425
• The 1-year US interest rate is 5 per annum
• No income or storage costs for gold
• Is there an arbitrage opportunity?
  

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• NOW
• Borrow 390 from the bank
• Buy gold at 390
• Short position in a forward contract
• IN ONE YEAR
• Sell gold at 425 (the forward price)
• reimburse 390 ? exp(0.05) 410
• ARBITRAGE PROFIT 15 ?
• NOTE THAT ARBITRAGE PROFIT AS LONG AS
• S0 ? exp(r ?T) lt F0

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2. Gold Another Arbitrage Opportunity?

• Suppose that
• The spot price of gold is US390
• The quoted 1-year forward price of gold is US390
• The 1-year US interest rate is 5 per annum
• No income or storage costs for gold
• Is there an arbitrage opportunity?

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• NOW
• Short sell gold and receive 390
• Make a 390 deposit at the bank
• Long position in a forward contract
• IN ONE YEAR
• Buy gold at 390 (the futures price)
• Terminal value on the bank account 390 ?
  exp(0.05) 410
• ARBITRAGE PROFIT 20 ?
• NOTE THAT ARBITRAGE PROFIT AS LONG AS
• S0 ? exp(r ?T) gt F0
• Therefore F0 has to be equal to S0 ? exp(r ?T)
  410

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Forward-Spot Parity Relationship

• No income, no storage costs and no arbitrage
• ? F SerT

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Synthetic Forward AssetBorrowing
Buy Asset
25
Synthetic Forward
ST
125
75
Borrowing
-25
Loss
11/22/2009
Simon Fraser University
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Forward-Spot Parity Implication

• Buying a forward contract is equivalent to buying
  the underlying asset financed by borrowing.
• Selling a forward contract is equivalent to
  shorting the underlying asset with complete
  lending of the proceeds.
• The forward price F contains no more information
  about the future asset price ST than that already
  contained in the current asset price S and the
  riskless return.

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When an Investment Asset Provides a Known Dollar
Income

• Consider a forward on a bond
• S0 900, F0 930
• Tbond 5 years, Tforward 1 year
• Coupon in 6 months 40
• Coupon in 12 months 40
• r(6 months) 9, r(12 months) 10
• NOW
• Borrow 900 (38.24 for 6 m and 861.76 for 12 m)
• Buy 1 bond at 900
• Short position in the forward

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• IN 6 MONTHS
• Receive first coupon and reimburse 40
• 40 38.24 ? exp(0.09 ? 1/2)
• IN 12 MONTHS
• Receive second coupon 40
• Sell the bond at 930 (forward price)
• Reimburse 861.76 ? exp(0.1) 952.39
• ARBITRAGE PROFIT 17.61 ?
• TO PREVENT AN ARBITRAGE PROFIT
• I2 F0 S0 I1exp(-r6m ? 0.5) exp(r12m ? 1)
  0
• F0 S0 I1exp(-r6m ? 0.5) I2exp(-r12m ? 1)
  exp(r12m ? 1)
• F0 (S0 I) exp(r ? T) where I is the PV of all
  future incomes

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When an Investment Asset Provides a Known Yield

• Yields Income expressed as a of asset price,
  usually measured by continuous compounding per
  year, and denoted by q
• Yields work just like interest rates
• e.g. Income in dollars generated by a yield q
  after T years is eqT
• Intuitively, we have
• with cash income F0 (S0 I)erT
• with yield F0 (S0 e-qT)erT S0 e(r-q)T

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Arbitrage Table
? F Se(r-q)T
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When an Investment Asset Provides a Known Yield

• EXAMPLE
• Consider a three-month futures on SP 500 stock
  index. Stocks underlying the index provide a 2
  dividend per annum (q 0.02).
• S0 1,100 points r 3
• F0 S0 e(r-q)T
• 1100 e(0.03-0.02)3/12 1102.75 points

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Valuing a Forward Contract

• Value of a forward at the time it is first
  entered is ZERO. Later it may be positive or
  negative, depending on the evolution of the
  underlying asset price.
• Sequence
• Past (when the first position is taken)
• Now (time 0, when we value the forward)
• Delivery date of the forward (time T)

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• K is delivery price in a forward contract (old
  F0)
• F is forward price that would apply to the
  contract today
• On time 0
• Contract I long position, forward price K, T,
  value ?
• Contract II long position, forward price F, T,
  value 0
• At time T payoff (I) ST K and payoff (II)
  ST F
• D payoff at time T F K (no risk)
• D price today (F K)e-rT
• As value II 0 today gt value I is (F K)e-rT
• Similarly, the value of a short forward contract
  is
• (K F )erT

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Example

• Stock is expected to pay dividend of 1 per share
  in two months and in five months. Stock price is
  50 and the riskfree rate of return is 8 pa
  (continuous compounding). Investor takes a short
  position in six-month forward contract on the
  stock.
• Determine forward price and initial value of
  forward contract.
• Three months later, the price of the stock is
  48. Determine the forward price and the value of
  the short position.

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Example

• Determine forward price and initial value of
  forward contract.
• I1 x exp(-0.08x2/12)1 x exp(-0.08x5/12)1.9540
• F0(50-1.9540) x exp(0.08x6/12)50.01
• Initial value by design is zero.
• Three months later, the price of the stock is
  48. Determine the forward price and the value of
  the short position.
• In three months Iexp(-0.08x2/12)0.9868
• F(48- 0.9868)exp(0.08x3/12)47.96
• f-(47.96 - 50.01)exp(0.08x3/12)2.01

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Forward vs. Futures Prices

• Forward and futures prices are usually assumed to
  be the same. 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
• A strong negative correlation implies the reverse

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Forward vs. Futures Prices (II)

• Other factors
• Taxes, Transaction costs, trading
• complexity, margins, risk of default,
• liquidity
• Empirical Studies
• Futures prices gt or Forwards prices

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Futures on Stock Index

• Index can be viewed as an investment asset paying
  a dividend yield
• The futures price and spot price relationship is
  therefore
• F0 S0 e(rq )T
• where q is the dividend yield on the portfolio
  represented by the index

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Index Arbitrage

• When F0gtS0e(r-q)T an arbitrageur buys the stocks
  underlying the index and takes a short position
  in futures
• When F0ltS0e(r-q)T an arbitrageur takes a long
  position in futures and shorts or sells the
  stocks underlying the index

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Index Arbitrage (II)

• Index arbitrage involves simultaneous trades in
  futures and many different stocks
• Very often a computer is used to generate the
  trades
• Occasionally (e.g., on Black Monday) simultaneous
  trades are not possible and the theoretical
  no-arbitrage relationship between F0 and S0 does
  not hold

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NYSE to remove trading curb born out of '87 crash

• Remove current buying and selling curbs -- called
  "trading collars" -- on brokerages using
  computer-assisted program trading when DJIA moves
  up or down more than 50 points.
• NYSE defines computer-assisted program trades as
  the buying or selling of a basket of at least 15
  stocks from the SP 500 Index valued at 1
  million or more.

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The Top Program Traders
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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

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Why the Relation Must Be True
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Example

• Suppose that the yen-denominated interest rate is
  2 and the dollar-denominated interest rate is
  6. The current exchange rate is 0.009 dollars
  per yen.
• Determine the one-year forward rate.

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Storage Costs

• Storage costs can be treated as negative income
• F0 (S0U )erT
• where U is the present value of the storage
  costs
• Alternatively F0 S0 e(ru )T
• where u is the storage cost per unit time as a
• percent of the asset value

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Example with Storage Costs

• Futures on gold, S0 450, T 1, r 0.07
• Storage cost 2/ounce (at the end of each year)
• F0 (450 2 e-0.07) e0.07 484.63
• If F0 500 in the market
• NOW Borrow 450, buy gold at 450, short futures
• IN ONE YEAR Sell gold at 500, pay storage 2,
  pay off loan 482.63
• ARBITRAGE PROFIT 500 2 482.63 15.37

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Example with Storage Costs (II)

• If F0 470 in the market
• Investor owns some gold
• NOW Sell gold at 450, long futures, invest 450
  at 7
• IN ONE YEAR Final value of investment 482.63,
  buy gold at 470
• ARBITRAGE PROFIT 482.63 470 12.63
• Investor does not own any gold
• NOW Short sell gold and get 450, long futures,
  invest 450 at 7
• IN ONE YEAR Final value of investment 482.63,
  buy gold at 470 and return it to owner

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Consumption vs. Investment Assets

• Investment assets are assets held by significant
  numbers of people purely for investment purposes
  (Examples individual stock, gold, silver)
• Consumption assets are assets held primarily for
  consumption (Examples copper, oil)

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Futures on Consumption Assets

• If F0 gt (S0U )erT gt borrow cash, buy asset,
  store it, short futures gt arbitrage opportunity
  ?
• If F0 lt (S0U )erT gt sell asset, invest
  proceeds, long futures
• But individuals and companies are reluctant to
  sell the commodity (and take a long position in
  the futures instead) because futures cannot be
  consumed whenever you want.

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Futures on Consumption Assets

• F0 ? (S0U )erT
• where U is the present value of the storage
  costs
• Alternatively,
• F0 ? S0 e(ru )T
• where u is the storage cost per unit time as a
  percent of the asset value

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The Convenience Yield

• The convenience yield, y, is the benefit provided
  when owning a physical commodity
• Not obtained when holding a futures
• We may decide to use the physical commodity right
  now

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The Cost of Carry

• The cost of carry, c, is the storage cost plus
  the interest costs less the income earned
• For an investment asset F0 S0ecT
• For a consumption asset F0 ? S0ecT
• The convenience yield on the consumption asset,
  y, is defined so that
• F0 S0 e(cy )T

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A No-Arbitrage Proof That Time Travel Is
Impossible

• Can you use finance theory to prove that the
  ability to travel through time does not exist now
  and it will never exist in the future.
• Hint Show how time travel would make available
  new arbitrage opportunities.