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: 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( ) = $409.50

Forward Pricing  Arbitrage strategy: NowAfter 1 year Borrow US$390Sell Gold under Forward Buy GoldReceive US$425 Short ForwardRepay US$ Initial cost of portfolio = 0 CF after one year = US$15.50 (riskless profit)

Forward Pricing  If forward price is US$390, reverse strategy: NowAfter 1 year Sell Gold Investment pays $ Invest US$390 Buy Gold under Forward Long ForwardPay US$390 Initial cost of portfolio = 0 CF after one year = US$19.50 (riskless profit)

Forward Pricing  If using continuous compounding, then: F 0 = S 0 e rT 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: F 0 = (S 0 – I)e rT where I is the present value of the income  If asset provides a known yield: F 0 = S 0 e (r–q)T where q is the average yield during the life of the contract (continuous compounding)

Pricing with Known Income  Suppose that: Coupon bearing bond is selling for $900 Coupon of $40 expected in four months 9-month forward price of same bond is $ and 9-month rates are 3% and 4%  What is the arbitrage strategy?

Pricing with Known Income  Arbitrage strategy: Borrow cash, buy bond, short forward PV of coupon = 40 e –0.03(4/12) = $39.60 Borrow $39.60 at 3% for four months Borrow remaining $ at 4% for nine months Repay e 0.04(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 F 0 is forward price that would apply to the contract today  Value of a long forward contract, ƒ, is ƒ = (F 0 – K)e –rT  Value of a short forward contract is ƒ = (K – F 0 )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 F 0 = S 0 e (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 F 0 > S 0 e (r–q)T an arbitrageur buys the stocks underlying the index and sells futures  When F 0 < S 0 e (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 r f is the foreign risk-free interest rate:

Currency Arbitrage

 Suppose that: 2-year rates in Australia and the US are 5% and 7% respectively Spot exchange rate is AUD/USD = year forward rate: 0.62 e (0.07–0.05)2 =  What if the forward rate was ?

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, for 1, x 0.63 = USD USD620 will grow to 620 e 0.07x2 = Pay USD under contract Profit = – = USD16.91

Consumption Assets 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

Cost of Carry  Cost of carry, c, is storage cost plus interest costs less income earned  For an investment asset F 0 = S 0 e cT  For a consumption asset F 0  S 0 e cT  Convenience yield on a consumption asset, y, is defined so that F 0 = S 0 e (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, F 0 e –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 = – F 0 e –rT + E(S T )e –kT  If securities are priced based on zero NPVs, then the PV should equate to zero F 0 = E(S T )e (r–k)T

Expected Future Spot Prices  If asset has no systematic risk, then k = r and F 0 is an unbiased estimate of S T positive systematic risk, then k > r and F 0 < E(S T ) negative systematic risk, then k E(S T )