Determination of Forward and Futures Prices Chapter 5 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Consumption vs Investment Assets Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver) Consumption assets are assets held primarily for consumption (Examples: copper, oil) Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Short Selling Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Short Selling (continued) At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends and other benefits the owner of the securities receives Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Short Selling: Example An investor shorts 500 shares in April when the price is $120 and close out position by buying themback in July when price is $100. Supose a dividend of $1 is paid in May. Short sale of share: Apr: Borrow 500 shares and sell at $120:+$60,000 May: Pay dividend : $- 500 July: Buy back 500 shares at $100 : -$50,000 Net Profit : $9,500

Exp. cont. Purchase of shares Apr: Purchase 500 shares for $120: -$60,000 May: Receive dividend : +$500 July: Sell 500 shares for $100 : +$50,000 Net Profit (Loss) : -$9,500

Notation for Valuing Futures and Forward Contracts Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

1. An Arbitrage Opportunity? Suppose that: The spot price of a non-dividend paying stock is $40 The 3-month forward price is $43 The 3-month US$ interest rate is 5% per annum Is there an arbitrage opportunity? Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 18

Yes Now; In 3 months; Borrow $40 at 5% for 3 months Buy 1 unit of asset Enter into forward contract to sell in 3 months for $43. In 3 months; Sell asset for $43 Repay loan with interest: 40e0.05x3/12 = $40.50 Profit = $43-$40.50 = $2.5.

2. Another Arbitrage Opportunity? Suppose that: The spot price of nondividend-paying stock is $43 The 3-month forward price is US$39 The 1-year US$ interest rate is 5% per annum Is there an arbitrage opportunity? Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 19

Yes Now; In 3 months; Short 1 unit of asset to realize $40 Invest $40 at 5% for 3 months Enter into foeward contract to buy asset in 3 months for $39. In 3 months; Buy asset for $39 Close out position Receive 40e0.05x3/12 = $40.50 Profit = 40.5 -39 = $1.5

The Forward Price If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then F0 = S0erT where r is the 1-year risk-free rate of interest. In our examples, S0 =40, T=0.25, and r=0.05 so that F0 = 40e0.05×0.25 = 40.50 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 20

If F0 > S0erT : Arbitrageurs can buy the asset and short forward contracts. 1. Borrow S0 at r for T years 2. Buy the asset 3. Short a forward contract on the asset. Profit= F0 - S0erT If F0 < S0erT : Arbitrageurs can short the asset and enter into long forward contracts on it. 1. Sell the asset for S0 2. Invest the proceed at r for T years 3. Take a long position in a forward contract Profit= S0erT - F0

If Short Sales Are Not Possible.. Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

When an Investment Asset Provides a Known Dollar Income F0 = (S0 – I )erT where I is the present value of the income during life of forward contract Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp. 5.2 (Pg 110) Consider a 10-month forward contract on a stock priice is $50. We assume that the risk-free rate of interest continuously compounded is 8% per annnum for all maturities. We also assume that dividends of $0.75 per share are expected after three months, six months, and nine months. Find the forward price.

When an Investment Asset Provides a Known Yield F0 = S0 e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding) Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp 5.3 (Pg.111) Consider a six-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once durng a six-month period. The risk-free rate of interest with continuous compounding is 10% per annum. The price is $25. Find the forward price.

Valuing a Forward Contract Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short forward contract is (K – F0 )e–rT Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp 5.4 (Pg 112) A long forward contract on a non-dividend paying stock was entered into some time ago. It currently has six months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24. Find the value of the forward contract.

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 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Stock Index F0 = S0 e(r–q )T Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F0 = S0 e(r–q )T where q is the average dividend yield on the portfolio represented by the index during life of contract Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Stock Index (continued) For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp. 5.5 (Pg. 115) Consider a three-month futures contract on the S&P 500. Suppose that the stocks undelying the index provide a dividend yield of 1% per annum, that the current value of the index is 1,300, and the continuously compounded risk-free interest rate is 5% per annum. Find the index futures price.

Index 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 or sells the stocks underlying the index Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Index Arbitrage (continued) Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

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 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp. 5.6 (Pg. 118) Suppose that the two-year interest rates in Australia and the US are 5% and 7% respectively, and the spot exchange rate between the AUD and the USD is o.62 USD PER AUD. Suppose that the two-year forward exchange rate is 0.63 Suppose that the two-year forward exchange rate is 0.66 Define the arbitrage strategies.

Futures on Consumption Assets F0  S0 e(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. Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

Exp. 5.8 (Pg. 121) Consider a one-year futures contract on gold. We assume no income and tht it costs $2 per ounce per year to store gold,with the payment being made at the end of the year. The spot price is $600 and the risk-free rate is 5% per annum for all maturities. Suppose the actual gold futures price is $700 Suppose the actual gold futures price is $610. Define the arbitrage opportunities.

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(c–y )T Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008