# Lecture 10. Purchase of shares April: Purchase 500 shares for \$120-\$60,000 May: Receive dividend +500 July: Sell 500 shares for \$100 per share +50,000.

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Lecture 10

Purchase of shares April: Purchase 500 shares for \$120-\$60,000 May: Receive dividend +500 July: Sell 500 shares for \$100 per share +50,000 Net profit = -\$9,500 Short Sale of shares April: Borrow 500 shares and sell for \$120+60,000 May: Pay dividend -\$500 July: Buy 500 shares for \$100 per share -\$50,000 Replace borrowed shares to close short position. Net profit = + 9,500

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 Futures Price Notation S0:S0:Spot price today F0:F0:Futures or forward price today T:T:Time until delivery date r:r:Risk-free interest rate for maturity T

The price of a non interest bearing asset futures contract. The price is merely the future value of the spot price of the asset.

Example IBM stock is selling for \$68 per share. The zero coupon interest rate is 4.5%. What is the likely price of the 6 month futures contract?

Example - continued If the actual price of the IBM futures contract is selling for \$70, what is the arbitrage transactions? NOW Borrow \$68 at 4.5% for 6 months Buy one share of stock Short a futures contract at \$70 Month 6Profit Sell stock for \$70+70.00 Repay loan at \$69.55-69.55 \$0.45

Example - continued If the actual price of the IBM futures contract is selling for \$65, what is the arbitrage transactions? NOW Short 1 share at \$68 Invest \$68 for 6 months at 4.5% Long a futures contract at \$65 Month 6Profit Buy stock for \$65-65.00 Receive 68 x e.5x.045 69.55 \$4.55

The price of a non interest bearing asset futures contract. The price is merely the future value of the spot price of the asset, less dividends paid. I = present value of dividends

Example IBM stock is selling for \$68 per share. The zero coupon interest rate is 4.5%. It pays \$.75 in dividends in 3 and 6 months. What is the likely price of the 6 month futures contract?

If an asset provides a known % yield, instead of a specific cash yield, the formula can be modified to remove the yield. q = the known continuous compounded yield

Example A stock index is selling for \$500. The zero coupon interest rate is 4.5% and the index is known to produce a continuously compounded dividend yield of 2.0%. What is the likely price of the 6 month futures contract?

The profit (or value) from a properly priced futures contract can be calculated from the current spot price and the original price as follows, where K is the delivery price in the contract (this should have been the original futures price. Long Contract ValueShort Contract Value

Example IBM stock is selling for \$71 per share. The zero coupon interest rate is 4.5%. What is the likely value of the 6 month futures contract, if it only has 3 months remaining? Recall the original futures price was 69.55.

Commodities require storage Storage costs money. Storage can be charged as either a constant yield or a set amount. The futures price of a commodity can be modified to incorporate both, as in a dividend yield. Futures price given constant yield storage cost Futures price given set price storage cost u =continuously compounded cost of storage, listed as a percentage of the asset price

Example The spot price of copper is \$3.60 per pound. The 6 month cost to store copper is \$0.10 per pound. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?

Example The spot price of copper is \$3.60 per pound. The annual cost to store copper is quoted as a continuously compounded yield of 0.5%. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?

Shortages in an asset may cause a lower than expected futures price. This lower price is the result of a reduction in the interest rate in the futures equation. The reduction is called the convenience yield or y.

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 5.18 The Cost of Carry (Page 117) The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F 0 = S 0 e cT For a consumption asset F 0 S 0 e cT The convenience yield on the consumption asset, y, is defined so that F 0 = S 0 e (c–y )T c can be thought of as the difference between the borrowing rate and the income earned on the asset. C = r - q

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