Embed Size (px)
Citation preview
DerivativesLecture 10
Short Sale ExamplePurchase of sharesApril: Purchase 500 shares for $120 -$60,000May: Receive dividend +500July: Sell 500 shares for $100 per share +50,000
Net profit = -$9,500
Short Sale of sharesApril: Borrow 500 shares and sell for $120 +60,000May: Pay dividend -$500July: 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, 6th Edition, Copyright © John C. Hull 2007
Futures Price Notation
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
Futures Price Calculation The price of a non interest bearing asset futures
contract. The price is merely the future value of the spot
price of the asset.
rTeSF 00
Futures Price Calculation
Example IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. What is the likely price of the 6 month futures contract?
55.69$
68
0
50.045.0
00
F
eF
eSF rT
Futures Price CalculationExample - 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 6 Profit Sell stock for $70 +70.00Repay loan at $69.55 -69.55
$0.45
Futures Price CalculationExample - 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 6 Profit Buy stock for $65 -65.00Receive 68 x e.5x.045 69.55
$4.55
Futures Price Calculation 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
rTeISF )( 00
Futures Price Calculation
Example IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. It pays $.75 in dividends in 3 and 6 months. What is the likely price of the 6 month futures contract?
47.1$
75.75.50.045.25.045.
Iee
I
04.68$
)47.168(
)(
0
50.045.0
00
F
eF
eISF rT
Futures Price Calculation 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
TqreSF )(00
Futures Price Calculation
Example A stock index is selling for $500. The zero coupon
interest rate is 4.5% and the index is known to produce a continuously compounded dividend yield of 2.0%. What is the likely price of the 6 month futures contract?
29.506$
500
0
50.)02.045(.0
)(00
F
eF
eSF Tqr
Futures Price Profit Calculation 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.
rTe
KFalue
)(V 0
Long Contract Value
rTe
FKalue
)(V 0
Short Contract Value
Futures Price Calculation
Example IBM stock is selling for $71 per share. The zero
coupon interest rate is 4.5%. What is the likely value of the 6 month futures contract, if it only has 3 months remaining? Recall the original futures price was 69.55.
80.71$
71
0
25.045.0
00
F
eF
eSF rT
22.2$
)55.6980.71(Value
25.045.
e
Futures Prices and Storage 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.
rTeUSF 00
Futures price given constant yield storage
cost
Futures price given set price storage cost
TureSF )(00
u =continuously compounded cost of storage, listed as a percentage of the asset pricerTe
UCost Storaget
Futures Prices and StorageExample The spot price of copper is $3.60 per pound. The 6 month cost to store
copper is $0.10 per pound. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?
76.3$
)098.60.3( 50.035.
00
e
eUSF rT098.
.1050.035.
eU
Futures Prices and StorageExample The spot price of copper is $3.60 per pound. The annual cost to store
copper is quoted as a continuously compounded yield of 0.5%. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?
67.3$
60.3 50.)005.035(.
)(00
e
eSF Tur
Convenience Yield 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.
TyureSF )(00
Fundamentals of Futures and Options Markets, 6th 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 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
c can be thought of as the difference between the borrowing rate and the income earned on the asset.
C = r - q
