Derivatives
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A derivative is a product with value derived from an underlying
asset.
Ask price – Market-maker asks for the high price
Bid price – Market-maker bids for the low price
Bid-Ask spread is part of the market-maker’s profit(market-maker
profit may also include commission from the sale)
Positions
 Short – You profit from declines in the underlying asset value
 Long – You profit from increases in the underlying asset value
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Forwards (Long Position)
 Enter a contract now for some future required payoff even if negative
 Can be paid now or at expiration
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Options – gives you the option to exercise at expiration
 Calls and Puts
Options
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Styles
 European – can only be exercised at expiration
 American – can be exercised at anytime
 Bermudan – can be exercised during specified times; rare
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Positions
 In-the-money – Payoff is positive right now
 At-the-money – Payoff is zero right now
 Out-of-the-money – Payoff is negative right now
Put-Call Parity
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The cost of buying a call and selling a put must equal the
price of today’s stock (or the present value of the forward
price) less the present value of the options’ strike price.
𝐶𝑎𝑙𝑙 𝐾, 𝑇 − 𝑃𝑢𝑡 𝐾, 𝑇 = 𝑃𝑉 𝐹𝑜,𝑇 − 𝑃𝑉 𝐾
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Synthetically Created Options (using put-call parity)
 Forwards, Bonds, Calls, and Puts
Risk Management
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Ways to reduce potential losses or securing a gain
Diversifiable risk can be hedged, while
nondiversifiable (systematic) risk cannot
Hedging
 Covered Call – writing a call plus long in the asset
 Covered Put – writing a put plus short in the asset
 Naked Option – writing an option without a position in
asset
Risk Management
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Cost to carry
 Difference between interest and dividend rates
 Cost for you to borrow and buy stock, then hold it
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(Reverse) Cash and Carry
 Short a forward contract and buy the asset
 Pays off if forward price is too high
Combining Options
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Synthetic forward
 Obtain the stock in future at price determined today
 Buy a call and sell a put at same strike price
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Spreads
 Bear
○ Buy call and sell higher call or buy put and sell higher put
○ Profit with increase, up to a limit
 Bull (opposite of bear)
○ Sell a call and buy a higher call or sell a put and buy a higher put
○ Profit with decline in price, to a limit
Combining Options
 Box – constant (often zero) payoff
○ Combination of long and short synthetic forwards or bull and bear spreads
○ No market risk, so only useful for borrowing or lending money
 Collars
○ Long put and short call with higher strike
○ Zero cost collar – Premiums are equal
○ Collared Stock – Long in stock and buy a collar
 Ratio
○ Buying and selling unequal numbers of options
○ Can be used for more complicated hedging strategies
Combining Options
 Straddles
○ Purchase call and put with same strike price
○ Profit with volatility in either direction
○ Write a straddle to bet on stability
 Strangles
○ Straddle with out-of-the-money options to reduce costs
○ Reduced profit with volatility, but lose less in the middle
 Butterfly spread
○ Write a straddle, then buy put and call on far sides for protection
○ Bets on stability while protecting against losses in either direction
○ Can be asymmetric to shift location of peak
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Pay Later Strategies
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Take the following premiums for one-year European options for an underlying
asset with a current spot price of $100. The risk-free annual effective rate of
interest is 8.5%.
Strike Price
Call
Put
$80
$28.34
$2.07
90
21.46
4.41
100
15.79
7.96
110
11.33
12.71
120
7.95
18.55
Determine the net financing cost (net premiums) of:
1. A 100-110 bull spread using call options
2. A 100-120 box spread
3. A ratio spread using 90 and 110-strike options, with a payoff of 20 at expiration price
110 and payoff of 0 at expiration price 120
4. A collar with a width of $10 using 90 and 100-strike options
5. A straddle using at-the-money options
6. An 80-120 strangle
7. A butterfly spread with a at-the-money straddle and insurance options out $10
Answers
1. $4.46
2. $18.43
3. -$12.53
4. -$11.38
5. $23.75
6. $10.02
7. -$8.01
Four ways to purchase a stock
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Outright purchase
 Receive now
 Pay now:
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𝑆0
Borrow to pay for the stock
 Receive now
 Pay later:
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𝑆0 𝑒 𝛿𝑡
Prepaid forward contract
 Receive in future
 Pay now:
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𝑆0 − 𝑃𝑉(𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠)
Forward contract
 Receive in future
 Pay in future:
Futures contracts
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Simply a standardized forward contract, sold in exchanges
Marked-to-market
 Changes in value are settled daily through parties
 Parties maintain margin accounts to cover these changes
Swaps
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Simply a series of forward contracts
Payment
 Prepaid - pay now
 Postpaid - pay at end
 Level annual payments - most common
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Types
 Commodity, eg. price of corn
 Interest rate
 Foreign currency
 Any of these could be deferred, or start in the future
Problem 1
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Samantha buys 100 shares of stock but changes her mind and
immediately sells the stock. The broker’s commission is $20 on
a purchase or sale. Samantha lost $70 on this transaction.
What was the difference between the bid and ask price per
share?
ASM p.487
Answer: $.30
Problem 2
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John short sells a stock for $10,000. The proceeds of the sale are
retained by the lender. (Ignore interest on the proceeds.) John
must deposit $5,000 with the lender as collateral. He earns 6%
effective on this haircut. At the end of one year, he closes his
short position by buying the stock for $8,000 and returning it
to the lender. A dividend of $500 was payable one day before he
covered the short. What was John’s effective rate of interest on
his investment?
ASM p.488
Answer: 36%
Problem 3
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Arnold buys a one-year 125-strike European call for a premium
of $16.86. He also sells a 100-strike call on the same underlying
asset for a premium of $31.93. The spot price at expiration is
$110. The effective annual interest rate is 3.5%. What is Arnold’s
total profit at expiration for the two options? ASM p.512
Answer: $5.60
Problem 4
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We are given the following:
 Forward Price = $163.13
 150-European Strike Call Premium = $23.86
 150-European Strike Put Premium = $11.79
 Determine the risk free rate. ASM p.577
Answer: 8.78%
Problem 5
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The current price of the stock is $72. The stock pays
continuous dividends at 2% and the continuous compounded
risk free interest rate is 6%. Determine the forward price in 1.5
years. ASM p.612
Answer: $49.38
Problem 6
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A stock has a current price of $65. A dividend of $3.25 is
expected to be paid in 6 months. The risk-free interest rate is
10% effective per annum. X is the forward price of a one-year
forward contact that has the stock as the underlying asset.
Determine X.
ASM p.612
Answer: $68.09
Problem 7
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Take these forward prices for forward contracts of Stock ABC:
Years to Exp.
Forward Price
1
$100
2
110
3
120
Take these spot rates of interest:
Term to maturity
Spot Rate
1
3.0%
2
3.5
3
3.8
X is the level swap price under a 3-year swap contract with the
same underlying asset. Determine X.
ASM p.630
Answer: $109.56
Problem 8
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Two interest rate forward contracts are available for interest
payments due 1 and 2 years from now. The forward interest
rates in these contracts are based on a one-year spot rate of 5%
and a 2-year spot rate of 5.5%. X is the level swap interest rate
in a 2-year interest rate swap contract that is equivalent to the
two forward contracts. Determine X.
ASM p.630
Answer: 5.49%