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Derivatives 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 Forwards (Long Position) Enter a contract now for some future required payoff even if negative Can be paid now or at expiration Options – gives you the option to exercise at expiration Calls and Puts

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Options Styles European – can only be exercised at expiration American – can be exercised at anytime Bermudan – can be exercised during specified times; rare 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

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Put-Call Parity 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. Synthetically Created Options (using put-call parity) Forwards, Bonds, Calls, and Puts

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Risk Management 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

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Risk Management Cost to carry Difference between interest and dividend rates Cost for you to borrow and buy stock, then hold it (Reverse) Cash and Carry Short a forward contract and buy the asset Pays off if forward price is too high

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Combining Options Synthetic forward Obtain the stock in future at price determined today Buy a call and sell a put at same strike price 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

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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

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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 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%. Determine the net financing cost (net premiums) of: 1. A bull spread using call options 2. A 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 A collar with a width of $10 using 90 and 100-strike options 5. A straddle using at-the-money options 6. An strangle 7. A butterfly spread with a at-the-money straddle and insurance options out $10 Strike PriceCallPut $80$28.34$

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Answers 1. $ $ $ $ $ $ $8.01

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Four ways to purchase a stock Outright purchase Receive now Pay now: Borrow to pay for the stock Receive now Pay later: Prepaid forward contract Receive in future Pay now: Forward contract Receive in future Pay in future:

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Futures contracts 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

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Swaps Simply a series of forward contracts Payment Prepaid - pay now Postpaid - pay at end Level annual payments - most common Types Commodity, eg. price of corn Interest rate Foreign currency Any of these could be deferred, or start in the future

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Problem 1 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

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Problem 2 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%

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Problem 3 Arnold buys a one-year 125-strike European call for a premium of $ He also sells a 100-strike call on the same underlying asset for a premium of $ 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

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Problem 4 We are given the following: Forward Price = $ European Strike Call Premium = $ European Strike Put Premium = $11.79 Determine the risk free rate. ASM p.577 Answer: 8.78%

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Problem 5 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

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Problem 6 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

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Problem 7 Take these forward prices for forward contracts of Stock ABC: Years to Exp.Forward Price 1$ Take these spot rates of interest: Term to maturitySpot Rate 13.0% 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

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Problem 8 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%

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