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## Exam FM/2 Review derivatives

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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**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**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**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**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**Combining Options**• Synthetic forward • Obtain the stock in future at price determined today • Buy a calland 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**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 • Pay Later Strategies**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 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**• 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:**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**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**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**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%**Problem 3**• 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**• 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**• 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**• 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**• 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**• 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%