EXAM FM/2 REVIEWDERIVATIVES

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

𝐶𝑎𝑙𝑙ሺ𝐾,𝑇ሻ− 𝑃𝑢𝑡ሺ𝐾,𝑇ሻ= 𝑃𝑉൫𝐹𝑜,𝑇൯− 𝑃𝑉ሺ𝐾ሻ

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 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

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 options2. A 100-120 box spread3. 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 1204. A collar with a width of $10 using 90 and 100-strike options5. A straddle using at-the-money options6. An 80-120 strangle7. A butterfly spread with a at-the-money straddle and insurance options out

$10

Strike Price Call Put$80 $28.34 $2.0790 21.46 4.41100 15.79 7.96110 11.33 12.71120 7.95 18.55

Answers1. $4.462. $18.433. -$12.534. -$11.385. $23.756. $10.027. -$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:

𝑆0 𝑆0𝑒𝛿𝑡 𝑆0 − 𝑃𝑉(𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠)

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.512Answer: $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 $1002 1103 120

Take these spot rates of interest:Term to maturity Spot Rate

1 3.0%2 3.53 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%