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**PSU Study Session Fall 2010 Dan Sprik**

Exam FM/2 PSU Study Session Fall 2010 Dan Sprik

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Forward Prices You can derive the price of some positions by using an arbitrage-free argument (assume that you canβt make money for free, then figure out what the position would have to cost to make this assumption true) πΉ 0,π‘ = π 0 π πβπΏ π‘

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**Prepaid Forward Prices**

Prepaid forward contract: receive stock at time T, but pay at time 0. Payment at time 0 is: Fpt,0 = S0 , no dividends Fpt,0 = S0 β PV(dividends) Fpt,0 = S0e-Ξ΄T (continuous dividends)

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Put-Call Parity Put Call Parity is the most important formula from Derivatives Markets. It says that buying a call and selling a put, each with strike price K, is the same as buying a prepaid forward and borrowing πΎ π βππ‘ dollars. How do we prove this? Show that payoffs are the same at time t, then infer that it must cost the same at time 0 to enter either position (otherwise arbitrage exists) Only valid for European options πͺβπ·= πΊ π π βπΉπ βπ² π βππ

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**Combining Options Bull spread: Buy C(K1) and sell C(K2), K2>K1**

Bear spread: Sell C(K1) and buy C(K2), K2>K1 Box spread: combining synthetic long and short forwards (or bull and bear spreads) Collar: Buy P(K1) and sell C(K2), K2>K1

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**Combining Options Straddle: Buy P(K1) and buy C(K1), at the money**

Written straddle: sell P(K1) and sell C(K2), at the money Strangle: Buy P(K1) and buy C(K1), out of the money Butterfly Spread: written strangle plus purchased out of the money call and put options.

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Hedging For seller β can lock in price with short forward or place a floor on price with a purchased put For buyer β can lock in price paid with a long forward or place a cap on price with a purchased call

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**Miscellaneous Cash and carry : Short forward and buy asset**

Reverse cash and carry: buy forward and short the asset Future contracts β marked to market and settled daily.

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**Mark-to-Market Example**

Dodi goes long on a 6 month futures contract on 100 units of stock index XYZ. She makes an initial deposit of 25,000 in her margin account, which is credited with interest at 5% effective per annum. The contract is marked to market and settled at the end of each week. The initial value of XYZ is The index futures price at the end of weeks 0, 1, 2, and 3 are 1000, 1002, 995, and 998 respectively. X is the amount in Dodiβs account at the end of 3 weeks. Determine X.

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Swap Contracts An agreement to exchange specified cash flows in the future A forward is like a single payment swap. Typically the price of a swap involves level payment at maturity dates. Notional Amount = quantity of asset being traded To find swap price: figure out sum of prepaid forward prices and find equivalent price-per-transaction that yields this same price Swaps involve implicit borrowing/lending

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**Vocabulary Cost of carry = rβΞ΄**

Credit risk = the risk that a loss will be experienced because of a default by the counterparty in a derivatives transaction Margin = the cash balance (or security deposit) required from a futures or options trader Maintenance Margin = When the balance in a traderβs margin account falls below the maintenance margin level, the trader receives a margin call requiring the account to be topped up to the initial margin level Margin Call = A request for extra margin when the balance in the margin account falls below the maintenance margin level

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Vocabulary (cont.) Covered Call = a short position in a call option on the asset combined with a long position in the asset Naked option = option that is combined with an offsetting position in the underlying stock Hedging = reducing the risk of your position Arbitrage = a trading strategy that takes advantage of securities being mispriced relative to each other Lease rate = Implied repo rate = implied interest rate r

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# 2 Seth buys a stock for 48 and at the same time buys a one year 48 strike European put for a premium of Laura buys a one year European call on the same stock for a premium of X. The annual effective risk free interest rate is 6.6%. Determine X (assume no dividends). A) B) C) D) E) 7.63

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# 5 Abbey buys a stock for 85 and writes an 85 strike one year European call on the same stock. The premium for an 85 strike one year put is The risk free annual rate of effective interest is 4.35%. X is the profit for a spot price at expiration of 92. Determine X to the nearest 0.10. A) B) C) D) E) 15.60

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# 13 You short an asset with a price of 50. At the same time, you write a 6-month put with an exercise price of 50, for The effective annual interest rate is 6%. Determine your profit as of the expiration date of the option, when the underlying asset price on that date is 48. A) 0 B) C) D) E) 4.82

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# 2 We are given the following information about the derivatives for a certain underlying asset: Forward price = 150 strike European call premium = 23.86 150 strike European put premium = 11.79 The risk free annual effective rate is X. Determine X. A) 8.07% B) 8.78% C) 9.19% D) 10.28% E) 11.39%

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# 11 You buy a call with an exercise price of 90 for a premium of 10, and you sell a call with an exercise of 105 for a premium of 3. Both options are on XYZ Corp stock, and both have 9-month maturities. The nominal annual interest rate is 8% convertible quarterly. What is the maximum possible profit, as of expiration date of the options, produced by this position? A) B) C) D) E) 15.00

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# 12 You buy a call with an exercise price of 90 for a premium of 10, and you sell a call with an exercise of 105 for a premium of 3. Both options are on XYZ Corp stock, and both have 9-month maturities. The nominal annual interest rate is 8% convertible quarterly. What is the price of XYZ Corp stock such that you would break even on your position, as of the expiration date of the options? A) B) C) D) E) 99.00

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# 2 Your company is considering insuring against decreases in the price of a product that it will sell a year from now. The current price of the product is 50 and the risk free annual effective rate of interest is 4%. The company has a choice of entering into a one year short forward contract with a forward price of 52 or buying a one year 50 strike European put with a premium of For what range of spot prices at expiration would the profit from the forward exceed the profit from the put? 46.86 to β B) 0 to C) 0 to 50 D) 0 to 52 E) 0 to 57.14

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# 6 A company sells a product at a current price of 500. The company hedges against price declines using a one year 500 strike European put option with a premium of The risk free annual effective rate of interest is 9%. X is the companyβs profit from the put option for a price of expiration of 490. Determine X. A) B) C) D) E)

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# 1 The current price of a stock is 84. A one year forward contract is entered into. It is expected that four quarterly dividends of 5 each will be paid on the stock starting 3 months from now. The 4th dividend will be paid the day before the expiration of the forward contract. The risk free interest rate is 6% compounded quarterly. What is the price of a prepaid forward contract? A) B) C) D) E) 69.80

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# 4 The current price of a stock is 72. The stock is expected to pay dividends continuously at a constant annual rate of 2%. The risk free force of interest is 6% per annum. X is the forward price of a 1.5 year forward contract. Determine X. A) B) C) D) E) 76.45

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# 3 The one year forward contract has a forward price of 90 and a two year forward contract has a forward price of 95. The yield curve is flat at 5% effective per annum. X is the implicit borrowing and lending that occurs at the end of one year under a two year swap contract with a level swap price. Determine X. A) B) C) D) E) 2.56

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