Call Option: An option, but not an obligation, to buy at a specified price. o European: Can be exercised only at expiration.

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1 I. Introduction: A derivative is a financial instrument that has a value determined by an underlying asset. Uses of derivatives: o Risk management o Speculation on prices o Reducing transaction costs o Avoiding taxes (regulatory arbitrage) Financial engineering is the construction of a given financial product from other products. Catastrophe bonds are bonds that the issuer does not have to redeem if a specified catastrophe took place. Diversifiable Risk: Risk that is unrelated to other risks. Derivatives are usually introduced in a market often result from increased price risk. Ask price is the price an investor pays to buy. Bid price is the price an investor can sell at. Bid-ask spread is the difference between bid and ask prices. Bid-ask bounce is the apparent fluctuation in prices because of bid-ask spread. A long position is equivalent to purchasing the asset, you make profit if prices go up = Lending money. A short position is the opposite of long; you make profit if prices go down. The lender in a short position requires a collateral = proceeds and a haircut to protect against increased prices. The lender also provides interest on the collateral; this is called repo rate or short rebate. II. Forward Contracts: A forward contract sets the following in relation to buying an asset: o The quantity and type of the underlying asset o The expiration date at which the contract will be carried over o The forward price that will be paid at the expiration date The buyer of the asset has a long position in the forward contract. The seller of the asset has a short position in the forward contract. Long payoff = Spot price Forward price Short payoff = Forward price Spot price Payoff Diagrams:

2 Adding a zero coupon bond to a position shifts the payoff diagram but leaves the profit diagram unaffected. Buying an asset = Long forward + purchase zero-coupon bond redeemable at forward price III. Call Option: An option, but not an obligation, to buy at a specified price. o European: Can be exercised only at expiration. o American: Can be exercised at any time during the life of the option. o Bermudan: Can be exercised during specified periods but not at any time. Purchased Call payoff = Max[0,(spot strike)] Purchased Call profit = Max[0,(spot strike)] FV(Premium) For a written call payoff & profit, multiply by -1 Purchased call = Insurance against increasing prices Payoff & Profit Diagrams: (For diagrams of a written call, invert about the x-axis) Long Call = Purchased Call. IV. Put Option: An option, but not an obligation, to sell at a specified price. o European: Can be exercised only at expiration. o American: Can be exercised at any time during the life of the option. o Bermudan: Can be exercised during specified periods but not at any time. CAUTION: Buyer of put option = seller of asset & Seller of put option = buyer of asset. Purchased Put payoff = Max[0,(Strike Spot)] Purchased Put profit = Max[0,(Strike Spot)] FV(Premium) For a written put, multiply by -1 Purchased put = Insurance against price drop Payoff & Profit diagrams: (For diagrams of a written put, invert about x-axis) Short Put = Purchased Put.

3 Moneyeness of options: o In-the-money: Immediate exercise would result in a positive payoff o Out-of-the-money: Immediate exercise would result in a negative payoff o At-the-money: Strike price = Current price V. Summary of Long & Short: Long position = Buying the asset: o Long Forward o Purchased Call o Written Put Short position: Selling the asset: o Short Forward o Written Call o Purchased Put VI. Options as Insurance: A long position needs insurance against dropping prices; buying a put option sets a minimum price and provides that insurance. Purchased put options are floors. A short position needs insurance against increasing prices; buying a call option sets a maximum price and provides that insurance. Purchased call options are caps. Equivalent Positions: o Long position + Purchased Put + Borrow PV(Strike) = Purchased Call o Short Position + Purchased Call + Lend PV(Strike) = Purchased Put Covered Call: Writing a call that is covered by a long position in the underlying asset. Covered Put: Writing a put that is covered by a short position in the underlying asset. Naked writing: Writing an option while you have no position in the underlying asset. VII. Synthetic Forward: A synthetic forward is constructed by purchasing a call option and selling a put option. The premium of a synthetic forward equals the difference between the cost of the call and the amount received for the put. Put-Call Parity: o Call(K, T) Put(K, T) = PVF, K VIII. Spreads & Collars: Bull Spread: Constructed of only calls or only puts. o Purchase an option at a lower price and write a similar option at a higher price. o Usually denoted as (A-B Bull Spread): A: Strike price of purchased option B: Strike price of written option Used to speculate on increasing prices Profit diagram has the following attributes: a. Floor at prices below A b. Cap at prices above B c. Increasing between A & B

4 o Profit diagram: Bear Spread: Exact opposite of Bull spread. o Sell options at prices lower than the buying prices. o Profit diagram is the mirror image about the x-axis of Bull spread. Box Spread: o Constructed by a synthetic long and a synthetic short. o Or purchasing a bull spread and a bear spread. Ratio Spread: o Constructed by purchasing (m) options at one price and selling (n) similar options at a different price o Profit Diagram: IX. Collars: Purchased Collar: o Purchase a put option and write a call option at a higher strike price. (Profit diagram of bear spread) Written Collar: o Write a put option and purchase a call option at a higher strike price. (profit diagram of bull spread)

5 Zero-Cost collar: o Constructed by buying and selling options resulting in a net price of 0. X. Speculating on Volatility: Spreads and collars are directional tools; their profit is determined by whether prices go up or down. Some financial tools rely on changes in price no matter if that was up or down and regardless of how much. Straddles: o Constructed by purchasing a put and a call with same strike price, both options are at-the-money. o Used to speculate on volatility being higher that the market assessment of volatility. o Greater changes in prices result in greater profit. o Profit diagram: Strangle: o Constructed the same way as a straddle, but with out-of-the-money options. o Lower losses when the price does not move but prices need to move further to make profit. o Profit diagram: Written Straddle: o Constructed by selling at-the-money options. o Used to speculate on volatility being lower than the market assessment. o Profit diagram is an inverted straddle diagram.

6 Butterfly Spread: o Constructed by writing a straddle and buying a strangle. o Profit diagram: Asymmetric butterfly spread: o Usually denoted as (a-b-c) asymmetric butterfly. o To know how much of each option you need: = K% You need to purchase K (c a) of the higher strike option. You need to purchase (1 K) (c a) of the lower strike price option. XI. XII. Summary in terms of price and volatility: If Volatility If Price Increase Irrelevant Decrease Decrease Buy Puts Sell Short Sell Calls Irrelevant Buy Straddle Why Bother, don t trade Sell Straddle Increase Buy Calls Buy Long Sell Puts Hedging: Reasons to hedge: o Shifting income to a non-taxable form. o Protecting against bankruptcy. o Increasing debt capacity. o Reduce financing costs. o Risk aversion. o Non-financial reasons. Reasons not to hedge: o Transaction costs. o Cost of expertise. o Monitoring transactions. o Taxes complications.

7 XIII. Buying a stock: There are four different ways to buying a stock: Method Payment Time Stock Delivery Time Price Outright Purchase 0 0 S 0 Fully-leveraged purchase T 0 S 0 x e rt Prepaid forward 0 T See below Forward Contract T T See below Pricing a prepaid forward contract: o If stock has no dividends: F, = S o If stock pays dividends: F, = S PV(Dividends) If dividends are paid continuously, we use δ = annualized compound daily dividend yield F, = S e Pricing a forward contract: o Forward price is the future value of the prepaid forward price. o Forward premium: Ration of forward price to spot price. o Annulaized forward premium = ln(, ) = r δ. This is also known as the cost of carry. o Where r = risk free rate and δ = dividend yield. XIV. Swaps: Swaps are contracts calling for an exchange of payment over time. To calculate the level payments for a swap: o Use spot rates for zero-coupon bonds to calculate present value of forward prices. o Solve for a series of level payments having the same present value.

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