Options and their Applications

2 Some Details about Option Contracts Options – Grants its owner the right, but not the obligation, to buy (if purchasing a “call” option) or sell (if purchasing a “put” option) a given asset on a specified date at a specified price (the “strike price”) at the origination of the contract. Options – Grants its owner the right, but not the obligation, to buy (if purchasing a “call” option) or sell (if purchasing a “put” option) a given asset on a specified date at a specified price (the “strike price”) at the origination of the contract. Key Features: Key Features: Calls allow you to bet on increases in the asset’s value. Calls allow you to bet on increases in the asset’s value. Puts allow you to bet on decreases in the asset’s value. Puts allow you to bet on decreases in the asset’s value. The option buyer pays a premium to acquire the option. The option buyer pays a premium to acquire the option. The seller of the option does have an obligation to buy/sell the asset. Much riskier than buying the option. The seller of the option does have an obligation to buy/sell the asset. Much riskier than buying the option. Options can be: “in-the-money”, “at-the-money”, and “out- of-the-money”. Options can be: “in-the-money”, “at-the-money”, and “out- of-the-money”. An option is a portfolio of forward contract and a riskless bond. Also, can create forwards from options! An option is a portfolio of forward contract and a riskless bond. Also, can create forwards from options!

3 Option Details (cont.) Call owner has the right to buy an asset at a strike price specified today at some time in the future. Call owner has the right to buy an asset at a strike price specified today at some time in the future. Put owner has the right to sell an asset at a strike price specified today at some time in the future. Put owner has the right to sell an asset at a strike price specified today at some time in the future. Call seller/writer has the obligation to sell an asset at a strike price specified today. Call seller/writer has the obligation to sell an asset at a strike price specified today. Put seller/writer has the obligation to buy an asset at a strike price specified today. Put seller/writer has the obligation to buy an asset at a strike price specified today. Two main types of options: European and American. Two main types of options: European and American. American option value always greater than/equal to a European option’s value. American option value always greater than/equal to a European option’s value.

4 Why Use Options? Allows you to protect against adverse price movements but still allows for upside potential. Allows you to protect against adverse price movements but still allows for upside potential. Easy to buy protection you need and sell off the coverage you don’t need (e.g., buy a put at X = 50 and sell a put at X = 20). Easy to buy protection you need and sell off the coverage you don’t need (e.g., buy a put at X = 50 and sell a put at X = 20). Options can be purchased on exchanges or in the OTC market (thus giving users a wider choice of markets). Options can be purchased on exchanges or in the OTC market (thus giving users a wider choice of markets). You can sell options and earn the premium to boost income or cushion losses. You can sell options and earn the premium to boost income or cushion losses.

5 Option Payoffs Option payoffs are non-linear. Option payoffs are non-linear. Owner’s Net Payoffs are similar to that of a portfolio of a forward contract and a riskless bond. Owner’s Net Payoffs are similar to that of a portfolio of a forward contract and a riskless bond. Call owner = C = max[0, (S – X)] - Premium Call owner = C = max[0, (S – X)] - Premium Put owner = P = max[0, (X – S)] - Premium Put owner = P = max[0, (X – S)] - Premium Main incentive of call/put seller is to earn the premium. Main incentive of call/put seller is to earn the premium. Call seller = Premium + min[0, (X – S)] Call seller = Premium + min[0, (X – S)] Put seller = Premium + min[0, (S – X)] Put seller = Premium + min[0, (S – X)]

6 Put-Call Parity Relation Portfolio 1 – own a Call (C) and a riskless bond (X*D, where D = a TVM discount factor). Portfolio 1 – own a Call (C) and a riskless bond (X*D, where D = a TVM discount factor). Portfolio 2 – own a Put (P) and a share of stock at the current spot price (S). Portfolio 2 – own a Put (P) and a share of stock at the current spot price (S). Note: Both portfolios have identical payoffs, thus they must have the same value (assuming “no arbitrage”). Note: Both portfolios have identical payoffs, thus they must have the same value (assuming “no arbitrage”). Can use the above relation to price a put option based on the call’s price: Can use the above relation to price a put option based on the call’s price: P = C – (S – X*D)

7 Inter-relationships between Options and Forwards Recall that there are six building blocks (two linear and four non-linear). Recall that there are six building blocks (two linear and four non-linear). We can use the non-linear payoffs of options to construct linear payoffs that are identical to a forward contract’s payoffs: We can use the non-linear payoffs of options to construct linear payoffs that are identical to a forward contract’s payoffs: (1) Long Call and Short Put = Long Forward: (1) Long Call and Short Put = Long Forward: C – P = F (2) Short Call and Long Put = Short Forward: (2) Short Call and Long Put = Short Forward: -C + P = -F

8 Factors affecting an Option’s Value Factor Call Owner Put Owner Current Price (S) +- Exercise Price (X) -+ Time to Expir. (T) ++/- Volatility (  ) ++ Risk-free rate (r f ) +-

C = S*[N(d 1 )] - Xe –r f T *[N(d 2 )] d 1 = d 2 = d 1 -  t See Spreadsheet File. Spreadsheet FileSpreadsheet File ln(S/X) + [r f + (  2 /2)]*T  t The Black-Scholes Option Pricing Model In the limit, the binomial method becomes the Black- Scholes OPM:

10 Assumptions underlying the Black-Scholes OPM Option is European-style. Option is European-style. Stock pays no dividend during the option’s term. Stock pays no dividend during the option’s term. Interest rates are constant. Interest rates are constant. No restrictions on short sales of stock. No restrictions on short sales of stock. Trading is done continuously and share prices follow a continuous Ito process. Trading is done continuously and share prices follow a continuous Ito process. The distribution of terminal stock prices is lognormal The distribution of terminal stock prices is lognormal

11 American Options Differ from European options because American options can be exercised at any time (not just at expiration). Differ from European options because American options can be exercised at any time (not just at expiration). If the stock pays no dividend, it is never optimal to exercise the option early (i.e., can sell it for S – X*D which is always greater than S – X). If the stock pays no dividend, it is never optimal to exercise the option early (i.e., can sell it for S – X*D which is always greater than S – X). For a dividend-paying stock, early exercise will be optimal only if the dividend is sufficiently large: For a dividend-paying stock, early exercise will be optimal only if the dividend is sufficiently large: Exercise early when Dividend > C – (S – X).

12 Using Options to Hedge Interest Rate Risk Interest Rate Cap: this contract represents a call option on an interest rate (rather than a bond price). Interest Rate Cap: this contract represents a call option on an interest rate (rather than a bond price). Interest Rate Floor: this contract represents a put option on an interest rate (betting that rates will fall). Interest Rate Floor: this contract represents a put option on an interest rate (betting that rates will fall). Interest Rate Collar: created by buying a cap and selling a floor (at different strike prices). You can use this when you want to lock in a range of interest rates. Interest Rate Collar: created by buying a cap and selling a floor (at different strike prices). You can use this when you want to lock in a range of interest rates.

13 Uses of FX Rate Options Used to protect against adverse movements in currencies that can affect the value of a firm’s international receivables, fixed assets, liabilities, etc. Used to protect against adverse movements in currencies that can affect the value of a firm’s international receivables, fixed assets, liabilities, etc. Typically used to hedge “firm commitments” and, increasingly, anticipated cash flows. Typically used to hedge “firm commitments” and, increasingly, anticipated cash flows. Protection (and its cost) will vary depending on whether the option is at- or out-of-the money. Protection (and its cost) will vary depending on whether the option is at- or out-of-the money. Puts and calls can be bought/sold to create combined payoffs that lock in a range of “acceptable” prices. Puts and calls can be bought/sold to create combined payoffs that lock in a range of “acceptable” prices.

14 Uses of FX Rate Options (cont.) Hedge Overseas Revenues/Assets against a weakening foreign currency (buy puts on the foreign currency and/or sell calls). Hedge Overseas Revenues/Assets against a weakening foreign currency (buy puts on the foreign currency and/or sell calls). Hedge Overseas Costs/Liabilities against a strengthening foreign currency (buy calls on the foreign currency and/or sell puts). Hedge Overseas Costs/Liabilities against a strengthening foreign currency (buy calls on the foreign currency and/or sell puts). Use the above strategies to protect cash flows and, more broadly, protect operating margins, gain a competitive advantage, and provide greater customer satisfaction. Use the above strategies to protect cash flows and, more broadly, protect operating margins, gain a competitive advantage, and provide greater customer satisfaction.

15 Uses of Commodity Options Options on commodities such as metals can protect against adverse price movements that can affect basic inputs and outputs of the production process. Options on commodities such as metals can protect against adverse price movements that can affect basic inputs and outputs of the production process. Hedging with commodity options can: Hedging with commodity options can: protect operating margins, protect operating margins, ensure cash flow is available for new investments, ensure cash flow is available for new investments, facilitate long-term planning and budgeting, and facilitate long-term planning and budgeting, and give the firm a pricing / service advantage. give the firm a pricing / service advantage. Option Spreads can be used to target the RANGE and/or COST of the commodity price protection. (See Spreadsheet File). Option Spreads can be used to target the RANGE and/or COST of the commodity price protection. (See Spreadsheet File).Spreadsheet FileSpreadsheet File

16 Uses of Equity Options Traditionally used by institutional equity investors and investment banks/dealers. Traditionally used by institutional equity investors and investment banks/dealers. Growing use of equity options by non-financial firms related to: Growing use of equity options by non-financial firms related to: Stock repurchase programs, Stock repurchase programs, Mergers, acquisitions, and divestitures, Mergers, acquisitions, and divestitures, Managing employee stock option programs, Managing employee stock option programs, Protecting an investment portfolio or enhancing yields. Protecting an investment portfolio or enhancing yields.