Options

1. Options Spring 2007 Lecture Notes 4.6.1 Readings:Mayo 28

2. Goals • Definitions • Options • Call option • Put option • Option strategies

3. Derivatives: Definition • Derivative: Any security whose payoff depends on any other security

4. Goals • Definitions • Options • Call option • Put option • Option strategies

5. Options • Two types • Call: Option to buy • Put: Option to sell • Parts: • Option price • Strike price • Expiration

6. Call Option • Option to purchase asset at the strike price • Horizon:(two types) • American: Anytime between now and the expiration date • European: On the expiration date only • Strike price: Price at which the security can be purchased

7. Example: Buying a call option on Amazon • Amazon share price = $100 • Purchase American call option • Option price = $5 • Strike price = $120 • Expiration = 2 months from now • Case A: price goes to $150 • Exercise option • Buy at $120, sell at $150 • Total = 150-120-5 = +$25 • Case B: price goes to $50 • Don’t exercise option • Total = -5 (lose entire investment)

8. Example: Writing (selling) a call option on Amazon • Amazon share price = $100 • Write American call option • Option price = $5 • Strike price = $120 • Expiration = 2 months from now • Case A: price goes to $150 • Purchaser exercises option • Buy at $150, sell at $120 • Total = -150+120+5 = -25 • Case B: price goes to $50 • Purchaser doesn’t exercise option • Total = +5

9. Options and Insurance • The writer is kind of selling insurance to the buyer • As long as the price doesn’t go up by too much ($20) the writer gets to pocket the $5 • Like an insurance premium • Danger: If price rises by large amount, option writer can lose lots of money

10. How do you lose big money with options? • Write (sell) a naked call on Amazon.com (p = 100), strike price = 150 • Sell for $5 • You feel very happy (+5) • Then Amazon goes to $250 • The other side of your option trade exercises the option • You must buy Amazon at $250, and sell it for $150

11. Option Terms • In-the-money • Stock price > call option strike price • At-the-money • Stock price = call option strike price • Out-of-the-money • Stock price < call option strike price

12. Intrinsic ValueValue of option if used today • Strike price = $100 Intrinsic Value 5 Stock Price 0 104 105 100

13. Option Pricing • Is it as easy as • (Price – strike price) when strike < stock price • 0 if strike is > stock price • Why does this get more complicated? • Have to consider today plus all days to the expiration date • Even though the price is in the zero value range today (out-of-the-money, it might move into the positive value range tomorrow

14. General Properties of an option price • Option value will be higher: • When the expiration date is farther in the future • When the stock price moves around more • (This is known as higher volatility)

15. Option Pricing • There are different formulas that try to take account of all this stuff • Black/Scholes is the most famous of these • Techniques used • Arbitrage • Stochastic calculus

16. Option Price (red) versusIntrinsic Value (black)Value of option if used today • Strike price = $100 Intrinsic Value 5 Stock Price 0 104 105 100

17. Goals • Definitions • Options • Call option • Put option • Option strategies • Real options

18. Put Option • Same as Call • Price • Strike price • Expiration • Difference: Option to Sell

19. Example: Put Value • Strike price = $100 Intrinsic Value 10 5 Stock Price 0 90 96 100 95

20. Goals • Definitions • Futures • Options • Call option • Put option • Option strategies

21. Options+Stocks • Holding option alone is known as holding a “naked option” • Holding option with the stock is known as a “covered option”

22. Insuring gains by buying a put option • Purchasing a put option on stock you already own sets a floor on what you can sell • Buy stock at 75, price rises to 100 • Lock in gains, buy put at strike = 100 • Gains will be at least 100-75 • Cost = price of the put option

23. Example 1: Buy Stock + Put • Strike price = $100 • How much would your portfolio (option + stock) be worth for different prices? Total Value 105 Stock Price 100 100 105

24. Example 2: Option Straddle • Purchase a put and call at the same strike price • Strategy makes money when stock price moves a lot (volatility is high)

25. Straddle Example • Current stock price = 100 • Purchase at-the-money call (strike = 100) for $2 • Purchase at-the-money put (strike = 100) for $3 • What is the total value of your option portfolio for different stock prices?

26. Straddle Performance • Lose money when no change in price • Price goes up: Call makes money • Price goes down: Put makes money • Strategy makes money when price moves a lot (depends on option prices)

27. Straddle Contingency GraphPlot of net $ gain as a function of stock price • Strike price = $100 • Option prices: call = $2, put = $3 Net Gain 1 0 Stock Price -5 94 95 105 106 100

28. Writing Call Options • Writing a naked call • Writing a naked put

29. Writing a Naked Call Option(1 share, option price = $5, strike = 100) +5 105 110 0 100 Stock Price -5

30. Writing a Covered Call Option(1 share, option price = $5, strike = 100, stock purchased at 100) +5 90 0 95 100 Stock Price -5

31. Other Combinations • Many other combinations are possible • As with futures, you can use options to reduce risk or increase risk if you want

32. Exotic Options • More complicated functions of prices • Often involve time path of prices • Ordinary options do not care about path • Example: Barrier option • “deactivates” if price crosses a barrier any time during a given period

33. Other Applications • Stock options • Investment options

34. Option Summary • Can be used to either reduce, or increase risk • Have insurance like characteristics • Derivatives as fire