Introduction to Options

1. Introduction to Options

2. Option – Definition • An option is a contract that gives the holder the right but not the obligationto buy or sell a defined asset called the underlying asset at a predetermined price within a specified period of time. • There are two fundamental types of options – call option and put option • A call option gives the holder the right to buy the asset at a predetermined price within a specified period of time. • A put option gives the holder the right to sellthe asset at a predetermined price within a specified period of time.

3. Option – Definitions Underlying asset – the asset on which the option contract is written. Strike price (or Exercise price) – the fixed, predetermined price at which the holder of the option can buy or sell the underlying asset. Option Premium – the price paid for the option Expiration date – the lat day on which the option can be exercised There are two types of option: European option – An option that can be exercised only on the expiration date itself. American option - An option that can be exercised at any time up to and including the expiration date.

4. Option – Definition • We say the seller writes an option. • If you buy an option , we say you have a long position in the option; and when you write an option, we say you have a short position in the option • Denoting • S – the value of the underlying asset • c – the value of a call option • p – the value of a put option • X – the exercise price • T – the expiration date • t – current time •  - T-t: time to expiration

5. Profit / Loss S X c Gain/Loss for a Buyer of a Call Option at the Expiration Date

6. Profit / Loss c S X Gain/Loss for a writer of a Call Option at the Expiration Date

7. Profit / Loss S X p Gain/Loss for a Buyer of a Put Option at the Expiration Date

8. Profit or Loss p S X Gain/Loss for a Seller of a Put Option at the Expiration Date

9. Intrinsic and Time Value • Option prices can be broken down into two components: intrinsic value and time value. • The Intrinsic value is the value of the option if it is immediately exercised or zero. • For calls: IVc = max(0, S0-X) and for puts: IVc = max(0, X- S0), as the stock price change IV may change as well. • The time value is difference between the option current value and the intrinsic value, reflecting the possibility that the option will create further gains in the future.

10. Intrinsic and Time Value • Call options also classified into: • At the money – when the current stock price is close to the strike price. • In the money – when the current stock price is much above the strike price. • Out of the money – when the current stock price is much below the strike price.

11. Option value Current Value Time value Intrinsic Value S X Out of the money At the money In the money

12. Option value Current Value Time value Intrinsic Value S X In the money At the money Out of the money

13. The Range of A Call Option’s Values At the expiration date:

14. Cash Flows At Expiration Trading Strategy Today (0) Buy one call option Sell short one share of stock Lend Net cash flow

15. Call Price Value Range A Call Option’s Upper and Lower Boundaries

16. The Range of A Put Option’s Values At the expiration date:

17. Cash Flows At Expiration Trading Strategy Today (0) Buy one put option Buy one share of stock Borrow Net cash flow

18. put Price Value Range A Put Option’s Upper and Lower Boundaries

19. The Factors that Affect on the Option Value Call Put

20. Option Strategies • Bull Spread • A bull spread is an option strategy designed to allow investors to profit if prices rise but to limit his losses if prices fall. • A bull spreadis employed by buying a call option with low strike price (XL) and writing a call option with high strike price (XH) • Numerical Example • Consider buying a call option X($45) at 8$ and writing a call option X($55) at $3.

21. Profit or Loss XL s XH cH-cL

22. Bull Spread • There is more than one way to implement a bull spread strategy: • Buy a Call at XL and write a call at XH. • Buy a put at XL and write a call at XH. • Buy a put at XL, write a call at XH and buy the stock.

23. Option Strategies • Bear Spread • A bear spread is an option strategy designed to allow investors to profit if prices fall but to limit his losses if prices rise. • A bear spread is employed by writing a call option with low strike price (XL) and buying a call option with high strike price (XH) • Numerical Example • Consider writing a call option X($45) at 8$ and buying a call option X($55) at $3.

24. Profit or Loss cL-cH XL s XH

25. Option Strategies • Long Straddle • A long straddle is an option strategy designed to investor who believes that something dramatic will happen to the stock price but he has no sure exactly which direction it will go . • A long straddleis employed by buying both put and call options at the same strike price. • Numerical Example • Consider buying a call option X($50) at $5 and buying a put option X($50) at $3.

26. Profit or Loss S X - (p+c)

27. Profit or Loss p+c S X • Short Straddle • A short straddle is an option strategy designed to investor who believes that stock price will be stable. • A short straddleis employed by writing both put and call options at the same strike price.

28. Long Butterfly • A Long Butterfly is an option strategy designed to investor who believes that stock price will be stable but to limit his losses if the price will be volatile. • A Butterflyis employed by buying a Call option with low strike price (XL) and a Call option with high strike price (XH), and write two Call options with medium strike price (XM).

29. Long Butterfly • Because of the non-linear relationship between the Call price and the strike price: C (C(XL)+ C(XL))/2 C(XM) X XL XM XH

30. Profit or Loss S XH 2CM-(CL+CH) XM XL • Long Butterfly • This implies that the premium balance is negative.

31. Put-Call Parity • The value of a call option and a put option on the same underlying asset, with the same exercise price and maturity, are related by simple formula called put-call parity. • The formula is derived by the no-arbitrage argument, using a strategy composed of the underlying asset, a put option, a call option, and a riskless asset. • At the expiration date, this strategy’s cash flow is expected to be zero in each event, and therefore, its value must be zero.

32. Cash Flows At Expiration Trading Strategy Today (0) Buy one call option Sell short one share of stock Lend Write (Sell) one put option Net cash flow

33. Numerical Example • The stock price is $100 and the risk-free interest rate is 5%. A Call and a put options with a strike price of $100 and 6 month to maturity are traded at $5 and $4, respectively. Show arbitrage strategy!

34. Cash Flows At Expiration Trading Strategy Today (0) Buy one call option Sell short one share of stock Write (Sell) one put option Lend $100/1.050.5=97.6 Net cash flow

35. Hedging with Options and Forward Contracts • A U.S. firm has been promised a payment of 1M£ in one month • The spot price is 1.8$/£. A Call and a Put options with a strike price of 1.82$/£ and one month to maturity are traded at $0.05 and $0.02, respectively. A forward contract with one month to maturity is traded at a forward rate of 1.83$/£. • The firm wants to ensure that it will not get less than $1.78 per one pound, but for each cent that the spot price will above1.82 it wants to gain one cent.

36. Buying a Put Option Long £ 1.82 -0.02 1.8 1.82

37. Selling a Forward and Buying a Call Option Long £ 1.83 -0.05 1.78 1.83 1.82

38. Binomial Option Pricing Model (BOPM) • The BOPM is a relatively simple way to price options and it is based on the following assumptions: • An efficient market. • Short Selling is allowed with a full used of the proceeds. • Borrowing and lending at the risk-free interest rate is permitted. • Future stock price will have one of two possible values. • The BOPM is developed in four steps

39. 0 T • Step 1: Determine Stock Price Distribution • The two possible future values of the stock are Su and Sd, where: • u and d are constants and satisfy:

40. 0 T • Step 2: Determine Option Price Distribution • Given the stock price distribution we can calculate the value of the call option at expiration date.

41. 0 T Numerical Example

42. Step 3: Create A Riskless Portfolio • As the stock and the option’s values are fully correlated, we can construct a riskless portfolio by holding the stock and the option in opposite direction with some proportion: • Writing one call option • Buying h shares of stock such that the portfolio's future cash flow will be identical in each state of nature:

43. Cash Flows At Expiration Trading Strategy Today (0) Writing 3 call option Buy 2 shares of stock Net cash flow • h is the number of shares we must buy for one call option we write.

44. Step 4: Solve for the Call Using NPV • The portfolio's value is the present value of its expected cash flows • As the portfolio’s future cash flows is known for certainty, the appropriate discount rate should be the risk-free interest rate.

45. Numerical Example • Consider the pervious example and suppose that the time to maturity is =1/4 year and the risk-free interest rate is 5%

46. Cash Flows At Expiration Trading Strategy Today (0) Write (sell) 3 call option Buy 2 shares of stock Borrow (200-24)=176 Net cash flow - × = - 0 . 25 176 ( 1 . 05 ) 178 . 2 • Arbitrage Opportunity • Suppose that the call option is traded at $8, which implies that it is overpriced

47. Cash Flows At Expiration Trading Strategy Today (0) Buy (sell) 3 call option Sell short 2 shares of stock Lend (200-18)=182 Net cash flow • Arbitrage Opportunity • Suppose that the call option is traded at $6, which implies that it is underpriced

48. The analytical Solution of the BOPM Substituting the hedging (h) equation in the pricing equation:

49. Substituting the Call pricing equation in the Put-Call-Parity:

50. The Multi-Period BOPM The number of choices in period n is equal to n+1: