1. Introduction to Options

2. Option – Definition An option is a contract that gives the holder the right but not the obligation to 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 sell the 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. Gain/Loss for a Buyer of a Call Option at the Expiration Date X S c Profit / Loss

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

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

8. X S p Profit or Loss 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: IV c = max(0, S 0 -X) and for puts: IV c = max(0, X- S 0 ), 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. X S Option value Intrinsic Value Time value Curren t Value Out of the money At the money In the money

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

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

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

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

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. Call Put The Factors that Affect on the Option Value

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 spread is employed by buying a call option with low strike price (X L ) and writing a call option with high strike price (X H ) Numerical Example Consider buying a call option X($45) at 8$ and writing a call option X($55) at $3.

21. XHXH s c H -c L Profit or Loss XLXL Bull SpreadShort Call X=$55 Long Call X=$45 STST -53-840 -53-845 03-350 53255 5-2760 5-121770

22. Bull Spread There is more than one way to implement a bull spread strategy: Buy a Call at X L and write a call at X H. Buy a put at X L and write a call at X H. Buy a put at X L, write a call at X H 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 (X L ) and buying a call option with high strike price (X H ) Numerical Example Consider writing a call option X($45) at 8$ and buying a call option X($55) at $3.

24. XHXH s c L -c H Profit or Loss XLXL Bear SpreadLong Call X=$55 Short Call X=$45 STST 5-3840 5-3845 0-3350 -5-3-255 -52-760 -512-1770

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 straddle is 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. Straddle Spread Long put X=$50 Long Call X=$50 STST 1217-530 27-540 -32-545 -8-3-550 -3 055 2-3560 12-31570 S Profit or Loss X - (p+c)

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

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 Butterfly is employed by buying a Call option with low strike price (X L ) and a Call option with high strike price (X H ), and write two Call options with medium strike price (X M ).

29. Long Butterfly Because of the non-linear relationship between the Call price and the strike price: C X XLXL XMXM XHXH (C(X L )+ C(XL))/2 C(X M )

30. Long Butterfly This implies that the premium balance is negative. S Profit or Loss XMXM 2C M -(C L +C H ) XLXL XHXH

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 $100/1.05 0.5 =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 above 1.82 it wants to gain one cent.

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

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

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

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

41. Numerical Example 0T

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

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 2.178)05.1(176 25.0  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:

51. The recursive solution n=2

55. The analytical solution n=2

57. Black-Scholes Option Pricing Model (B&S) The B&S Model is based on a creation of a fully hedged portfolio; thus we can you a risk-free interest rate to discount the cash flow. The model assumptions are: 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. The stock price is Log-Normal distributed, which implies that the stock returns are Normally distributed.

58. B&S Formula N(.) is the cumulative area of the standard normal distribution at the value d. e is the base of natural logarithms and is approximately equal to 2.7128. r is the continuously compounded, annual risk-free interest rate. d N(d)

59. B&S Formula N(d 1 ) is also known as delta  - the sensitivity of the option value to changes in the stock value, or the hedging ratio - the number of shares we must buy for each one call option we write to create a riskless portfolio (a fully hedged portfolio). N(d2) is the probability of exercising the option: The deeper is the option in the money the higher the option exercising probability.

60.  is the standard deviation of stock returns reflecting the volatility of stock price. The higher the stock price volatility the higher the option value. Thus, we need five variables – X, S, , r,  - to calculate the price of a call option.

61. Numerical Example Consider the following data: What is the value of a call option?

62. Numerical Example

63. B&S Formula for Currency Options r L – The local risk-free interest rate r F – The foreign risk-free interest rate

64. Numerical Example Consider the following data: the spot $/£ exchange rate is S 0 = $1.6, the US interest rate is r L = 2%, the UK interest rate is r F = 4%, the exchange rate volatility is  =10%, and the time to maturity is  =1/4. What is the value of a call option and a put option with strike price of X=$1.55?

65. Numerical Example