Principles of option pricing

1. Principles of option pricing

2. Option A contract that gives the holder the right - not the obligation -  to buy (call), or to sell (put) a specified amount of the underlying asset, at a set exchange rate and expiration date.

3. Glossary of terms The investor buying the option is called the buyer or holder. The investor selling the option is called the writer or seller. When the holder of the option decides to buy (sell) the asset at maturity, it is said that he/she is exercising the option. The asset to be bought or sold is called the underlying asset. Since the holder enjoys a privilege - the option to buy or sell - he/she must pay a premium to acquire the option. The price agreed upon for buying or sellingthe underlying asset is called exercise price or strike price.

4. Glossary of terms (con’t) Options are traded on options exchanges. The number of outstanding option contracts at any time is called open interest.

5. American vs. European options American options can be exercised at any time during their life span European options can be exercised only at maturity

6. Option valuation basics Like with any other financial asset, the option premium or market value or option price is a function of future expected cash flows.

7. Notation C: price of an American call c: price of an European call P: price of an American put p: price of an European put E: exercise or strike price S: stock price before maturity ST: stock price at maturity T: time to maturity r: risk-free rate

8. Boundaries to option prices: Call options. At expiration: C = max[0, (ST -E)] Before expiration Upper bound: A call cannot sell for more than the stock: C < S and c < S Lower bound: C > = max[0, (S -E)] c > = max[0, (S - E/(1+r)T)]

9. What happens if this relationship is not satisfied?

10. Boundaries to option prices: Arbitrage Assume the Exxon December 26 call struck at $80 sells for $1. It is now December 17. The stock of Exxon is at $83/share. The risk-free rate is 6%. If the option is American, buy the call for $1, exercise it and make $3 Arbitrage profit = $2

11. What if the option is European? Construct an arbitrage portfolio

12. Arbitrage portfolio to exploit boundary violations for European call options

13. Arbitrage portfolio to exploit boundary violations for European call options

14. Arbitrage portfolio to exploit boundary violations for European call options

15. Arbitrage portfolio to exploit boundary violations for European call options

16. Arbitrage portfolio to exploit boundary violations for European call options

17. Arbitrage portfolio to exploit boundary violations for European call options

18. Arbitrage portfolio to exploit boundary violations for European call options

19. Arbitrage portfolio to exploit boundary violations for European call options

20. Arbitrage portfolio to exploit boundary violations for European call options

21. Arbitrage portfolio to exploit boundary violations for European call options

22. Analysis We have created a riskless portfolio: the terminal cash flow is zero, regardless of the stock price, while the up-front cash flow is positive. We made $2.116 in pure arbitrage profits.

23. Boundaries to option prices: Put options. At expiration: P = max[0, (E -ST)] Before expiration Upper bound: A put cannot sell for more than the stock: P < S and p < S Lower bound: P > = max[0, (E - S)] p > = max[0, (E/(1+r)T -S)]

24. Boundary violations By now, we know that if price boundaries are violated, we might be able to construct an arbitrage portfolio.

25. Boundary violations: American put options Assume the Exxon December 26 put struck at $80 sells for $2. It is now December 17. The stock of Exxon is at $75/share. The risk-free rate is 6%. If the option is American, we can buy it for $2 and exercise it. Arbitrage profit = $3.

26. Arbitrage portfolio to exploit boundary violations for European put options

27. Arbitrage portfolio to exploit boundary violations for European put options

28. Arbitrage portfolio to exploit boundary violations for European put options

29. Arbitrage portfolio to exploit boundary violations for European put options

30. Arbitrage portfolio to exploit boundary violations for European put options

31. Arbitrage portfolio to exploit boundary violations for European put options

32. Arbitrage portfolio to exploit boundary violations for European put options

33. Arbitrage portfolio to exploit boundary violations for European put options

34. Arbitrage portfolio to exploit boundary violations for European put options

35. Analysis We have created a riskless portfolio: the terminal cash flow is zero, regardless of the stock price, while the up-front cash flow is positive. We made $2.8835 in pure arbitrage profits.

36. Three important concepts Intrinsic value - how much the call is worth if exercised. Market value, price, or premium - the price at which the call can be sold/purchased in the market. Time value - the difference between premium and intrinsic value

37. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

38. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

39. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

40. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

41. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

42. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

43. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

44. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

45. S - E S E A Snapshot of the intrinsic value of the American Call: S - E

46. E - S E S E A Snapshot of the intrinsic value of the American Put: E - S

47. E - S S E A Snapshot of the intrinsic value of the American Put: E - S E

48. E - S S E A Snapshot of the intrinsic value of the American Put: E - S E

49. E - S S E A Snapshot of the intrinsic value of the American Put: E - S E

50. E - S S E A Snapshot of the intrinsic value of the American Put: E - S E