1. Fundamentals of Futures and Options Markets, 7th Ed, Global Edition. Ch 13, Copyright © John C. Hull 2010 Valuing Stock Options Chapter 12+13 1

2. 2 Sub-Topics Binomial model of options pricing Black-Scholes-Merton (BSM) model of options pricing Pricing options on individual stocks and indices Pricing options on currencies Pricing options on interest rates Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

3. 3 Introduction Two methods for pricing options Binomial model: a discrete-time option pricing model Black-Scholes-Merton model: a continuous time option pricing model Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

4. 4 Binomial model of options pricing One-step binomial model The binomial model limits the price moves of the underlying asset to one of only two possible new prices A one-period model limits the time over which the price move occurs to one period, at the end of which the underlying asset moves to one of two possible prices and simultaneously the option expires We assume that arbitrage profits are arbitraged away to reveal an arbitrage-free price Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

5. 5 Binomial model of options pricing One-step binomial model You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. What is the value of the call option at expiry if the stock price is $22? What is the value of the call option at expiry if the stock price is $18? What volume of stock makes the portfolio riskless? What is the future value of the portfolio? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

6. 6 Binomial model of options pricing One-step binomial model Stock Price = $22 Stock Price = $18 Stock price = $20 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

7. 7 Binomial model of options pricing One-step binomial model What is the value of the call option at expiry if the stock price is $22? What is the value of the call option at expiry if the stock price is $18? Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

8. 8 Binomial model of options pricing One-step binomial model What volume of stock makes the portfolio riskless? What is the future value of the portfolio? The portfolio is riskless so we would expect it to have the same value in either scenario. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

9. 9 Binomial model of options pricing One-step binomial model You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa, continuously compounded. What is the current value of the portfolio? What is the current value of the call option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

10. 10 Binomial model of options pricing One-step binomial model What is the current value of the portfolio? Riskless portfolios earn the risk-free rate of return, hence the present value of the portfolio equals the future value discounted at the risk-free rate of return. What is the current value of the call option? The current value of the portfolio also equals the value of the stock plus the value of the option, hence Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

11. 11 Binomial model of options pricing Generalised one-step binomial model S 0 =stock price f = price of option S 0 u =stock price moves up S 0 d =stock price moves down fu = price of option if stock price moves up fd= price of option if stock price moves down Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

12. 12 Binomial model of options pricing Generalised one-step binomial model A derivative lasts for time T and is dependent on a stock Su ƒ u Sd ƒ d SƒSƒ Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

13. 13 Binomial model of options pricing Generalised one-step binomial model Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when Su  – ƒ u = Sd  – ƒ d or Su  – ƒ u Sd  – ƒ d Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

14. 14 Binomial model of options pricing Generalised one-step binomial model Value of the portfolio at time T is Su  – ƒ u = Sd  – ƒ d Value of the portfolio today is (Su  – ƒ u )e –rT Another expression for the portfolio value today is S 0  – f Hence ƒ = S 0  – ( Su  – ƒ u )e –rT Substituting for  we obtain ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

15. 15 Binomial model of options pricing One-step binomial model You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa, continuously compounded. What is the current value of the call option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

16. 16 Binomial model of options pricing One-step binomial model What is the current value of the call option? The probability of an up movement: The value of the option: Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

17. 17 Binomial model of options pricing Illustrate how to arbitrage an anomaly You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa, continuously compounded. How would you profit from an arbitrage if the option was quoted at $1.00? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

18. 18 Binomial model of options pricing Illustrate how to arbitrage an anomaly How would you profit from an arbitrage if the option was quoted at $1.00? If the option is selling at $1.00 and it should be selling at $0.633, it is overpriced. Sell the option and buy the stock. The number of units of stock bought per option sold: Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

19. 19 Binomial model of options pricing Illustrate how to arbitrage an anomaly How would you profit from an arbitrage if the option was quoted at $1.00? If we sell 1,000 calls and buy 250 shares, this would require borrowing, at the risk-free rate, funds equal to: ie borrow $4,000 At expiry the portfolio will equal: The return on the investment will equal: Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

20. 20 Binomial model of options pricing Risk-neutral valuation The variables p and (1  – p ) can be interpreted as the risk-neutral probabilities of up and down movements In a risk-neutral world all individuals are indifferent to risk and hence require no compensation for risk, therefore the expected return on all securities is equal to the risk- free interest rate. The value of a derivative is its expected payoff in a risk- neutral world discounted at the risk-free rate Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

21. 21 Binomial model of options pricing One-step binomial model You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In 3 months it will either be $22 or $18. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa, continuously compounded. What is the current value of the call option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

22. 22 Binomial model of options pricing One-step binomial model What is the current value of the call option? In a risk-neutral world the expected return on a stock must equal the risk-free rate At the end of three months, the call option has a 0.6523 probability of being worth 1 and a 0.3477 probability of being worth zero. Its expected future value therefore is: Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

23. 23 Binomial model of options pricing One-step binomial model What is the current value of the call option? In a risk-neutral world the expected future value should be discounted at the risk-free rate to get the present value Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

24. 24 Binomial model of options pricing Two-step binomial model: Call option You have a long position in a stock and a short position in a call option on the stock. The current price of the stock is $20. In consecutive 3-month periods there is an equal chance it will either rise by 10% or fall by 10%. The 3-month call option has a strike price of $21. The risk-free rate of interest is 12% pa continuously compounding. What is the value of the option at nodes B and C? What is the value of the option at node A? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

25. 25 Binomial model of options pricing Two-step binomial model: Call option 20 A B C D F E Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

26. 26 Binomial model of options pricing Two-step binomial model: Call option The value of the stock at nodes D, E and F: 20 A 22 18 24.2 19.8 16.2 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

27. 27 Binomial model of options pricing Two-step binomial model: Call option The value of the option at nodes D, E and F: 20 A 22 18 24.2 3.2 19.8 0.0 16.2 0.0 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

28. 28 Binomial model of options pricing Two-step binomial model: Call option The value of the option at nodes B and C: 20 A 22 2.0257 18 0.0 24.2 3.2 19.8 0.0 16.2 0.0 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

29. 29 Binomial model of options pricing Two-step binomial model: Call option The value of the option at node A: 20 1.2823 22 2.0257 18 0.0 24.2 3.2 19.8 0.0 16.2 0.0 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

30. 30 Binomial model of options pricing Generalised two-step binomial model f fufu fdfd f uu f dd f ud p 1-p p p Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

31. 31 Binomial model of options pricing Generalised two-step binomial model The value of an option using the generalised two-step binomial model can be calculated Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

32. 32 Binomial model of options pricing Two-step binomial model: Put option A two-year European put has a strike of $52 on a stock whose current price is $50. There are two time steps of one year, in each the stock price either moves up by 20% or down by 20%. The risk-free rate of interest is 5% pa continuously compounding. What is the value of the option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

33. 33 Binomial model of options pricing Two-step binomial model: Put option What is the value of the option? 50 60 1.4147 40 9.4636 72 0 48 4 32 20 0.6282 1-0.6282 0.6282 1-0.6282 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

34. 34 Binomial model of options pricing Two-step binomial model: Put option What is the value of the option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

35. 35 Binomial model of options pricing American options In valuing American options The value of the option at the final nodes remains the same as for European options The value of the option at earlier nodes is the greater of: The expected payoff discounted at the risk-free rate The payoff from early exercise: Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

36. 36 Binomial model of options pricing Two-step binomial model: American A two-year American put has a strike of $52 on a stock whose current price is $50. There are two time steps of one year, in each the stock price either moves up by 20% or down by 20%. The risk-free rate of interest is 5% pa continuously compounding. What is the value of the option? Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

37. 37 Binomial model of options pricing Two-step binomial model: American What is the value of the option? 50 60 1.4147 40 12 72 0 48 4 32 20 0.6282 1-0.6282 0.6282 1-0.6282 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

38. 38 Binomial model of options pricing Two-step binomial model: American What is the value of the option? 50 5.0894 60 1.4147 40 12 72 0 48 4 32 20 0.6282 1-0.6282 0.6282 1-0.6282 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

39. 39 Binomial model of options pricing Delta Delta (  ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock In a multi-step binomial tree the value of  varies from node to node Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

40. 40 Binomial model of options pricing Determining u and d In practice u and d are determined from the stock price volatility: where  is the volatility and  t is the length of the time step This is the approach used by Cox, Ross, and Rubinstein Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

41. 41 Binomial model of options pricing Options on various assets The price on options on various assets, calculated using the binomial model, is similar except for the calculation of p : where a equals e r  t for a non dividend paying stock or bond e (r-q)  t for a dividend paying stock or index e (r-rf)  t for a currency 1 for a futures contract Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

42. 42 Black-Scholes-Merton model of options pricing Explain the assumptions of the model The returns of the underlying asset are continuously compounding and are normally distributed, ie they are log-normally distributed There are no riskless arbitrage opportunities Investors can borrow and lend at the risk-free rate, which in the short term is constant The volatility of the underlying is known and constant There are no taxes or transaction costs There are no cashflows on the underlying The options are European Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

43. Introduction We look at the standard approach to pricing options where we focus on European options which can only be exercised at a specific time. A call option gives the buyer the right to buy the asset at time T for the strike price K so at time T Value of a Call = max(S T - K, 0) A put option gives the buyer the right to sell the asset at time T for the strike price K so at time T Value of a Put = max( K - S T, 0) What should these values be at earlier times? 43

44. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Introduction The value of a call option c has 3 parts The intrinsic value is the value if the option was exercised at time t which is (S t - K ) The time value of money on the strike price is the difference the strike price and its present value which shows how much we save by paying K at time T not now ( K - K e - rT ) The insurance I shows how much investors are willing to pay to limit future losses So c = (S t - K ) + ( K - K e - rT ) + I = S t - K e - rT + I. 44

45. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Introduction In the formula for the value of a call option c = S t - K e - rT + I we know K, r and T but we do not know what the share price S t will be at any future date. The best we can do is to make assumptions about how share prices change over time and what this tells us about the probability distribution of possible S t values i.e. what type of distribution and what mean and variance the S t values have. We use these assumed values in our formula for c 45

46. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 How Share Prices Move Many studies have shown share prices S i have a skewed probability distribution like the lognormal distribution shown in Fig 13.1 p 290. For values with a lognormal distribution, the logs of these values ln(S i ) have the normal distribution shown in Fig 13.2 p 291. If a share does not have dividends then its continuous rate of return u i is defined as the log of the ratio of the current price & the previous price u i = ln (S i / S i-1 ) = ln (S i ) - ln (S i-1 ) 46

47. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 How Share Prices Move As both ln (S i ) and ln (S i-1 ) are normally distributed so too is their difference u i. Using this result in the Black-Scholes model it is assumed that - Returns on a share (  S / S) over short time periods are normally distributed - Returns in different periods are independent - In 1 period the returns have mean  and standard deviation . - In  t periods the returns have mean  t and variance of  2  t 47

48. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 How Share Prices Move If S T is the share price at time T and it has a lognormal distribution then it will have Mean E(S T ) = S 0 e  T Variance Var(S T ) = S 0 2 e 2  T See next slide For the long term continuous returns ln (S T / S 0 ) Mean  -  2 /2 Variance  2 See Ex 13.2: Confidence Limits for Stock Returns 48

49. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Lognormal Distribution 49

50. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Expected Return p 293 From the CAPM we know the expected return  that investors require depends upon the riskiness of an asset & The level of interest rates like r The value of an option is not affected by  but there is an issue you need to be aware of. While the return in a short period  t is  t the return with continuous compounding over long periods R has a different mean from  namely E(R) =  –    50

51. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Expected Return To see why suppose the  t are 1 day periods with 250 trading days in a year then  t = 1/250 If the mean daily return is  (1/250) the mean yearly return should be  … but it is not!! The yearly return over a period of T years with continuous compounding R is given by For this R value we find E( R) =  –    51

52. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Expected Return This difference reflects the difference between arithmetic and geometric means Geometric means are always lower because they are less affected by extreme values This is illustrated in the next snapshot 52

53. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 53 Mutual Fund Returns (See Business Snapshot 13.1 on page 294) If Returns are 15%, 20%, 30%, -20% and 25% Their arithmetic mean of these returns is 14% (15 + 20 + 30 - 20 + 25) / 5 = 14 The actual value of $100 after 5 yrs is 100 x 1.15 x 1.2 x 1.3 x 0.8 x 1.25 = $179.40 With 14% returns we should have 100 x 1.14 5 = 192.54 The actual return is the geometric mean 12.4% 100 x 1.124 5 = 179.40

54. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Volatility The value of the insurance component I of c depends upon the riskiness of the call option which depends upon the volatility. The volatility which is the standard deviation of the continuously compounded rate of return is  in 1 year & in period  t If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day? 54

55. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued where there are 252 trading days in one year and 1 day is a period of  t = (1/252) If  = 25% p.a. the volatility for 1 day is = 25 x 0.063 = 1.575% 55

56. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Estimating Volatility from Historical Data (page 295-298) 1. Take observations S 0, S 1,..., S n on the variable at end of each trading day 2. Define the continuously compounded daily return as: 3. Calculate the standard deviation, s, of the u i ´s (This is for daily returns) 4. The historical volatility per yearly estimate is: 56

57. Estimating Volatility from Historical Data (Calculating  ) To find the mean for u i we use the formula To find the variance for u i we use the formula To find the standard deviation  we find the square root of the variance Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 57

58. Estimating Volatility from Historical Data (If there are Dividends) Dividends are usually paid twice a year. In those periods where there are no dividends we use the same formula for daily returns u i which is u i = ln (S i / S i-1 ) When dividends D are paid the formula changes to u i = ln ([S i + D]/ S i-1 ) The formulae for the mean and variance are the same as when there are no dividends Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 58

59. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Concepts Underlying Black-Scholes Key Assumptions Share prices have a lognormal distribution with mean  and standard deviation  All assets are perfectly divisible and have zero trading costs There are no dividends in the time to maturity There are no riskless arbitrage opportunities Security trading is continuous Investors can borrow or lend at a constant risk-free rate r. 59

60. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Concepts Underlying Black-Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty (see p 298) The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate 60

61. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Concepts Underlying Black-Scholes To obtain this formula we set up a portfolio containing shares and options The option price & the stock price depend on the same underlying source of uncertainty and move in a well defined way as c rises when S rises & p falls when S rises We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty as it gives a fixed return 61

62. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Concepts Underlying Black-Scholes The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate In the example on p 298-299 and Fig 13.3 we see that c and S change in the following way  c = 0.4  S Here the riskless portfolio contains A long position in 40 shares A short position in 100 call options N.B. If this relationship changes however we would have to rebalance 62

63. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The Black-Scholes Formulas (See page 299-300) 63

64. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 The N(x) Function The other terms have all been used before but in addition to the d terms there is a new term N ( d ) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than d as shown in Fig 13.4 The tables for N(d) at the end of the book and at the back of the Formula sheet The use of the Black-Scholes formula is demonstrated in Ex 13.4 p 301 64

65. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Properties of Black-Scholes Formula As S 0 becomes very large both d 1 and d 2 also become large and both N ( d 1 ) and N ( d 2 ) are both now close to 1, the area under the Normal curve. With N ( d 1 ) and N (d 2 ) values close to 1 we find from our option value formulae that c tends to S 0 – Ke - rT and p tends to zero As S 0 becomes very small now c tends to zero and p tends to Ke -rT – S 0 65

66. Black-Scholes-Merton model of options pricing The BSM model Example 10.9 The stock price six months from the expiration of a European option is $42, the exercise price is $40, the risk-free interest rate is 10% per annum, and the volatility is 20% per annum. What is the value of the option if it is a call? What is the value of the option if it is a put? 66 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

67. Black-Scholes-Merton model of options pricing The BSM model Example 10.9 What is the value of the option if it is a call? Using tables: N(0.7693) = 0.7791, N(0.6278) = 0.7349, N(-0.7693) = 0.2209 and N(-0.6278) = 0.2651 The price of the European call option is $4.76. 67 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

68. Black-Scholes-Merton model of options pricing The BSM model Example 10.9 What is the value of the option if it is a put? Using tables: N(0.7693) = 0.7791, N(0.6278) = 0.7349, N(-0.7693) = 0.2209 and N(-0.6278) = 0.2651 The price of the European put option is $0.81. 68 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010

69. Risk-Neutral Valuation A key result in the pricing of derivatives is the risk-neutral valuation principle which says Any security dependent on other traded securities can be valued on the assumption that investors are risk neutral While investors may not actually be risk-neutral we can assume they are when we derive the prices of derivatives. This means All expected returns are equal to r We can use r as our discount rate everywhere 69

70. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Applying Risk-Neutral Valuation Derivatives can be valued with the following procedure 1. Assume that the expected return from an asset is the risk-free rate 2. Calculate the expected payoff from the derivative 3. Discount at the risk-free rate 70

71. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Valuing a Forward Contract with Risk- Neutral Valuation In the case of forward contracts if we have a long contract then Payoff at expiry is S T – K Expected payoff in a risk-neutral world is S 0 e rT – K The value of the forward contract f is given by the present value of the expected payoff f = e - rT [S 0 e rT – K] = S 0 – Ke - rT. (which is the same as equation 5.5). 71

72. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price The is a one-to-one correspondence between prices and implied volatilities Traders and brokers often quote implied volatilities rather than dollar prices The CBOE publishes the SPX VIX which shows the implied volatility for a range of 30-day put and call options on the S&P 500 72

73. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Implied Volatility An index value of 15 means an implied volatility of about 15% In the futures contracts on the VIX one contract is 1000 times the index How futures contracts on the VIX work is shown in Ex 13.5 p 304 In the graph in Fig 13.5 on p 305 we can see how the VIX is usually somewhere between 10 and 20 but during the GFC it got up to 80. 73

74. The VIX Index of S&P 500 Implied Volatility; Jan. 2004 to Sept. 2009 Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 74

75. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black- Scholes-Merton formula Only dividends with ex-dividend dates during life of option should be included The “dividend” should be the expected reduction in the stock price on the ex- dividend date Look at Ex 13.6 p 306 75

76. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 American Calls An American call on a non-dividend-paying stock should never be exercised early An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date 76

77. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Black’s Approximation for Dealing with Dividends in American Call Options This procedure is illustrated in Ex 13.7 p 307 Here we set the American price equal to the maximum of two European prices: 1. The 1st European price is for an option maturing at the same time as the American option 2. The 2nd European price is for an option maturing just before the final ex-dividend date 77

78. Fundamentals of Futures and Options Markets, 7th Ed, Ch 13, Copyright © John C. Hull 2010 Valuing Employee Stock Options p 306-309 Please read Ch 13.11 very carefully as this material is examinable. 78