1. OPTION PRICING MODEL Black-Scholes (BS) OPM. Cox, Ross, and Rubinstien Binomial Option Pricing Model: BOPM Black-Scholes (BS) OPM. Cox, Ross, and Rubinstien Binomial Option Pricing Model: BOPM

2. BOPM Model is based on constructing a replicating portfolio (RP). The RP is a portfolio whose cash flows match the cash flows of a call option. By the law of one price, two assets with the same cash flows will in equilibrium be equally priced; if not, arbitrage opportunities would exist. Model is based on constructing a replicating portfolio (RP). The RP is a portfolio whose cash flows match the cash flows of a call option. By the law of one price, two assets with the same cash flows will in equilibrium be equally priced; if not, arbitrage opportunities would exist.

3. SINGLE-PERIOD BOPM The single-period BOPM assumes there is one period to expiration (T) and two possible states at T -- up state and a down state. The model is based on the following assumptions: The single-period BOPM assumes there is one period to expiration (T) and two possible states at T -- up state and a down state. The model is based on the following assumptions:

4. ASSUMPTION 1 In one period the underlying stock will either increase to equal a proportion u times its initial value (So) or decrease to equal a proportion d times its initial value. Example: assume current stock price is $100 and u = 1.1 and d =.95. In one period the underlying stock will either increase to equal a proportion u times its initial value (So) or decrease to equal a proportion d times its initial value. Example: assume current stock price is $100 and u = 1.1 and d =.95.

5. Binomial Stock Movement

6. Assumption 2 Assume there is a European call option on the stock that expires at the end of the period. Example: X = $100. Assume there is a European call option on the stock that expires at the end of the period. Example: X = $100.

7. Binomial Call Movement

8. Assumption 3 Assume there is a risk-free security in which investors can go short or long. Rf = period riskfree rate. rf = (1 + Rf). Example: Rf =.05 and rf = 1.05. Assume there is a risk-free security in which investors can go short or long. Rf = period riskfree rate. rf = (1 + Rf). Example: Rf =.05 and rf = 1.05.

9. Replicating portfolio The replicating portfolio consist of buying Ho shares of the stock at So and borrowing Bo dollars. Replicating Portfolio is therefore a leveraged stock purchase. Given the binomial stock price movements and the rate on the risk-free security, the RP’s two possible values at T are known. The replicating portfolio consist of buying Ho shares of the stock at So and borrowing Bo dollars. Replicating Portfolio is therefore a leveraged stock purchase. Given the binomial stock price movements and the rate on the risk-free security, the RP’s two possible values at T are known.

10. Replicating Portfolio:

11. Constructing the RP The RP is formed by solving for the Ho and Bo values (Ho* and Bo*) which make the two possible values of the replicating portfolio equal to the two possible values of the call (Cu and Cd). Mathematically, this requires solving simultaneously for the Ho and Bo which satisfy the following two equations. The RP is formed by solving for the Ho and Bo values (Ho* and Bo*) which make the two possible values of the replicating portfolio equal to the two possible values of the call (Cu and Cd). Mathematically, this requires solving simultaneously for the Ho and Bo which satisfy the following two equations.

12. Solve for Ho and Bo where: Equations:

13. Solution Equations:

14. Example Equations

15. Example A portfolio consisting of Ho* =.6667 shares of stock and debt of Bo* = $60.32, would yield a cash flow next period of $10 if the stock price is $10 and a cash flow of 0 if the stock price is $95. These cash flows match the possible cash flow of the call option. A portfolio consisting of Ho* =.6667 shares of stock and debt of Bo* = $60.32, would yield a cash flow next period of $10 if the stock price is $10 and a cash flow of 0 if the stock price is $95. These cash flows match the possible cash flow of the call option.

16. RP’s Cashflows End-of-Period CF:

17. Law of One Price By the law of one price, two assets which yield the same CFs, in equilibrium are equally priced. Thus, the equilibrium price of the call is equal to the value of the RP. By the law of one price, two assets which yield the same CFs, in equilibrium are equally priced. Thus, the equilibrium price of the call is equal to the value of the RP.

18. Equilibrium Call Price BOPM Call Price:

19. Arbitrage The equilibrium price of the call is governed by arbitrage. If the market price of the call is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the call and long (short) in the replicating portfolio. The equilibrium price of the call is governed by arbitrage. If the market price of the call is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the call and long (short) in the replicating portfolio.

20. Arbitrage: Overpriced Call Example: Market call price = $7.35 Strategy: –Short the Call –Long RP Buy Ho*=.6667 shares at $100 per share. Borrow Bo* = $60.32. Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95. Example: Market call price = $7.35 Strategy: –Short the Call –Long RP Buy Ho*=.6667 shares at $100 per share. Borrow Bo* = $60.32. Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95.

21. Initial Cashflow Short call Long Ho shares at So Borrow Bo CFo Short call Long Ho shares at So Borrow Bo CFo $7.35 -$66.67=(.6667)$100 $60.32 $1 $7.35 -$66.67=(.6667)$100 $60.32 $1

22. End-of-Period CF

23. Arbitrage: underpriced Call Example: Market call price = $5.35 Strategy: –Long In call –Short RP Sell Ho*=.6667 shares short at $100 per share. Invest Bo* = $60.32 in riskfree security. Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95. Example: Market call price = $5.35 Strategy: –Long In call –Short RP Sell Ho*=.6667 shares short at $100 per share. Invest Bo* = $60.32 in riskfree security. Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95.

24. Initial Cashflow Long call Short Ho shares at So Invest Bo in RF CFo Long call Short Ho shares at So Invest Bo in RF CFo -$5.35 $66.67=(.6667)$100 -$60.32 $1 -$5.35 $66.67=(.6667)$100 -$60.32 $1

25. End-of-Period CF

26. Conclusion When the market price of the call is equal to $6.35, the arbitrage is zero. Hence, arbitrage ensures that the price of the call will be equal to the value of the replicating portfolio. When the market price of the call is equal to $6.35, the arbitrage is zero. Hence, arbitrage ensures that the price of the call will be equal to the value of the replicating portfolio.

27. Alternative Equation By substituting the expressions for Ho* and Bo* into the equation for Co*, the equation for the equilibrium call price can be alternatively expressed as:

28. BOPM Equation Alternative Equation:

29. BOPM Equation Example:

30. Note In the alternative expression, p is defined as the risk-neutral probability of the stock increasing in one period. The bracket expression can be thought of as the expected value of the call price at T. Thus, the call price can be thought of as the present value of the expected value of the call price. In the alternative expression, p is defined as the risk-neutral probability of the stock increasing in one period. The bracket expression can be thought of as the expected value of the call price at T. Thus, the call price can be thought of as the present value of the expected value of the call price.

31. Realism To make the BOPM more realistic, we need to –extend the model from a single period to multiple periods, and –estimate u and d. To make the BOPM more realistic, we need to –extend the model from a single period to multiple periods, and –estimate u and d.

32. Multiple-Period BOPM In the multiple-period BOPM, we subdivide the period to expiration into a number of subperiods, n. –As we increase n (the number of subperiods), we increase the number of possible stock prices at T, which is more realistic, and we make the length of each period smaller, making the assumption of a binomial process more realistic. In the multiple-period BOPM, we subdivide the period to expiration into a number of subperiods, n. –As we increase n (the number of subperiods), we increase the number of possible stock prices at T, which is more realistic, and we make the length of each period smaller, making the assumption of a binomial process more realistic.

33. Two-Period Example Using our previous example, suppose we subdivide the period to expiration into two periods. Assume: – u = 1.0488, d =.9747, – So = $100, n = 2, – Rf =.025, X = $100 Using our previous example, suppose we subdivide the period to expiration into two periods. Assume: – u = 1.0488, d =.9747, – So = $100, n = 2, – Rf =.025, X = $100

35. Method for Pricing Call Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. Move to period 1 and use single-period model to price the call at each node. Move to period one and use single-period model to price the call in current period. Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. Move to period 1 and use single-period model to price the call at each node. Move to period one and use single-period model to price the call in current period.

36. Step 1: Find IV at Expiration Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. –Cuu = Max[110-100,0] = 10 –Cud = Max[102.23-100,0] = 2.23 –Cdd = Max[95-100,0] = 0 Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. –Cuu = Max[110-100,0] = 10 –Cud = Max[102.23-100,0] = 2.23 –Cdd = Max[95-100,0] = 0

37. Step 2: Find Cu and Cd Move to preceding period (period 1) and determine the price of the call at each stock price using the single-period model. For Su = $104.88, determine Cu using single-period model for that period. For Sd = $97.47, determine Cd using single-period model for that period.

38. At Su = $104.88, Cu = $7.32 Using Single-Period Model

39. At Sd = $97.47, Cd = $1.48 Using Single-Period Model

40. Step 3: Find Co Substitute the Cu and Cd values (determined in step 2) into the equations for Ho* and Bo*, then determine the current value of the call using the single-period model.

41. At So = $100, Co = $5.31 Using Single-Period Model

43. Point: Multiple-Period Model Whether it is two periods or 1000, the multiple-period model determines the price of a call by determining all of the IVs at T, then moving to each of the preceding periods and using the single-period model to determine the call prices at each node. Such a model is referred to as a recursive model -- Mechanical. Whether it is two periods or 1000, the multiple-period model determines the price of a call by determining all of the IVs at T, then moving to each of the preceding periods and using the single-period model to determine the call prices at each node. Such a model is referred to as a recursive model -- Mechanical.

44. Point: Arbitrage Strategy Like the single-period model, arbitrage ensures the equilibrium price. The arbitrage strategies underlying the multiperiod model are more complex than the single-period model, requiring possible readjustments in subsequent periods. For a discussion of multiple-period arbitrage strategies, see JG, pp. 158-163. Like the single-period model, arbitrage ensures the equilibrium price. The arbitrage strategies underlying the multiperiod model are more complex than the single-period model, requiring possible readjustments in subsequent periods. For a discussion of multiple-period arbitrage strategies, see JG, pp. 158-163.

45. Point: Impact of Dividends The model does not factor in dividends. If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and may make early exercise worthwhile if the call is America.

46. Point: Adjustments for Dividends and American Options –The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates. See JG, pp.192-196. –The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV. See JG, pp. 196-199. –The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates. See JG, pp.192-196. –The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV. See JG, pp. 196-199.

47. Estimating u and d The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the statistical characteristics of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

48. Binomial Process The binomial process that we have described for stock prices yields after n periods a distribution of n+1possible stock prices. This distribution is not normally distributed because the left-side of the distribution has a limit at zero (I.e. we cannot have negative stock prices) The distribution of stock prices can be converted into a distribution of logarithmic returns, gn:

49. Binomial Process The distribution of logarithmic returns can take on negative values and will be normally distributed if the probability of the stock increasing in one period (q) is.5. The next figure shows a distribution of stock prices and their corresponding logarithmic returns for the case in which u = 1.1, d =.95, and So = 100.

51. Binomial Process Note: When n = 1, there are two possible prices and logarithmic returns:

52. Binomial Process When n = 2, there are three possible prices and logarithmic returns:

53. Binomial Process Note: When n = 1, there are two possible prices and logarithmic returns; n = 2, there are three prices and rates; n = 3, there are four possibilities. The probability of attaining any of these rates is equal to the probability of the stock increasing j times in n period: pnj. In a binomial process, this probability is

54. Binomial Distribution Using the binomial probabilities, the expected value and variance of the logarithmic return after one period are.022 and.0054:

55. Binomial Distribution The expected value and variance of the logarithmic return after two periods are.044 and.0108:

56. Binomial Distribution Note: The parameter values (expected value and variance) after n periods are equal to the parameter values for one period time the number of periods:

57. Binomial Distribution Note: If q =.5, then skewness is zero.

58. Binomial Distribution Note: The expected value and variance of the logarithmic return are also equal to

59. Deriving the formulas for u and d The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the expected value and variance of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

60. Deriving the formulas for u and d

61. Derivation of u and d formulas Solution:

62. Annualized Mean and Variance  e and V e are the mean and variance for a length of time equal to the option’s expiration. Often the annualized mean and variance are used. The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12). For an example, see JG, pp. 167-168. If annualized parameters are used in the formulas for u and d, then they must be multiplied by the proportion t, where t is the time to expiration expressed as a proportion of a year.  e and V e are the mean and variance for a length of time equal to the option’s expiration. Often the annualized mean and variance are used. The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12). For an example, see JG, pp. 167-168. If annualized parameters are used in the formulas for u and d, then they must be multiplied by the proportion t, where t is the time to expiration expressed as a proportion of a year.

63. Equations

64. Terms t Time to expiration as a proportion of a year. Annualized variance of the stock’s logarithmic return. Annualized mean of the stock’s logarithmic return.

65. Example: JG, pp. 168-169. Using historical quarterly stock price data, suppose you estimate the stock’s quarterly mean and variance to be 0 and.004209. The annualized mean and variance would be 0 and.016836. If the number of subperiods to an expiration of one quarter (t=.25) is n = 6, then u = 1.02684 and d =.9739.

66. Estimated Parameters: Estimates of u and d:

67. Example: Working Back The estimated annualized mean and variance are.044 and.0108. If the expiration is one year ( t = 1), number of subperiods to expiration is one (n = 1, h = 1 year), then u = 1.159 and d =.94187. If the expiration is one year (t = 1), the number of subperiods to expiration is 2 (n = 2, h = ½ year), then u = 1.1 and d =.95.

68. Call Price The BOPM computer program (provided to each student) was used to value a $100 call option expiring in one quarter on a non- dividend paying stock with the above annualized mean (0) and variance (.016863), current stock price of $100, and annualized RF rate of 9.27%.

69. BOPM Values

70. BOPM Values Second Example

71. u and d for Large n In the u and d equations, as n becomes large, or equivalently, as the length of the period becomes smaller, the impact of the mean on u and d becomes smaller. For large n, u and d can be estimated as:

72. Summary of the BOPM The BOPM is based on the law of one price, in which the equilibrium price of an option is equal to the value of a replicating portfolio constructed so that it has the same cash flows as the option. The BOPM derivation requires: Derivation of single-period model. Specification of the mechanics of the multiple- period model. Estimation of u and d. The BOPM is based on the law of one price, in which the equilibrium price of an option is equal to the value of a replicating portfolio constructed so that it has the same cash flows as the option. The BOPM derivation requires: Derivation of single-period model. Specification of the mechanics of the multiple- period model. Estimation of u and d.

73. BOPM and the B-S OPM –The BOPM for large n is a practical, realistic model. –As n gets large, the BOPM converges to the B- S OPM. That is, for large n the equilibrium value of a call derived from the BOPM is approximately the same as that obtained by the B-S OPM. –The math used in the B-S OPM is complex but the model is simpler to use than the BOPM. –The BOPM for large n is a practical, realistic model. –As n gets large, the BOPM converges to the B- S OPM. That is, for large n the equilibrium value of a call derived from the BOPM is approximately the same as that obtained by the B-S OPM. –The math used in the B-S OPM is complex but the model is simpler to use than the BOPM.

74. B-S OPM Formula B-S Equation:

75. Terms: –T = time to expiration, expressed as a proportion of the year. –R = continuously compounded annual RF rate. –R = ln(1+Rs), Rs = simple annual rate. –  = annualized standard deviation of the logarithmic return. –N(d) = cumulative normal probabilities. –T = time to expiration, expressed as a proportion of the year. –R = continuously compounded annual RF rate. –R = ln(1+Rs), Rs = simple annual rate. –  = annualized standard deviation of the logarithmic return. –N(d) = cumulative normal probabilities.

76. N(d) term N(d) is the probability that deviations less than d will occur in the standard normal distribution. The probability can be looked up in standard normal probability table (see JG, p.217) or by using the following:

77. N(d) term

78. B-S OPM Example ABC call: X = 50, T =.25, S = 45,  =.5, Rf =.06

79. B-S Features Model specifies the correct relations between the call price and explanatory variables:

80. Dividend Adjustments The B-S model can be adjusted for dividends using the pseudo-American model. The model selects the maximum of two B-S-determined values:

81. Implied Variance –The only variable to estimate in the B-S OPM (or equivalently, the BOPM with large n) is the variance. This can be estimated using historical averages or an implied variance technique. –The implied variance is the variance which makes the OPM call value equal to the market value. The software program provided each student calculates the implied variance. –The only variable to estimate in the B-S OPM (or equivalently, the BOPM with large n) is the variance. This can be estimated using historical averages or an implied variance technique. –The implied variance is the variance which makes the OPM call value equal to the market value. The software program provided each student calculates the implied variance.

82. B-S Empirical Study Black-Scholes Study (1972): Conducted efficient market study in which they simulated arbitrage positions formed when calls were mispriced (C* not = to Cm). They found some abnormal returns before commission costs, but found they disappeared after commission costs. Galai found similar results. Black-Scholes Study (1972): Conducted efficient market study in which they simulated arbitrage positions formed when calls were mispriced (C* not = to Cm). They found some abnormal returns before commission costs, but found they disappeared after commission costs. Galai found similar results.

83. MacBeth-Merville Studies MacBeth and Merville compared the prices obtained from the B-S OPM to observed market prices. They found: –the B-S model tended to underprice in-the- money calls and overprice out-of-the money calls. –the B-S model was good at pricing on-the- money calls with some time to expiration. MacBeth and Merville compared the prices obtained from the B-S OPM to observed market prices. They found: –the B-S model tended to underprice in-the- money calls and overprice out-of-the money calls. –the B-S model was good at pricing on-the- money calls with some time to expiration.

84. Bhattacharya Studies Bhattacharya (1980) examined arbitrage portfolios formed when calls were mispriced, but assumed the positions were closed at the OPM values and not market prices. Found: B-S OPM was correctly specified. Bhattacharya (1980) examined arbitrage portfolios formed when calls were mispriced, but assumed the positions were closed at the OPM values and not market prices. Found: B-S OPM was correctly specified.

85. Conclusion Empirical studies provide general support for the B-S OPM as a valid pricing model, especially for near-the-money options. The overall consensus is that the B-S OPM is a useful model. Today, the OPM may be the most widely used model in the field of finance. Empirical studies provide general support for the B-S OPM as a valid pricing model, especially for near-the-money options. The overall consensus is that the B-S OPM is a useful model. Today, the OPM may be the most widely used model in the field of finance.