1. FIN 545 PROF. ROGERS SPRING 2011 Option valuation

2. PRINCIPLES OF OPTION VALUATION ASSOCIATED AUDIO CONTENT = APPROX 20 MINUTES (15 SLIDES) Segment 1

3. Basic Notation and Terminology Symbols  S 0 (stock price)  X (exercise price)  T (time to expiration = (days until expiration)/365)  r (risk-free rate)  S T (stock price at expiration)  C(S 0,T,X), P(S 0,T,X)

4. Principles of Call Option Pricing Minimum Value of a Call  C(S 0,T,X)  0 (for any call)  For American calls:  C a (S 0,T,X)  Max(0,S 0 - X)  Concept of intrinsic value: Max(0,S 0 - X)  Concept of time value of option  C(S,T,X) – Max(0,S – X)  For example, what is the time value of a call option trading at $5 with exercise price of $20 when the the underlying asset is trading at $22.75?

5. Principles of Call Option Pricing (continued) Maximum Value of a Call  C(S 0,T,X)  S 0  “Right” but not “obligation” can never be more valuable than underlying asset (and will typically be worth less).  Option values are always equal to a percentage of the underlying asset’s value!

6. Principles of Call Option Pricing (continued) Effect of Time to Expiration  More time until expiration, higher option value!  Volatility is related to time (we’ll see this in binomial and Black-Scholes models).  Calls allow buyer to invest in other assets, thus a pure time value of money effect.

7. Principles of Call Option Pricing (continued) Effect of Exercise Price  Lower exercise prices on call options with same underlying and time to expiration always have higher values!

8. Principles of Call Option Pricing (continued) Lower Bound of a European Call  C e (S 0,T,X)  Max[0,S 0 - X(1+r) -T ]  A call option can never be worth less than the difference between the underlying’s value and the present value of the exercise price on the call (or zero, if this difference is negative).

9. Principles of Call Option Pricing (continued) American Call Versus European Call  C a (S 0,T,X)  C e (S 0,T,X)  If there are no dividends on the stock, an American call will never be exercised early (unless there are complicating factors…we’ll discuss employee options eventually).  Rather than exercise, better to sell the call in the market.  Options are worth more alive than dead!  If no dividends, the value of the American call and identical European call should be equal.  If dividend is sufficiently large to invoke potential for early exercise, this “early exercise option” is a source of additional value for an American call (vs. the equivalent European).

10. Principles of Put Option Pricing Minimum Value of a Put  P(S 0,T,X)  0 (for any put)  For American puts:  P a (S 0,T,X)  Max(0,X - S 0 )  Concept of intrinsic value: Max(0,X - S 0 )

11. Principles of Put Option Pricing (continued) Maximum Value of a Put  P e (S 0,T,X)  X(1+r) -T  European put option value must be no more than the present value of the exercise price of the put option.  P a (S 0,T,X)  X  American put option value is bounded above by the exercise price.  No “present value effect” because of potential for early exercise (more on this shortly).

12. Principles of Put Option Pricing (continued) The Effect of Time to Expiration  Same effect as call options: more time, more value!

13. Principles of Put Option Pricing (continued) Effect of Exercise Price  Raising exercise price of put options increases value!

14. Principles of Put Option Pricing (continued) Lower Bound of a European Put  P e (S 0,T,X)  Max(0,X(1+r) -T - S 0 )  The value of a put option cannot be less than the difference between the present value of the put option’s exercise price and the underlying’s value (or zero if this difference is negative).

15. Principles of Put Option Pricing (continued) American Put Versus European Put  P a (S 0,T,X)  P e (S 0,T,X) Early Exercise of American Puts  There is typically the probability of a sufficiently low stock price occurring that will make it optimal to exercise an American put early.  Dividends on the stock reduce the likelihood of early exercise.

16. Principles of Put Option Pricing (continued) Put-Call Parity  Form portfolios A and B where the options are European.  Portfolio A: Buy share of stock; buy put option on stock with exercise price X, and maturity date T  Portfolio B: Buy call option on stock with exercise price X, and maturity date T; buy risk-free bond with face value X and maturity date T  The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that  S 0 + P e (S 0,T,X) = C e (S 0,T,X) + X(1+r) -T  Equation illustrates “put-call parity.”  Equation can be rearranged to offer various interpretations.

17. Put-Call Parity Example Price of underlying asset (S) = $19.50 Premium for call option on underlying asset with exercise price = $20 and 3 months until expiration = $2.50 Premium for put option on underlying asset with exercise price = $20 and 3 months until expiration = $1.50 Risk-free rate = 5% Does put-call parity hold? Which option is overpriced? What would be the trading strategy?

19. VALUING OPTIONS WITH BINOMIAL MODELS ASSOCIATED AUDIO CONTENT = APPROX 38 MINUTES (12 SLIDES) Segment 2

20. One-Period Binomial Model Conditions and assumptions  One period, two outcomes (states)  S = current stock price  u = 1 + return if stock goes up  d = 1 + return if stock goes down  r = risk-free rate Value of European call at expiration one period later  C u = Max(0,Su - X) or  C d = Max(0,Sd - X)

21. One-Period Binomial Model (continued) This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors. The probabilities of the up and down moves are never specified. They are irrelevant to the option price.

22. One-Period Binomial Model (continued) An Illustrative Example  Let S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07  First find the values of C u, C d, h, and p:  C u = Max(0,100(1.25) - 100) = Max(0,125 - 100) = 25  C d = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0  h = (25 - 0)/(125 - 80) = 0.556  p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6  Then insert into the formula for C:

23. Student exercises Calculate the option values if the following changes are made to the prior example:  S = 110  S = 90  u = 1.40, d = 0.70  u = 1.15, d = 0.90  r = 10%  r = 4%

24. (Return to text slide)

25. Two-Period Binomial Model n We now let the stock go up another period so that it ends up Su 2, Sud or Sd 2. n The option expires after two periods with three possible values:

26. Two-Period Binomial Model (continued) After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values The price of the call today will be ????

27. Two-Period Binomial Model (continued)

28. An Illustrative Example  Let S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07  Su 2 = 100(1.25) 2 = 156.25  Sud = 100(1.25)(0.80) = 100  Sd 2 = 100(0.80) 2 = 64  The call option prices at maturity are as follows:

29. Two-Period Binomial Model (continued) The two values of the call at the end of the first period are

30. Two-Period Binomial Model (continued) Therefore, the value of the call today is

31. Student exercises Calculate the option values if the following changes are made to the prior example:  S = 110  S = 90  u = 1.40, d = 0.70  u = 1.125, d = 0.90, r = 3.5%

32. ABSENCE OF ARBITRAGE AND OPTION VALUATION ASSOCIATED AUDIO CONTENT = APPROX 24 MINUTES (10 SLIDES) Segment 3

33. The “no-arbitrage” concept Important point: d < 1 + r < u to prevent arbitrage We construct a hedge portfolio of h shares of stock and one short call. Current value of portfolio:  V = hS - C At expiration the hedge portfolio will be worth  V u = hSu - C u  V d = hSd - C d  If we are hedged, these must be equal. Setting V u = V d and solving for h gives (see next page!)

34. One-Period Binomial Model (continued) These values are all known so h is easily computed. The variable, h, is called “hedge ratio.” Since the portfolio is riskless, it should earn the risk- free rate. Thus  V(1+r) = V u (or V d ) Substituting for V and V u  (hS - C)(1+r) = hSu - C u And the theoretical value of the option is

35. No-arbitrage condition C = hS – [(hSu – C u )(1 + r) -1 ] Solving for C provides the same result as we determined in our earlier example! Can alternatively substitute Sd and C d into equation If the call is not priced “correctly”, then investor could devise a risk-free trading strategy, but earn more than the risk-free rate….arbitrage profits!

36. 1-period binomial model risk-free portfolio example A Hedged Portfolio  Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares.  Value of investment: V = 556($100) - 1,000($14.02) $41,580. (This is how much money you must put up.)  Stock goes to $125  Value of investment = 556($125) - 1,000($25) = $44,500  Stock goes to $80  Value of investment = 556($80) - 1,000($0) = $44,480 (difference from 44,500 is due to rounding error)

37. One-Period Binomial Model (continued) An Overpriced Call  Let the call be selling for $15.00  Your amount invested is 556($100) - 1,000($15.00) = $40,600  You will still end up with $44,500, which is a 9.6% return.  Everyone will take advantage of this, forcing the call price to fall to $14.02 You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.

38. An Underpriced Call  Let the call be priced at $13  Sell short 556 shares at $100 and buy 1,000 calls at $13. This will generate a cash inflow of $42,600.  At expiration, you will end up paying out $44,500.  This is like a loan in which you borrowed $42,600 and paid back $44,500, a rate of 4.46%, which beats the risk- free borrowing rate. One-Period Binomial Model (continued)

39. 2-period binomial model risk-free portfolio example A Hedge Portfolio  Call trades at its theoretical value of $17.69.  Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701  So  Buy 701 shares at $100 for $70,100  Sell 1,000 calls at $17.69 for $17,690  Net investment: $52,410

40. Two-Period Binomial Model (continued) A Hedge Portfolio (continued)  The hedge ratio then changes depending on whether the stock goes up or down  What is the hedge ratio if “up”?  What is the hedge ratio if “down”?  Describe how you alter your portfolio in each circumstance.  In each case, you wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.

41. Two-Period Binomial Model (continued) A Hedge Portfolio (continued)  Your wealth then grows by 7% from the end of the first period to the end of the second.  Conclusion: If the option is correctly priced and you maintain the appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.

42. Two-Period Binomial Model (continued) A Mispriced Call in the Two-Period World  If the call is underpriced, you buy it and short the stock, maintaining the correct hedge over both periods. You end up borrowing at less than the risk-free rate.  If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.

43. EXTENSIONS OF THE BINOMIAL MODEL VALUATION PROCESS ASSOCIATED AUDIO CONTENT = APPROX 33 MINUTES (16 SLIDES) Segment 4

44. Extensions of the binomial model Early exercise (American options) Put options Call options with dividends Real option examples

45. Pricing Put Options Same procedure as calls but use put payoff formula at expiration. Using our prior example, put prices at expiration are

46. Pricing Put Options (continued) The two values of the put at the end of the first period are

47. Pricing Put Options (continued) Therefore, the value of the put today is

48. Pricing Put Options (continued) Let us hedge a long position in stock by purchasing puts. The hedge ratio formula is the same except that we ignore the negative sign:  Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.

49. Pricing Put Options (continued) Stock goes from 100 to 125. We now have  299 shares at $125 + 1,000 puts at $0.0 = $37,375  This is a 7% gain over $34,930. The new hedge ratio is  So sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.

50. Pricing Put Options (continued) Stock goes from 100 to 80. We now have  299 shares at $80 + 1,000 puts at $13.46 = $37,380  This is a 7% gain over $34,930. The new hedge ratio is  So buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.

51. Pricing Put Options (continued) Stock goes from 125 to 156.25. We now have  Bond worth $37,375(1.07) = $39,991  This is a 7% gain.  Stock goes from 125 to 100. We now have  Bond worth $37,375(1.07) = $39,991  This is a 7% gain.

52. Pricing Put Options (continued)  Stock goes from 80 to 100. We now have  1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain  Stock goes from 80 to 64. We now have  1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain

53. Early Exercise & American Puts Now we must consider the possibility of exercising the put early. At time 1 the European put values were  P u = 0.00 when the stock is at 125  P d = 13.46 when the stock is at 80  When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace P u = 13.46 with P u = 20. The value of the put today is higher at

54. Call options and dividends One way to incorporate dividends is to assume a constant yield, , per period. The stock moves up, then drops by the rate .  Using the same parameters from our earlier 2-period binomial example (now incorporate 10% dividend yield to be paid at the first “up” or “down” node).  The call prices at expiration are

55. Calls and dividends (continued) The European call prices after one period are The European call value at time 0 is

56. Will an American call option be exercised early? If there are no dividends, a call option should never be exercised early (NOTE: not true for employee options later!) If the call is American, when the stock is at 125, it pays a dividend of $12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. Therefore, a rational investor would exercise early (to capture the “high” dividend! In the example we just completed, replace the binomial value of C u = $22.78 with C u = $25. Thus, if the the call is American, at time 0 its value is $14.02. The $12.77 value on the prior page would be the value of the European call option.

57. Calls and dividends Alternatively, we can specify that the stock pays a specific dollar dividend at time 1. Unfortunately, the tree no longer recombines. We can still calculate the option value but the tree grows large very fast. Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.

58. Calls and dividends Yet another alternative (and preferred) specification is to subtract the present value of the dividends from the stock price and let the adjusted stock price follow the binomial up and down factors. The tree now recombines and we can easily calculate the option values following the same procedure as before.

59. Real options An application of binomial option valuation methodology to corporate financial decision making. Consider an oil exploration company  Traditional NPV analysis assumes that decision to operate is “binding” through the life of the project.  Real options analysis adds “flexibility” by allowing management to consider abandonment of project if oil prices drop too low.  If “option” adds value to the project, then Project value = NPV of project + value of real options  See spreadsheet example.

60. VALUING OPTIONS WITH BLACK-SCHOLES- MERTON ASSOCIATED AUDIO CONTENT = APPROX 37 MINUTES (15 SLIDES) Segment 5

61. Origins of the Black-Scholes-Merton Formula Brownian motion and the works of Einstein, Bachelier, Wiener, Itô Black, Scholes, Merton and the 1997 Nobel Prize

62. Black-Scholes-Merton Model as the Limit of the Binomial Model Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available. Demonstrate (using spreadsheet model) that binomial valuation converges toward Black- Scholes valuation. The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.

63. Assumptions of the Model Stock prices behave randomly and evolve according to a lognormal distribution.  A lognormal distribution means that the log (continuously compounded) return is normally distributed. The risk-free rate and volatility of the log return on the stock are constant throughout the option’s life There are no taxes or transaction costs The options are European

64. A Nobel Formula The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions. The model is derived with complex mathematics but is easily understandable. The formula is (if no dividends on underlying stock):

65. A Nobel Formula (continued)  where  N(d 1 ), N(d 2 ) = cumulative normal probability   = annualized standard deviation (volatility) of the continuously compounded return on the stock  r c = continuously compounded risk-free rate

66. A Nobel Formula (continued) A Digression on Using the Normal Distribution  The familiar normal, bell-shaped curve (Figure 5.5)Figure 5.5  See Table 5.1 for determining the normal probability for d 1 and d 2. This gives you N(d 1 ) and N(d 2 ).Table 5.1

67. A Nobel Formula (continued) A Numerical Example  S 0 = 30, X = 30, r = 4%, T = 0.5,  = 0.40.  See spreadsheet for calculations.  C = $3.65.

68. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula  Interpretation of the Formula  The concept of risk neutrality, risk neutral probability, and its role in pricing options  The option price is the discounted expected payoff, Max(0,S T - X). We need the expected value of S T - X for those cases where S T > X.

69. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  Interpretation of the Formula (continued)  The first term of the formula is the expected value of the stock price given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.  The second term is the expected value of the payment of the exercise price at expiration.

70. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Black-Scholes-Merton Formula and the Lower Bound of a European Call  Recall that the lower bound would be  The Black-Scholes-Merton formula always exceeds this value as seen by letting S 0 be very high and then let it approach zero.

71. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Formula When T = 0  At expiration, the formula must converge to the intrinsic value.  It does but requires taking limits since otherwise it would be division by zero.  Must consider the separate cases of S T  X and S T < X.

72. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Formula When S 0 = 0  Here the company is bankrupt so the formula must converge to zero.  It requires taking the log of zero, but by taking limits we obtain the correct result.

73. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Formula When  = 0  Again, this requires dividing by zero, but we can take limits and obtain the right answer  If the option is in-the-money as defined by the stock price exceeding the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.

74. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Formula When X = 0  Call price converges to the stock price.  Here both N(d 1 ) and N(d 2 ) approach 1.0 so by taking limits, the formula converges to S 0.

75. A Nobel Formula (continued) Characteristics of the Black-Scholes-Merton Formula (continued)  The Formula When r = 0  A zero interest rate is not a special case and no special result is obtained.

76. “GREEKS” OF BLACK-SCHOLES ASSOCIATED AUDIO CONTENT = APPROX 20 MINUTES (8 SLIDES) Segment 6

77. Variables in the Black-Scholes-Merton Model The Stock Price  Let S  then C . See Figure 5.6.Figure 5.6  This effect is called the delta, which is given by N(d 1 ).  Measures the change in call price over the change in stock price for a very small change in the stock price.  Delta ranges from zero to one. See Figure 5.7 for how delta varies with the stock price.Figure 5.7  The delta changes throughout the option’s life. See Figure 5.8. Figure 5.8

78. Variables in the Black-Scholes-Merton Model (continued) The Stock Price (continued)  Delta hedging/delta neutral: holding shares of stock and selling calls to maintain a risk-free position  The number of shares held per option sold is the delta, N(d 1 ).  As the stock goes up/down by $1, the option goes up/down by N(d 1 ). By holding N(d 1 ) shares per call, the effects offset.  The position must be adjusted as the delta changes.

79. Variables in the Black-Scholes-Merton Model (continued) The Stock Price (continued)  Delta hedging works only for small stock price changes. For larger changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:

80. Variables in the Black-Scholes-Merton Model (continued) The Stock Price (continued)  The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.  See Figure 5.9 for gamma vs. the stock priceFigure 5.9  See Figure 5.10 for gamma vs. timeFigure 5.10

81. Variables in the Black-Scholes-Merton Model (continued) The Exercise Price  Let X , then C   The exercise price does not change in most options so this is useful only for comparing options differing only by a small change in the exercise price.

82. Variables in the Black-Scholes-Merton Model (continued) The Risk-Free Rate  Let r  then C  See Figure 5.11. The effect is called rhoFigure 5.11  See Figure 5.12 for rho vs. stock price.Figure 5.12

83. Variables in the Black-Scholes-Merton Model (continued) The Volatility or Standard Deviation  The most critical variable in the Black-Scholes-Merton model because the option price is very sensitive to the volatility and it is the only unobservable variable.  Let  , then C  See Figure 5.13.Figure 5.13  This effect is known as vega.  See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.Figure 5.14

84. Variables in the Black-Scholes-Merton Model (continued) The Time to Expiration  Calculated as (days to expiration)/365  Let T , then C . See Figure 5.15. This effect is known as theta:Figure 5.15  See Figure 5.16 for theta vs. the stock priceFigure 5.16  Chance’s spreadsheet BSMbin7e.xls calculates the delta, gamma, vega, theta, and rho for calls and puts (thus can use as useful check).

85. EXTENSIONS TO BLACK-SCHOLES ASSOCIATED AUDIO CONTENT = APPROX 27 MINUTES (8 SLIDES) Segment 7

86. Black-Scholes-Merton Model When the Stock Pays Dividends Known Discrete Dividends  Assume a single dividend of D t where the ex-dividend date is time t during the option’s life.  Subtract present value of dividends from stock price.  Adjusted stock price, S, is inserted into the B-S-M model:  See Table 5.3 for example.Table 5.3  The Excel spreadsheet BSMbin7e.xls allows up to 50 discrete dividends.

87. Black-Scholes-Merton Model When the Stock Pays Dividends (continued) Continuous Dividend Yield  Assume the stock pays dividends continuously at the rate of .  Subtract present value of dividends from stock price. Adjusted stock price, S, is inserted into the B-S model.  See Table 5.4 for example.Table 5.4  This approach could also be used if the underlying is a foreign currency, where the yield is replaced by the continuously compounded foreign risk-free rate.  The Excel spreadsheet BSMbin7e.xls permit you to enter a continuous dividend yield.

88. Black-Scholes-Merton Model and Some Insights into American Call Options Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life Table 5.5 The value obtained upon exercise is compared to the ex-dividend value of the option. High dividends and low time value lead to early exercise. Chance’s Excel spreadsheet will calculate the American call price using the binomial model.

89. Estimating the Volatility Historical Volatility  This is the volatility over a recent time period.  Collect daily, weekly, or monthly returns on the stock.  Convert each return to its continuously compounded equivalent by taking ln(1 + return). Calculate variance.  Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12 (monthly returns). Take square root.

90. Estimating the Volatility (continued) Implied Volatility  This is the volatility implied when the market price of the option is set to the model price.  Figure 5.17 illustrates the procedure. Figure 5.17  Substitute estimates of the volatility into the B-S-M formula until the market price converges to the model price.  A short-cut for at-the-money options is

91. Estimating the Volatility (continued) Implied Volatility (continued)  Interpreting the Implied Volatility  The relationship between the implied volatility and the time to expiration is called the term structure of implied volatility. See Figure 5.18.Figure 5.18  The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained. Figure 5.19  The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.Figure 5.20

92. Put Option Pricing Models Restate put-call parity with continuous discounting Substituting the B-S-M formula for C above gives the B-S-M put option pricing model N(d 1 ) and N(d 2 ) are the same as in the call model.

93. Put Option Pricing Models (continued) The Black-Scholes-Merton price does not reflect early exercise. A binomial model would be necessary to get an accurate price. effect of the input variables on the Black-Scholes- Merton put formula. Chance’s spreadsheet also calculates put prices and Greeks.

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