Chapter 5: Option Pricing Models:The Black-Scholes Model: Chapter 5: Option Pricing Models: The Black-Scholes Model When I first saw the formula I knew enough about it to know that this is the answer. This solved the ancient problem of risk and return in the stock market. It was recognized by the profession for what it was as a real tour de force.
Merton Miller
Trillion Dollar Bet, PBS, February, 2000
Important Concepts in Chapter 5: Important Concepts in Chapter 5 The Black-Scholes option pricing model
The relationship of the model’s inputs to the option price
How to adjust the model to accommodate dividends and put options
The concepts of historical and implied volatility
Hedging an option position
Origins of the Black-Scholes Formula: Origins of the Black-Scholes Formula Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
Black, Scholes, Merton and the 1997 Nobel Prize
The Black-Scholes Model as the Limit of the Binomial Model: The Black-Scholes 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.
Consider the AOL June 125 call option. Figure 5.1, p. 131 shows the model price for an increasing number of time steps.
The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
The Assumptions of the Model: The Assumptions of the Model Stock Prices Behave Randomly and Evolve According to a Lognormal Distribution.
See Figure 5.2a, p. 134, 5.2b, p. 135 and 5.3, p. 136 for a look at the notion of randomness.
A lognormal distribution means that the log (continuously compounded) return is normally distributed. See Figure 5.4, p. 137.
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 Stock Pays No Dividends
The Options are European
A Nobel Formula: A Nobel Formula The Black-Scholes 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
A Nobel Formula (continued): A Nobel Formula (continued) where
N(d1), N(d2) = cumulative normal probability
s = annualized standard deviation (volatility) of the continuously compounded return on the stock
rc = continuously compounded risk-free rate
A Nobel Formula (continued): A Nobel Formula (continued) A Digression on Using the Normal Distribution
The familiar normal, bell-shaped curve (Figure 5.5, p. 139)
See Table 5.1, p. 140 for determining the normal probability for d1 and d2. This gives you N(d1) and N(d2).
A Nobel Formula (continued): A Nobel Formula (continued) A Numerical Example
Price the AOL June 125 call
S0 = 125.9375, X = 125, rc = ln(1.0456) = .0446, T = .0959, s = .83.
See Table 5.2, p. 141 for calculations. C = $13.21.
Familiarize yourself with the accompanying software
Excel: bsbin3.xls. See Software Demonstration 5.1. Note the use of Excel’s =normsdist() function.
Windows: bsbwin2.2.exe. See Appendix 5.B.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes 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,ST - X). We need the expected value of ST - X for those cases where ST > X.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes 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.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes Formula (continued)
The Black-Scholes Formula and the Lower Bound of a European Call
Recall from Chapter 3 that the lower bound would be
The Black-Scholes formula always exceeds this value as seen by letting S0 be very high and then let it approach zero.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes 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 ST X and ST < X.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes Formula (continued)
The Formula When S0 = 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.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes 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.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes Formula (continued)
The Formula When X = 0
From Chapter 3, the call price should converge to the stock price.
Here both N(d1) and N(d2) approach 1.0 so by taking limits, the formula converges to S0.
A Nobel Formula (continued): A Nobel Formula (continued) Characteristics of the Black-Scholes Formula (continued)
The Formula When rc = 0
A zero interest rate is not a special case and no special result is obtained.
The Variables in the Black-Scholes Model: The Variables in the Black-Scholes Model The Stock Price
Let S , then C . See Figure 5.6, p. 148.
This effect is called the delta, which is given by N(d1).
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, p. 149 for how delta varies with the stock price.
The delta changes throughout the option’s life. See Figure 5.8, p. 150.
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes 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(d1).
As the stock goes up/down by $1, the option goes up/down by N(d1). By holding N(d1) shares per call, the effects offset.
The position must be adjusted as the delta changes.
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes 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:
For our AOL June 125 call,
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Stock Price (continued)
If the stock goes from 125.9375 to 130, the delta is predicted to change from .569 to .569 + (130 - 125.9375)(.0121) = .6182. The actual delta at a price of 130 is .6171. So gamma captures most of the change in delta.
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, p. 152 for gamma vs. the stock price
See Figure 5.10, p. 153 for gamma vs. time
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes 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.
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Risk-Free Rate
Take ln(1 + discrete risk-free rate from Chapter 3).
Let rc , then C . See Figure 5.11, p. 154. The effect is called rho
In our example,
If the risk-free rate goes to .12, the rho estimates that the call price will go to (.12 - .0446)(5.57) = .42. The actual change is .43.
See Figure 5.12, p. 155 for rho vs. stock price.
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Volatility or Standard Deviation
The most critical variable in the Black-Scholes model because the option price is very sensitive to the volatility and it is the only unobservable variable.
Let s , then C . See Figure 5.13, p. 156.
This effect is known as vega.
In our problem this is
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Volatility or Standard Deviation (continued)
Thus if volatility changes by .01, the call price is estimated to change by 15.32(.01) = .15
If we increase volatility to, say, .95, the estimated change would be 15.32(.12) = 1.84. The actual call price at a volatility of .95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
See Figure 5.14, p. 157 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Time to Expiration
Calculated as (days to expiration)/365
Let T , then C . See Figure 5.15, p. 158. This effect is known as theta:
In our problem, this would be
The Variables in the Black-Scholes Model (continued): The Variables in the Black-Scholes Model (continued) The Time to Expiration (continued)
If one week elapsed, the call price would be expected to change to (.0959 - .0767)(-68.91) = -1.32. The actual call price with T = .0767 is 12.16, a decrease of 1.39.
See Figure 5.16, p. 159 for theta vs. the stock price
Note that your spreadsheet bsbin3.xls and your Windows program bsbwin2.2 calculate the delta, gamma, vega, theta, and rho for calls and puts.
The Black-Scholes Model When the Stock Pays Dividends: The Black-Scholes Model When the Stock Pays Dividends Known Discrete Dividends
Assume a single dividend of Dt 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 model:
See Table 5.3, p. 160 for example.
The Excel spreadsheet bsbin3.xls allows up to 50 discrete dividends. The Windows program bsbwin2.2 allows up to three discrete dividends.
The Black-Scholes Model in the Presence of 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, p. 161 for example.
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 bsbin3.xls and Windows program bsbwin2.2 permit you to enter a continuous dividend yield. The Black-Scholes Model in the Presence of Dividends (continued)
The Black-Scholes Model and Some Insights into American Call Options: The Black-Scholes Model and Some Insights into American Call Options Table 5.5, p. 163 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
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.
Your Excel spreadsheet bsbin3.xls and Windows program bsbwin2.2 will calculate the American call price using the binomial model.
Estimating the Volatility: 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. See Table 5.6, p. 166-167 for example with AOL.
Your Excel spreadsheet hisv2.xls will do these calculations. See Software Demonstration 5.2.
Estimating the Volatility (continued): 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, p. 168 illustrates the procedure.
Substitute estimates of the volatility into the B-S formula until the market price converges to the model price. See Table 5.7, p. 169 for the implied volatilities of the AOL calls.
A short-cut for at-the-money options is
Estimating the Volatility (continued): Estimating the Volatility (continued) Implied Volatility (continued)
For our AOL June 125 call, this gives
This is quite close; the actual implied volatility is .83.
Appendix 5.A shows a method to produce faster convergence.
Estimating the Volatility (continued): 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, p. 170.
The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19, p. 171. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
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, p. 172.
Put Option Pricing Models: Put Option Pricing Models Restate put-call parity with continuous discounting
Substituting the B-S formula for C above gives the B-S put option pricing model
N(d1) and N(d2) are the same as in the call model.
Put Option Pricing Models (continued): Put Option Pricing Models (continued) Note calculation of put price:
The Black-Scholes price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
See Table 5.8, p. 175 for the effect of the input variables on the Black-Scholes put formula.
Your software also calculates put prices and Greeks.
Managing the Risk of Options: Managing the Risk of Options Here we talk about how option dealers hedge the risk of option positions they take.
Assume a dealer sells 1,000 AOL June 125 calls at the Black-Scholes price of 13.5512 with a delta of .5692. Dealer will buy 569 shares and adjust the hedge daily.
To buy 569 shares at $125.9375 and sell 1,000 calls at $13.5512 will require $58,107.
We simulate the daily stock prices for 35 days, at which time the call expires.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) The second day, the stock price is 120.5442. There are now 34 days left. Using bsbin3.xls, we get a call price of 10.4781 and delta of .4999. We have
Stock worth 569($120.5442) = $68,590
Options worth -1,000($10.4781) = -$10,478
Total of $58,112
Had we invested $58,107 in bonds, we would have had $58,107e.0446(1/365) = $58,114.
Table 5.9, pp. 178-179 shows the remaining outcomes. We must adjust to the new delta of .4999. We need 500 shares so sell 69 and invest the money ($8,318) in bonds.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) At the end of the second day, the stock goes to 106.9722 and the call to 4.7757. The bonds accrue to a value of $8,319. We have
Stock worth 500($106.9722) = $53,486
Options worth -1,000($4.7757) = -$4,776
Bonds worth $8,319 (includes one days’ interest)
Total of $57,029
Had we invested the original amount in bonds, we would have had $58,107e.0446(2/365) = $58,121. We are now short by over $1,000.
At the end we have $56,540, a shortage of $1,816.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) What we have seen is the second order or gamma effect. Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3772 with a delta of .5086 and gamma of .0123. Our original June 125 call has a gamma of .0121. The stock gamma is zero.
We shall use the symbols 1, 2, 1 and 2. We use hS shares of stock and hC of the June 130 calls.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) The delta hedge condition is
hS(1) - 1,0001 + hC 2 = 0
The gamma hedge condition is
-1,0001 + hC 2 = 0
We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
hC = 1,000(.0121/.0123) = 984
hS = 1,000(.5692 - (.0121/.0123).5086) = 68
So buy 68 shares, sell 1,000 June 125s, buy 985 June 130s.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) The initial outlay will be
68($125.9375) - 1,000($13.5512) + 985($11.3772) = $6,219
At the end of day one, the stock is at 120.5442, the 125 call is at 10.4781, the 130 call is at 8.6344. The portfolio is worth
68($120.5442) - 1,000($10.4781) + 985($8.6344) = $6,224
It should be worth $6,219e.0446(1/365) = $6,220.
The new deltas are .4999 and .4384 and the new gammas are .0131 and .0129.
Managing the Risk of Options (continued): Managing the Risk of Options (continued) The new values are 1,012 of the 130 calls so we buy 27. The new number of shares is 56 so we sell 12. Overall, this generates $1,214, which we invest in bonds.
The next day, the stock is at $106.9722, the 125 call is at $4.7757 and the 130 call is at $3.7364. The bonds are worth $1,214. The portfolio is worth
56($106.9722) - 1,000($4.7757) + 1,012($3.7364) + $1,214 = $6,210.
The portfolio should be worth $6,219e.0446(2/365) = $6,221.
Continuing this, we end up at $6,589 and should have $6,246, a difference of $343. We are much closer than when only delta hedging.
Summary : Summary See Figure 5.21, p. 182 for the relationship between call, put, underlying asset, risk-free bond, put-call parity, and Black-Scholes call and put option pricing models.
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility: Appendix 5.A: A Shortcut to the Calculation of Implied Volatility This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be * and the corresponding Black-Scholes price be C(*). The initial guess should be
You then compute C(1*). If it is not close enough, you make the next guess.
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued): Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued) Given the ith guess, the next guess should be
where d1 is computed using 1*. Let us illustrate using the AOL June 125 call. C() = 13.50. The initial guess is
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued): Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued) At a volatility of .4950, the Black-Scholes value is 8.41. The next guess should be
where .1533 is d1 computed from the Black-Scholes model using .4950 as the volatility and 2.5066 is the square root of 2. Now using .8260, we obtain a Black-Scholes value of 13.49, which is close enough to 13.50. So .83 is the implied volatility.
Appendix 5.B: The BSBWIN2.2 Windows Software: Appendix 5.B: The BSBWIN2.2 Windows Software
Slide49: (Return to text slide)
Slide50: (Return to text slide)
Slide51: (Return to text slide)
Slide52: (Return to text slide)
Slide53: (Return to text slide)
Slide54: (Return to text slide)
Slide55: (Return to text slide)
Slide56: (Return to text slide)
Slide57: (Return to text slide)
Slide58: (Return to text slide)
Slide59: (Return to text slide)
Slide60: (Return to text slide)
Slide61: (Return to text slide)
Slide62: (Return to text slide)
Slide63: (Return to text slide)
Slide64: (Return to text slide)
Slide65: (Return to text slide)
Slide66: (Return to text slide)
Slide67: (Return to text slide)
Slide68: (Return to text slide)
Slide69: (Return to text slide)
Slide70: (Return to text slide)
Slide71: (To continue) (Return to text slide 31)
Slide72: (Return to text slide 31) ( To previous slide)
Slide73: (Return to text slide)
Slide74: (Return to text slide)
Slide75: (Return to text slide)
Slide76: (Return to text slide)
Slide77: (Return to text slide)
Slide78: (Return to text slide)
Slide79: (To continue) (Return to text slide 38)
Slide80: (To previous slide) (Return to text slide 38)
Slide81: (Return to text slide)