Chapter 5: Option Pricing Models: The Black-Scholes Model

D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed.

Ch. 5: 1

Chapter 5: Option Pricing Models:The Black-Scholes Model

When I first saw the formula I knew enough about it to know When I first saw the formula I knew enough about it to know that this is the answer. This solved the ancient problem of that this is the answer. This solved the ancient problem of risk and return in the stock market. It was recognized by the risk and return in the stock market. It was recognized by the profession for what it was as a real tour de force.profession for what it was as a real tour de force.

Merton MillerMerton Miller

Trillion Dollar BetTrillion Dollar Bet, PBS, February, 2000, PBS, February, 2000


Ch. 5: 2

Important Concepts in Chapter 5

The Black-Scholes option pricing modelThe Black-Scholes option pricing model The relationship of the model’s inputs to the option priceThe relationship of the model’s inputs to the option price How to adjust the model to accommodate dividends and How to adjust the model to accommodate dividends and

put optionsput options The concepts of historical and implied volatilityThe concepts of historical and implied volatility Hedging an option positionHedging an option position


Ch. 5: 3

Origins of the Black-Scholes Formula

Brownian motion and the works of Einstein, Bachelier, Brownian motion and the works of Einstein, Bachelier, Wiener, ItôWiener, Itô

Black, Scholes, Merton and the 1997 Nobel PrizeBlack, Scholes, Merton and the 1997 Nobel Prize


Ch. 5: 4

The Black-Scholes Model as the Limit of the Binomial Model

Recall the binomial model and the notion of a dynamic Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are risk-free hedge in which no arbitrage opportunities are available.available.

Consider the AOL June 125 call option. Consider the AOL June 125 call option. Figure 5.1, p. 131Figure 5.1, p. 131 shows the model price for an increasing number of time shows the model price for an increasing number of time steps.steps.

The binomial model is in discrete time. As you decrease The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous the length of each time step, it converges to continuous time.time.


Ch. 5: 5

The Assumptions of the Model

Stock Prices Behave Randomly and Evolve According to a Stock Prices Behave Randomly and Evolve According to a Lognormal Distribution. Lognormal Distribution. See See Figure 5.2a, p. 134Figure 5.2a, p. 134, , 5.2b, p. 1355.2b, p. 135 and and 5.3, p. 1365.3, p. 136 for for

a look at the notion of randomness.a look at the notion of randomness. A lognormal distribution means that the log A lognormal distribution means that the log

(continuously compounded) return is normally (continuously compounded) return is normally distributed. See distributed. See Figure 5.4, p. 137Figure 5.4, p. 137..

The Risk-Free Rate and Volatility of the Log Return on the The Risk-Free Rate and Volatility of the Log Return on the Stock are Constant Throughout the Option’s LifeStock are Constant Throughout the Option’s Life

There Are No Taxes or Transaction CostsThere Are No Taxes or Transaction Costs The Stock Pays No DividendsThe Stock Pays No Dividends The Options are EuropeanThe Options are European


Ch. 5: 6

A Nobel Formula

The Black-Scholes model gives the correct formula for a The Black-Scholes model gives the correct formula for a European call under these assumptions.European call under these assumptions.

The model is derived with complex mathematics but is The model is derived with complex mathematics but is easily understandable. The formula iseasily understandable. The formula is

Tódd

Tó

/2)Tó(r/X)ln(Sd

where

)N(dXe)N(dSC

12

2c0

1

2Tr

10c

−=

++=

−= −


Ch. 5: 7

A Nobel Formula (continued)

wherewhere N(dN(d11), N(d), N(d22) = cumulative normal probability) = cumulative normal probability = annualized standard deviation (volatility) of the

continuously compounded return on the stock rc = continuously compounded risk-free rate


Ch. 5: 8


A Digression on Using the Normal DistributionA Digression on Using the Normal Distribution The familiar normal, bell-shaped curve (The familiar normal, bell-shaped curve (

Figure 5.5, p. 139Figure 5.5, p. 139)) See See Table 5.1, p. 140Table 5.1, p. 140 for determining the normal for determining the normal

probability for dprobability for d11 and d and d22. This gives you N(d. This gives you N(d11) and ) and

N(dN(d22).).


Ch. 5: 9


A Numerical ExampleA Numerical Example Price the AOL June 125 callPrice the AOL June 125 call SS00 = 125.9375, X = 125, r = 125.9375, X = 125, rcc = ln(1.0456) = .0446, T = ln(1.0456) = .0446, T

= .0959, = .0959, = .83. = .83. SeeSee Table 5.2, p. 141Table 5.2, p. 141 for calculations. C = $13.21. for calculations. C = $13.21. Familiarize yourself with the accompanying softwareFamiliarize yourself with the accompanying software

Excel: bsbin3.xls. See Software Demonstration Excel: bsbin3.xls. See Software Demonstration 5.1. Note the use of Excel’s =normsdist() function.5.1. Note the use of Excel’s =normsdist() function.

Windows: bsbwin2.2.exe. SeeWindows: bsbwin2.2.exe. See Appendix 5.B.Appendix 5.B.


Ch. 5: 10


Characteristics of the Black-Scholes FormulaCharacteristics of the Black-Scholes Formula Interpretation of the FormulaInterpretation of the Formula

The concept of risk neutrality, risk neutral The concept of risk neutrality, risk neutral probability, and its role in pricing optionsprobability, and its role in pricing options

The option price is the discounted expected payoff, The option price is the discounted expected payoff, Max(0,SMax(0,STT - X). We need the expected value of S - X). We need the expected value of STT - -

X for those cases where SX for those cases where STT > X. > X.


Ch. 5: 11


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) Interpretation of the Formula (continued)Interpretation of the Formula (continued)

The first term of the formula is the expected value The first term of the formula is the expected value of the stock price given that it exceeds the exercise of the stock price given that it exceeds the exercise price times the probability of the stock price price times the probability of the stock price exceeding the exercise price, discounted to the exceeding the exercise price, discounted to the present.present.

The second term is the expected value of the The second term is the expected value of the payment of the exercise price at expiration.payment of the exercise price at expiration.


Ch. 5: 12


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Black-Scholes Formula and the Lower Bound of a The Black-Scholes Formula and the Lower Bound of a

European CallEuropean Call Recall from Chapter 3 that the lower bound would Recall from Chapter 3 that the lower bound would

bebe

The Black-Scholes formula always exceeds this The Black-Scholes formula always exceeds this value as seen by letting Svalue as seen by letting S00 be very high and then let be very high and then let

it approach zero.it approach zero.

)XeSMax(0, Tr0

c−−


Ch. 5: 13


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Formula When T = 0The Formula When T = 0

At expiration, the formula must converge to the At expiration, the formula must converge to the intrinsic value.intrinsic value.

It does but requires taking limits since otherwise it It does but requires taking limits since otherwise it would be division by zero.would be division by zero.

Must consider the separate cases of SMust consider the separate cases of STT X and S X and STT < <

X.X.


Ch. 5: 14


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Formula When SThe Formula When S00 = 0 = 0

Here the company is bankrupt so the formula must Here the company is bankrupt so the formula must converge to zero.converge to zero.

It requires taking the log of zero, but by taking It requires taking the log of zero, but by taking limits we obtain the correct result.limits we obtain the correct result.


Ch. 5: 15


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Formula When The Formula When = 0 = 0

Again, this requires dividing by zero, but we can Again, this requires dividing by zero, but we can take limits and obtain the right answertake limits and obtain the right answer

If the option is in-the-money as defined by the stock If the option is in-the-money as defined by the stock price exceeding the present value of the exercise price exceeding the present value of the exercise price, the formula converges to the stock price price, the formula converges to the stock price minus the present value of the exercise price. minus the present value of the exercise price. Otherwise, it converges to zero.Otherwise, it converges to zero.


Ch. 5: 16


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Formula When X = 0The Formula When X = 0

From Chapter 3, the call price should converge to From Chapter 3, the call price should converge to the stock price.the stock price.

Here both N(dHere both N(d11) and N(d) and N(d22) approach 1.0 so by taking ) approach 1.0 so by taking

limits, the formula converges to Slimits, the formula converges to S00..


Ch. 5: 17


Characteristics of the Black-Scholes Formula (continued)Characteristics of the Black-Scholes Formula (continued) The Formula When rThe Formula When rcc = 0 = 0

A zero interest rate is not a special case and no A zero interest rate is not a special case and no special result is obtained.special result is obtained.


Ch. 5: 18

The Variables in the Black-Scholes Model

The Stock PriceThe Stock Price Let S Let S then C then C . See. See Figure 5.6, p. 148Figure 5.6, p. 148.. This effect is called theThis effect is called the delta, which is given by N(ddelta, which is given by N(d11).).

Measures the change in call price over the change in Measures the change in call price over the change in stock price for a very small change in the stock price.stock price for a very small change in the stock price.

Delta ranges from zero to one. See Delta ranges from zero to one. See Figure 5.7, p. 149Figure 5.7, p. 149 for how delta varies with the stock price.for how delta varies with the stock price.

The delta changes throughout the option’s life. See The delta changes throughout the option’s life. See Figure 5.8, p. 150Figure 5.8, p. 150..


Ch. 5: 19

The Variables in the Black-Scholes Model (continued) The Stock Price (continued)The Stock Price (continued)

Delta hedging/delta neutral: holding shares of stock Delta hedging/delta neutral: holding shares of stock and selling calls to maintain a risk-free positionand selling calls to maintain a risk-free position

The number of shares held per option sold is the The number of shares held per option sold is the delta, N(ddelta, N(d11).).

As the stock goes up/down by $1, the option goes As the stock goes up/down by $1, the option goes up/down by N(dup/down by N(d11). By holding N(d). By holding N(d11) shares per call, ) shares per call,

the effects offset.the effects offset. The position must be adjusted as the delta changes.The position must be adjusted as the delta changes.


Ch. 5: 20


Delta hedging works only for small stock price Delta hedging works only for small stock price changes. For larger changes, the delta does not changes. For larger changes, the delta does not accurately reflect the option price change. This risk is accurately reflect the option price change. This risk is captured by the gamma:captured by the gamma:

For our AOL June 125 call,For our AOL June 125 call,

T2óS

eGamma Call

0

/2d21

π

−

=

.0121 .09592(3.14159)83)125.9375(.

eGamma Call

/2).(

==− 217420


Ch. 5: 21


If the stock goes from 125.9375 to 130, the delta is If the stock goes from 125.9375 to 130, the delta is predicted to change from .569 to .569 + (130 - predicted to change from .569 to .569 + (130 - 125.9375)(.0121) = .6182. The actual delta at a price 125.9375)(.0121) = .6182. The actual delta at a price of 130 is .6171. So gamma captures most of the change of 130 is .6171. So gamma captures most of the change in delta.in delta.

The larger is the gamma, the more sensitive is the The larger is the gamma, the more sensitive is the option price to large stock price moves, the more option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.This makes it more difficult to hedge.

See See Figure 5.9, p. 152Figure 5.9, p. 152 for gamma vs. the stock price for gamma vs. the stock price See See Figure 5.10, p. 153Figure 5.10, p. 153 for gamma vs. time for gamma vs. time


Ch. 5: 22

The Variables in the Black-Scholes Model (continued) The Exercise PriceThe Exercise Price

Let X 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.


Ch. 5: 23

The Variables in the Black-Scholes Model (continued) The Risk-Free RateThe Risk-Free Rate

Take ln(1 + discrete risk-free rate from Chapter 3).Take ln(1 + discrete risk-free rate from Chapter 3). Let rLet rcc 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.

)N(dTXeRho Call 2Trc−=

57.5)4670(..0959)125e(Rho Call 59)-.0446(.09 ==


Ch. 5: 24

The Variables in the Black-Scholes Model (continued) The Volatility or Standard DeviationThe Volatility or Standard Deviation

The most critical variable in the Black-Scholes model The most critical variable in the Black-Scholes model because the option price is very sensitive to the because the option price is very sensitive to the volatility and it is the only unobservable variable.volatility and it is the only unobservable variable.

Let Let , then C See See Figure 5.13, p. 156Figure 5.13, p. 156.. This effect is known as vega. This effect is known as vega.

In our problem this isIn our problem this is2

eTS vegaCall

/2-d0

21

π=

15.322(3.14159)

e.0959125.9375 vegaCall

/2-0.17422

==


Ch. 5: 25

The Variables in the Black-Scholes Model (continued) The Volatility or Standard Deviation (continued)The Volatility or Standard Deviation (continued)

Thus if volatility changes by .01, the call price is Thus if volatility changes by .01, the call price is estimated to change by 15.32(.01) = .15estimated to change by 15.32(.01) = .15

If we increase volatility to, say, .95, the estimated If we increase volatility to, say, .95, the estimated change would be 15.32(.12) = 1.84. The actual call change would be 15.32(.12) = 1.84. The actual call price at a volatility of .95 would be 15.39, which is an price at a volatility of .95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.linearity of the call price with respect to the volatility.

See See Figure 5.14, p. 157Figure 5.14, p. 157 for the vega vs. the stock price. for the vega vs. the stock price. Notice how it is highest when the call is approximately Notice how it is highest when the call is approximately at-the-money.at-the-money.


Ch. 5: 26

The Variables in the Black-Scholes Model (continued) The Time to ExpirationThe Time to Expiration

Calculated as (days to expiration)/365Calculated as (days to expiration)/365 Let T Let T , then C , then C . See . See Figure 5.15, p. 158Figure 5.15, p. 158. This effect . This effect

is known as theta:is known as theta:

In our problem, this would beIn our problem, this would be

)N(dXer T22

eS- thetaCall 2

Trc

/2d0 c

21

−−

−=π

68.91- (.4670)e(.0446)125

(.0959)2(3.14159)2

.83)e125.9375(0- thetaCall

9).0446(.095

/2(.1742)2

=−

=

−

−


Ch. 5: 27

The Variables in the Black-Scholes Model (continued) The Time to Expiration (continued)The Time to Expiration (continued)

If one week elapsed, the call price would be expected to If one week elapsed, the call price would be expected to change to (.0959 - .0767)(-68.91) = -1.32. The actual change to (.0959 - .0767)(-68.91) = -1.32. The actual call price with T = .0767 is 12.16, a decrease of 1.39.call price with T = .0767 is 12.16, a decrease of 1.39.

See See Figure 5.16, p. 159Figure 5.16, p. 159 for theta vs. the stock price for theta vs. the stock price Note that your spreadsheet bsbin3.xls and your Note that your spreadsheet bsbin3.xls and your

Windows program bsbwin2.2 calculate the delta, Windows program bsbwin2.2 calculate the delta, gamma, vega, theta, and rho for calls and puts.gamma, vega, theta, and rho for calls and puts.


Ch. 5: 28

The Black-Scholes Model When the Stock Pays Dividends Known Discrete DividendsKnown Discrete Dividends

Assume a single dividend of DAssume a single dividend of Dtt where the ex-dividend where the ex-dividend date is time t during the option’s life.date is time t during the option’s life.

Subtract present value of dividends from stock price. Subtract present value of dividends from stock price. Adjusted stock price, SAdjusted stock price, S, is inserted into the B-S model:, is inserted into the B-S model:

See See Table 5.3, p. 160Table 5.3, p. 160 for example. for example. The Excel spreadsheet bsbin3.xls allows up to 50 The Excel spreadsheet bsbin3.xls allows up to 50

discrete dividends. The Windows program bsbwin2.2 discrete dividends. The Windows program bsbwin2.2 allows up to three discrete dividends.allows up to three discrete dividends.

trt00

ceDSS −−=


Ch. 5: 29

Continuous Dividend YieldContinuous Dividend Yield Assume the stock pays dividends continuously at the rate of Assume the stock pays dividends continuously at the rate of .. Subtract present value of dividends from stock price. Adjusted Subtract present value of dividends from stock price. Adjusted

stock price, Sstock price, S, is inserted into the B-S model., is inserted into the B-S model.

See See Table 5.4, p. 161Table 5.4, p. 161 for example. for example. This approach could also be used if the underlying is a foreign This approach could also be used if the underlying is a foreign

currency, where the yield is replaced by the continuously currency, where the yield is replaced by the continuously compounded foreign risk-free rate.compounded foreign risk-free rate.

The Excel spreadsheet bsbin3.xls and Windows program The Excel spreadsheet bsbin3.xls and Windows program bsbwin2.2 permit you to enter a continuous dividend yield.bsbwin2.2 permit you to enter a continuous dividend yield.

The Black-Scholes Model in the Presence of Dividends (continued)

T00 eSS c−=


Ch. 5: 30

The Black-Scholes Model and Some Insights into American Call Options

Table 5.5, p. 163Table 5.5, p. 163 illustrates how the early exercise decision illustrates how the early exercise decision is made when the dividend is the only one during the is made when the dividend is the only one during the option’s lifeoption’s life

The value obtained upon exercise is compared to the ex-The value obtained upon exercise is compared to the ex-dividend value of the option.dividend value of the option.

High dividends and low time value lead to early exercise.High dividends and low time value lead to early exercise. Your Excel spreadsheet bsbin3.xls and Windows program Your Excel spreadsheet bsbin3.xls and Windows program

bsbwin2.2 will calculate the American call price using the bsbwin2.2 will calculate the American call price using the binomial model.binomial model.


Ch. 5: 31

Estimating the Volatility

Historical VolatilityHistorical Volatility This is the volatility over a recent time period.This is the volatility over a recent time period. Collect daily, weekly, or monthly returns on the stock.Collect daily, weekly, or monthly returns on the stock. Convert each return to its continuously compounded Convert each return to its continuously compounded

equivalent by taking ln(1 + return). Calculate equivalent by taking ln(1 + return). Calculate variance.variance.

Annualize by multiplying by 250 (daily returns), 52 Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12 (monthly returns). Take square (weekly returns) or 12 (monthly returns). Take square root. See root. See Table 5.6, p. 166-167Table 5.6, p. 166-167 for example with AOL. for example with AOL.

Your Excel spreadsheet hisv2.xls will do these Your Excel spreadsheet hisv2.xls will do these calculations. See Software Demonstration 5.2.calculations. See Software Demonstration 5.2.


Ch. 5: 32

Estimating the Volatility (continued)

Implied VolatilityImplied Volatility This is the volatility implied when the market price of This is the volatility implied when the market price of

the option is set to the model price.the option is set to the model price. Figure 5.17, p. 168Figure 5.17, p. 168 illustrates the procedure. illustrates the procedure. Substitute estimates of the volatility into the B-S Substitute estimates of the volatility into the B-S

formula until the market price converges to the model formula until the market price converges to the model price. See price. See Table 5.7, p. 169Table 5.7, p. 169 for the implied volatilities for the implied volatilities of the AOL calls.of the AOL calls.

A short-cut for at-the-money options isA short-cut for at-the-money options is

T(0.398)S

C

0

≈


Ch. 5: 33


Implied Volatility (continued)Implied Volatility (continued) For our AOL June 125 call, this givesFor our AOL June 125 call, this gives

This is quite close; the actual implied volatility is .83.This is quite close; the actual implied volatility is .83. Appendix 5.A shows a method to produce faster Appendix 5.A shows a method to produce faster

convergence.convergence.

0.8697 .0959.9375(0.398)125

13.50 ==


Ch. 5: 34


Implied Volatility (continued)Implied Volatility (continued) Interpreting the Implied VolatilityInterpreting the Implied Volatility

The relationship between the implied volatility and the time to The relationship between the implied volatility and the time to expiration is called the term structure of implied volatility. expiration is called the term structure of implied volatility. See See Figure 5.18, p. 170Figure 5.18, p. 170..

The relationship between the implied volatility and the The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. exercise price is called the volatility smile or volatility skew. Figure 5.19, p. 171Figure 5.19, p. 171. These volatilities are actually supposed to . These volatilities are actually supposed to be the same. This effect is puzzling and has not been be the same. This effect is puzzling and has not been adequately explained.adequately explained.

The CBOE has constructed indices of implied volatility of The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See and Nasdaq (VXN). See Figure 5.20, p. 172Figure 5.20, p. 172..


Ch. 5: 35

Put Option Pricing Models

Restate put-call parity with continuous discountingRestate put-call parity with continuous discounting

Substituting the B-S formula for C above gives the B-S put Substituting the B-S formula for C above gives the B-S put option pricing modeloption pricing model

N(dN(d11) and N(d) and N(d22) are the same as in the call model.) are the same as in the call model.

Tr00e0e

cXeSX)T,,(SC),,(P −+−=XTS

)]N(d[1S)]N(d[1XeP 102Trc −−−= −


Ch. 5: 36

Put Option Pricing Models (continued) Note calculation of put price:Note calculation of put price:

The Black-Scholes price does not reflect early exercise and, thus, is The Black-Scholes price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market 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. 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.With n = 100, we obtained 12.11.

See See Table 5.8, p. 175Table 5.8, p. 175 for the effect of the input variables on the Black- for the effect of the input variables on the Black-Scholes put formula.Scholes put formula.

Your software also calculates put prices and Greeks.Your software also calculates put prices and Greeks.

12.09 .5692] 125.9375[1

.4670] [1125eP 59(.0446).09

=−−−= −


Ch. 5: 37

Managing the Risk of Options Here we talk about how option dealers hedge the risk of Here we talk about how option dealers hedge the risk of

option positions they take.option positions they take. Assume a dealer sells 1,000 AOL June 125 calls at the Assume a dealer sells 1,000 AOL June 125 calls at the

Black-Scholes price of 13.5512 with a delta of .5692. Black-Scholes price of 13.5512 with a delta of .5692. Dealer will buy 569 shares and adjust the hedge daily.Dealer will buy 569 shares and adjust the hedge daily. To buy 569 shares at $125.9375 and sell 1,000 calls at To buy 569 shares at $125.9375 and sell 1,000 calls at

$13.5512 will require $58,107.$13.5512 will require $58,107. We simulate the daily stock prices for 35 days, at which We simulate the daily stock prices for 35 days, at which

time the call expires.time the call expires.


Ch. 5: 38

Managing the Risk of Options (continued) The second day, the stock price is 120.5442. There are The second day, the stock price is 120.5442. There are

now 34 days left. Using bsbin3.xls, we get a call price of now 34 days left. Using bsbin3.xls, we get a call price of 10.4781 and delta of .4999. We have10.4781 and delta of .4999. We have Stock worth 569($120.5442) = $68,590Stock worth 569($120.5442) = $68,590 Options worth -1,000($10.4781) = -$10,478Options worth -1,000($10.4781) = -$10,478 Total of $58,112Total of $58,112 Had we invested $58,107 in bonds, we would have had Had we invested $58,107 in bonds, we would have had

$58,107e$58,107e.0446(1/365).0446(1/365) = $58,114. = $58,114. Table 5.9, pp. 178-179Table 5.9, pp. 178-179 shows the remaining outcomes. shows the remaining outcomes.

We must adjust to the new delta of .4999. We need 500 We must adjust to the new delta of .4999. We need 500 shares so sell 69 and invest the money ($8,318) in bonds.shares so sell 69 and invest the money ($8,318) in bonds.


Ch. 5: 39

Managing the Risk of Options (continued) At the end of the second day, the stock goes to 106.9722 and 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 the call to 4.7757. The bonds accrue to a value of $8,319. We havehave Stock worth 500($106.9722) = $53,486Stock worth 500($106.9722) = $53,486 Options worth -1,000($4.7757) = -$4,776Options worth -1,000($4.7757) = -$4,776 Bonds worth $8,319 (includes one days’ interest)Bonds worth $8,319 (includes one days’ interest) Total of $57,029Total of $57,029 Had we invested the original amount in bonds, we would Had we invested the original amount in bonds, we would

have had $58,107ehave had $58,107e.0446(2/365).0446(2/365) = $58,121. We are now short by = $58,121. We are now short by over $1,000.over $1,000.

At the end we have $56,540, a shortage of $1,816.At the end we have $56,540, a shortage of $1,816.


Ch. 5: 40

Managing the Risk of Options (continued) What we have seen is the second order or gamma effect. What we have seen is the second order or gamma effect.

Large price changes, combined with an inability to trade Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.continuously result in imperfections in the delta hedge.

To deal with this problem, we must gamma hedge, i.e., To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at another option. Let us use the June 130 call, selling at 11.3772 with a delta of .5086 and gamma of .0123. Our 11.3772 with a delta of .5086 and gamma of .0123. Our original June 125 call has a gamma of .0121. The stock original June 125 call has a gamma of .0121. The stock gamma is zero.gamma is zero.

We shall use the symbols We shall use the symbols 11, , 22, , 11 and and 22. We use h. We use hSS

shares of stock and hshares of stock and hCC of the June 130 calls. of the June 130 calls.


Ch. 5: 41

Managing the Risk of Options (continued) The delta hedge condition isThe delta hedge condition is

hhSS(1) - 1,000(1) - 1,00011 + h + hC C 2 2 = 0 = 0

The gamma hedge condition isThe gamma hedge condition is -1,000-1,00011 + h + hCC 22 = 0 = 0

We can solve the second equation and get hWe can solve the second equation and get hCC and then substitute and then substitute

back into the first to get hback into the first to get hSS. Solving for h. Solving for hCC and h and hSS, we obtain, we obtain

hhCC = 1,000(.0121/.0123) = 984 = 1,000(.0121/.0123) = 984

hhSS = 1,000(.5692 - (.0121/.0123).5086) = 68 = 1,000(.5692 - (.0121/.0123).5086) = 68

So buy 68 shares, sell 1,000 June 125s, buy 985 June 130s.So buy 68 shares, sell 1,000 June 125s, buy 985 June 130s.


Ch. 5: 42

Managing the Risk of Options (continued) The initial outlay will beThe initial outlay will be

68($125.9375) - 1,000($13.5512) + 985($11.3772) = 68($125.9375) - 1,000($13.5512) + 985($11.3772) = $6,219$6,219

At the end of day one, the stock is at 120.5442, the 125 call is 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 at 10.4781, the 130 call is at 8.6344. The portfolio is worth 68($120.5442) - 1,000($10.4781) + 985($8.6344) = 68($120.5442) - 1,000($10.4781) + 985($8.6344) =

$6,224$6,224 It should be worth $6,219eIt should be worth $6,219e .0446(1/365).0446(1/365) = $6,220. = $6,220. The new deltas are .4999 and .4384 and the new gammas The new deltas are .4999 and .4384 and the new gammas

are .0131 and .0129.are .0131 and .0129.


Ch. 5: 43

Managing the Risk of Options (continued) The new values are 1,012 of the 130 calls so we buy 27. 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, The new number of shares is 56 so we sell 12. Overall, this generates $1,214, which we invest in bonds.this generates $1,214, which we invest in bonds.

The next day, the stock is at $106.9722, the 125 call is at 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 $4.7757 and the 130 call is at $3.7364. The bonds are worth $1,214. The portfolio is worthworth $1,214. The portfolio is worth 56($106.9722) - 1,000($4.7757) + 1,012($3.7364) + 56($106.9722) - 1,000($4.7757) + 1,012($3.7364) +

$1,214 = $6,210.$1,214 = $6,210. The portfolio should be worth $6,219eThe portfolio should be worth $6,219e.0446(2/365).0446(2/365) = $6,221. = $6,221. Continuing this, we end up at $6,589 and should have Continuing this, we end up at $6,589 and should have

$6,246, a difference of $343. We are much closer than $6,246, a difference of $343. We are much closer than when only delta hedging.when only delta hedging.


Ch. 5: 44

Summary See See Figure 5.21, p. 182Figure 5.21, p. 182 for the relationship between call, for the relationship between call,

put, underlying asset, risk-free bond, put-call parity, and put, underlying asset, risk-free bond, put-call parity, and Black-Scholes call and put option pricing models.Black-Scholes call and put option pricing models.


Ch. 5: 45

Appendix 5.A: A Shortcut to the Calculation of Implied Volatility

This technique developed by Manaster and Koehler gives a This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given starting point and guarantees convergence. Let a given volatility be volatility be ** and the corresponding Black-Scholes price and the corresponding Black-Scholes price be C(be C(**). The initial guess should be). The initial guess should be

You then compute C(You then compute C(11**). If it is not close enough, you ). If it is not close enough, you

make the next guess.make the next guess.

⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛=T2

Tr XS

ln c0*

1σ


Ch. 5: 46

Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)

Given the iGiven the ithth guess, the next guess should be guess, the next guess should be

where dwhere d11 is computed using is computed using 11**. Let us illustrate using the . Let us illustrate using the

AOL June 125 call. C(AOL June 125 call. C() = 13.50. The initial guess is) = 13.50. The initial guess is

[ ]TS

2e)C()C(

0

/2d*i*

i*

1i

21 πσσ

σσ−

−=+

.4950 .0959

29).0446(.095

125125.9375

ln * =⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛=1σ


Ch. 5: 47

Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)

At a volatility of .4950, the Black-Scholes value is 8.41. At a volatility of .4950, the Black-Scholes value is 8.41. The next guess should beThe next guess should be

where .1533 is dwhere .1533 is d11 computed from the Black-Scholes model computed from the Black-Scholes model

using .4950 as the volatility and 2.5066 is the square root using .4950 as the volatility and 2.5066 is the square root of 2of 2ππ. Now using .8260, we obtain a Black-Scholes value . Now using .8260, we obtain a Black-Scholes value of 13.49, which is close enough to 13.50. So .83 is the of 13.49, which is close enough to 13.50. So .83 is the implied volatility. implied volatility.

[ ].8260

.0959125.9375

(2.5066)e13.508.41.4950

/2(.1533)*2

2

=−

−=σ


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Appendix 5.B: The BSBWIN2.2 Windows Software


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Documents

Chapter 5: Option Pricing Models: The Black-Scholes Model