chapter 5: option pricing models: the black-scholes-merton model

Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.

Ch. 5: 1

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

Good theories, like Black-Scholes-Merton, provide a Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common effect of possible causes. They give you a common language with which to quantify and communicate your language with which to quantify and communicate your feelings about value.feelings about value.

Emanuel DermanEmanuel Derman

The Journal of DerivativesThe Journal of Derivatives, Winter, 2000, p. 64, Winter, 2000, p. 64


Ch. 5: 2

Important Concepts

The Black-Scholes-Merton option pricing modelThe Black-Scholes-Merton 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-Merton 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

Black-Scholes-Merton 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 DCRB June 125 call option. Consider the DCRB June 125 call option. Figure 5.1, p. 127Figure 5.1, p. 127 shows the model price for an increasing shows the model price for an increasing number of time steps.number of time 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

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. 130Figure 5.2a, p. 130, , 5.2b, p. 1305.2b, p. 130 and and 5.3, p. 1315.3, p. 131 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. 132Figure 5.4, p. 132..

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

Tσdd

Tσ

/2)Tσ(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. 134Figure 5.5, p. 134)) See See Table 5.1, p. 135Table 5.1, p. 135 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 DCRB June 125 callPrice the DCRB June 125 call SS00 = 125.94, X = 125, r = 125.94, X = 125, rcc = ln(1.0456) = 0.0446, = ln(1.0456) = 0.0446,

T = 0.0959, T = 0.0959, = 0.83. = 0.83. SeeSee Table 5.2, p. 136Table 5.2, p. 136 for calculations. C = $13.21. for calculations. C = $13.21. Familiarize yourself with the accompanying softwareFamiliarize yourself with the accompanying software

Excel: BSMbin7e.xls. See Software Demonstration Excel: BSMbin7e.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: BSMbwin7e.exe. SeeWindows: BSMbwin7e.exe. See Appendix 5.B.Appendix 5.B.


Ch. 5: 10


Characteristics of the Black-Scholes-Merton FormulaCharacteristics of the Black-Scholes-Merton 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 - X). We need the expected value of

SSTT - X for those cases where S - X for those cases where STT > X. > X.


Ch. 5: 11


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

Bound of a European CallBound of a European Call Recall from Chapter 3 that the lower bound would Recall from Chapter 3 that the lower bound would

bebe

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

let it approach zero.let it approach zero.

)XeSMax(0, Tr0

c


Ch. 5: 13


Characteristics of the Black-Scholes-Merton Formula Characteristics of the Black-Scholes-Merton Formula (continued)(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 X and

SSTT < X. < X.


Ch. 5: 14


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

Effect of Variables on Option PricingEffect of Variables on Option Pricing

Ce Pe Ca PaVariable

S0

XTrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +


Ch. 5: 19

Variables in the Black-Scholes-Merton Model

The Stock PriceThe Stock Price Let S Let S then C then C . See. See Figure 5.6, p. 142Figure 5.6, p. 142.. 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. 143Figure 5.7, p. 143 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. 143Figure 5.8, p. 143..


Ch. 5: 20

Variables in the Black-Scholes-Merton 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: 21


The Stock Price (continued)The Stock Price (continued) 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 DCRB June 125 call,For our DCRB June 125 call,

T2σS

eGamma Call

0

/2d21

0.0123 0.09592(3.14159)3)125.94(0.8

eGamma Call

/2)1742.0( 2


Ch. 5: 22


The Stock Price (continued)The Stock Price (continued) If the stock goes from 125.94 to 130, the delta is If the stock goes from 125.94 to 130, the delta is

predicted to change from 0.569 to 0.569 + (130 - predicted to change from 0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the of 130 is 0.6171. So gamma captures most of the change in delta.change 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. 145Figure 5.9, p. 145 for gamma vs. the stock price for gamma vs. the stock price See See Figure 5.10, p. 145Figure 5.10, p. 145 for gamma vs. time for gamma vs. time


Ch. 5: 23


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: 24


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. 147. The effect

is called rho

In our example,

If the risk-free rate goes to 0.12, the rho estimates that the call price will go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43.

See Figure 5.12, p. 147 for rho vs. stock price.

)N(dTXeRho Call 2Trc

57.5)4670.0(.0959)125e0(Rho Call 0959)-0.0446(0.


Ch. 5: 25


The Volatility or Standard DeviationThe Volatility or Standard Deviation The most critical variable in the Black-Scholes-Merton The most critical variable in the Black-Scholes-Merton

model because the option price is very sensitive to the model 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. 148Figure 5.13, p. 148.. 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

15.322(3.14159)

e0.0959125.94 vegaCall

/2-0.17422


Ch. 5: 26


The Volatility or Standard Deviation (continued)The Volatility or Standard Deviation (continued) Thus if volatility changes by 0.01, the call price is Thus if volatility changes by 0.01, the call price is

estimated to change by 15.32(0.01) = 0.15estimated to change by 15.32(0.01) = 0.15 If we increase volatility to, say, 0.95, the estimated If we increase volatility to, say, 0.95, the estimated

change would be 15.32(0.12) = 1.84. The actual call change would be 15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an price at a volatility of 0.95 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. 149Figure 5.14, p. 149 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: 27


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. 150Figure 5.15, p. 150. 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

68.91- (0.4670)5e(0.0446)12

(0.0959)2(3.14159)2

3)e125.94(0.8- thetaCall

59).0446(0.090

/2(0.1742)2


Ch. 5: 28


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 (0.0959 - 0.0767)(-68.91) = -1.32. The change to (0.0959 - 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of actual call price with T = 0.0767 is 12.16, a decrease of 1.39.1.39.

See See Figure 5.16, p. 150Figure 5.16, p. 150 for theta vs. the stock price for theta vs. the stock price Note that your spreadsheet BSMbin7e.xls and your Note that your spreadsheet BSMbin7e.xls and your

Windows program BSMbwin7e.exe calculate the delta, Windows program BSMbwin7e.exe calculate the delta, gamma, vega, theta, and rho for calls and puts.gamma, vega, theta, and rho for calls and puts.


Ch. 5: 29

Black-Scholes-Merton 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 date is time where the ex-dividend date is time

t during the option’s life.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-M model:, is inserted into the B-S-M model:

See See Table 5.3, p. 152Table 5.3, p. 152 for example. for example. The Excel spreadsheet BSMbin7e.xls allows up to 50 discrete The Excel spreadsheet BSMbin7e.xls allows up to 50 discrete

dividends. The Windows program BSMbwin7e.exe allows up to dividends. The Windows program BSMbwin7e.exe allows up to three discrete dividends.three discrete dividends.

trt00

ceDSS


Ch. 5: 30

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. 153Table 5.4, p. 153 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 BSMbin7e.xls and Windows program The Excel spreadsheet BSMbin7e.xls and Windows program BSMbwin7e.exe permit you to enter a continuous dividend yield.BSMbwin7e.exe permit you to enter a continuous dividend yield.

Black-Scholes-Merton Model When the Stock Pays Dividends (continued)

T00 eSS c


Ch. 5: 31

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

Table 5.5, p. 154Table 5.5, p. 154 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 BSMbin7e.xls and Windows Your Excel spreadsheet BSMbin7e.xls and Windows

program BSMbwin7e.exe will calculate the American call program BSMbwin7e.exe will calculate the American call price using the binomial model.price using the binomial model.


Ch. 5: 32

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. 156Table 5.6, p. 156 for example with DCRB. for example with DCRB.

Your Excel spreadsheet Hisv7e.xls will do these Your Excel spreadsheet Hisv7e.xls will do these calculations. See Software Demonstration 5.2.calculations. See Software Demonstration 5.2.


Ch. 5: 33

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. 158Figure 5.17, p. 158 illustrates the procedure. illustrates the procedure. Substitute estimates of the volatility into the B-S-M Substitute estimates of the volatility into the B-S-M

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. 159Table 5.7, p. 159 for the implied volatilities for the implied volatilities of the DCRB calls.of the DCRB 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: 34


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

This is quite close; the actual implied volatility is 0.83.This is quite close; the actual implied volatility is 0.83. Appendix 5.A shows a method to produce faster Appendix 5.A shows a method to produce faster

convergence.convergence.

0.8697 0.0959.94(0.398)125

13.50


Ch. 5: 35


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. 160Figure 5.18, p. 160..

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. 161Figure 5.19, p. 161. 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. 163Figure 5.20, p. 163..


Ch. 5: 36

Put Option Pricing Models

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

Substituting the B-S-M formula for C above gives the Substituting the B-S-M formula for C above gives the B-S-M put option pricing modelB-S-M put option 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: 37

Put Option Pricing Models (continued)

Note calculation of put price:Note calculation of put price:

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

See See Table 5.8, p. 165Table 5.8, p. 165 for the effect of the input variables on the Black- for the effect of the input variables on the Black-Scholes-Merton put formula.Scholes-Merton put formula.

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

12.08 .5692] 0 125.94[1

.4670] 0 [1125eP 0959(0.0446)0.


Ch. 5: 38

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 DCRB June 125 calls at the Assume a dealer sells 1,000 DCRB June 125 calls at the Black-Scholes-Merton price of 13.5533 with a delta of Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge 0.5692. Dealer will buy 569 shares and adjust the hedge daily.daily. To buy 569 shares at $125.94 and sell 1,000 calls at To buy 569 shares at $125.94 and sell 1,000 calls at

$13.5533 will require $58,107.$13.5533 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: 39

Managing the Risk of Options (continued)

The second day, the stock price is 120.4020. There are The second day, the stock price is 120.4020. There are now 34 days left. Using BSMbin7e.xls, we get a call price now 34 days left. Using BSMbin7e.xls, we get a call price of 10.4078 and delta of 0.4981. We haveof 10.4078 and delta of 0.4981. We have Stock worth 569($120.4020) = $68,509Stock worth 569($120.4020) = $68,509 Options worth -1,000($10.4078) = -$10,408Options worth -1,000($10.4078) = -$10,408 Total of $58,101Total of $58,101 Had we invested $58,107 in bonds, we would have had Had we invested $58,107 in bonds, we would have had

$58,107e$58,107e0.0446(1/365)0.0446(1/365) = $58,114. = $58,114. Table 5.9, pp. 168-169Table 5.9, pp. 168-169 shows the remaining outcomes. shows the remaining outcomes.

We must adjust to the new delta of 0.4981. We need 498 We must adjust to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.shares so sell 71 and invest the money ($8,549) in bonds.


Ch. 5: 40


At the end of the second day, the stock goes to 126.2305 and the call At the end of the second day, the stock goes to 126.2305 and the call to 13.3358. The bonds accrue to a value of $8,550. We haveto 13.3358. The bonds accrue to a value of $8,550. We have Stock worth 498($126.2305) = $62,863Stock worth 498($126.2305) = $62,863 Options worth -1,000($13.3358) = -$13,336Options worth -1,000($13.3358) = -$13,336 Bonds worth $8,550 (includes one days’ interest)Bonds worth $8,550 (includes one days’ interest) Total of $58,077Total of $58,077 Had we invested the original amount in bonds, we would have had Had we invested the original amount in bonds, we would have had

$58,107e$58,107e0.0446(2/365)0.0446(2/365) = $58,121. We are now short by over $44. = $58,121. We are now short by over $44. At the end we have $59,762, a excess of $1,406.At the end we have $59,762, a excess of $1,406.


Ch. 5: 41


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 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock original June 125 call has a gamma of 0.0121. The stock has a delta of 1.0 and a gamma of 0.0.has a delta of 1.0 and a gamma of 0.0.

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: 42


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 hhSS(0) -1,000(0) -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 and then substitute back into the first to get hsubstitute back into the first to get hSS. Solving for h. Solving for hCC and and hhSS, we obtain, we obtain hhCC = 1,000(0.0121/0.0123) = 985 = 1,000(0.0121/0.0123) = 985 hhSS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68 = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 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.


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The initial outlay will beThe initial outlay will be 68($125.94) - 1,000($13.5533) + 985($11.3792) = 68($125.94) - 1,000($13.5533) + 985($11.3792) =

$6,219$6,219 At the end of day one, the stock is at 120.4020, the 125 At the end of day one, the stock is at 120.4020, the 125

call is at 10.4078, the 130 call is at 8.5729. The portfolio call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth is worth 68($120.4020) - 1,000($10.4078) + 985($8.5729) 68($120.4020) - 1,000($10.4078) + 985($8.5729)

= $6,224= $6,224 It should be worth $6,219eIt should be worth $6,219e0.0446(1/365)0.0446(1/365) = $6,220. = $6,220. The new deltas are 0.4981 and 0.4366 and the new The new deltas are 0.4981 and 0.4366 and the new

gammas are 0.0131 and 0.0129.gammas are 0.0131 and 0.0129.


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The new values are 1,013 of the 130 calls so we buy 1013 - 985 = The new values are 1,013 of the 130 calls so we buy 1013 - 985 = 28. The new number of shares is 56 so we sell 68 – 56 = 12. 28. The new number of shares is 56 so we sell 68 – 56 = 12. Overall, this generates 12($120.402) – 28($8.5729) = $1,205, Overall, this generates 12($120.402) – 28($8.5729) = $1,205, which we invest in bonds.which we invest in bonds.

The next day, the stock is at $126.2305, the 125 call is at The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth$1,205. The portfolio is worth 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205

= $6,222.= $6,222. The portfolio should be worth $6,219eThe portfolio should be worth $6,219e0.0446(2/365)0.0446(2/365) = $6,221. = $6,221. Continuing this, we end up at $6,267 and should have $6,246, a Continuing this, we end up at $6,267 and should have $6,246, a

difference of $21. We are much closer than when only delta difference of $21. We are much closer than when only delta hedging.hedging.


Ch. 5: 45

Hedging in Practice

Traders usually ensure that their portfolios are delta-Traders usually ensure that their portfolios are delta-neutral at least once a dayneutral at least once a day

Whenever the opportunity arises, they improve gamma Whenever the opportunity arises, they improve gamma and vegaand vega

As portfolio becomes larger hedging becomes less As portfolio becomes larger hedging becomes less expensiveexpensive


Ch. 5: 46

Summary

See See Figure 5.21, p. 172Figure 5.21, p. 172 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-Merton call and put option pricing models.Black-Scholes-Merton call and put option pricing models.


Ch. 5: 47

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- and the corresponding Black-Scholes-Merton price be C(Merton price 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.

T

2Tr

X

Sln c

0*1


Ch. 5: 48

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

DCRB June 125 call. C(DCRB 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

0.4950 0.0959

2959)0.0446(0.0

125

125.9375ln *

1


Ch. 5: 49

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

At a volatility of 0.4950, the Black-Scholes-Merton value At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should beis 8.41. The next guess should be

where 0.1533 is dwhere 0.1533 is d11 computed from the Black-Scholes- computed from the Black-Scholes-

Merton-Merton model using 0.4950 as the volatility and Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 22.5066 is the square root of 2. Now using 0.8260, we . Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility. close enough to 13.50. So 0.83 is the implied volatility.

0.8260

0.0959125.9375

(2.5066)e13.508.41.49500

/2(0.1533)*2

2


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Appendix 5.B: The BSMbwin7e.exe Windows Software


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(To continue) (Return to text slide 38)


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chapter 5: option pricing models: the black-scholes-merton model

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