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15-1

Chapter 15Option Valuation

• Put-Call Parity• The Black-Scholes-Merton Option Pricing Model• Varying the Option Price Input Values• Measuring the Impact of Input Changes on Option

Prices• Implied Standard Deviations• Hedging a Stock Portfolio with Stock Index

Options• Implied Volatility Skews• Summary & Conclusions

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15-2

Put-Call Parity

TreKSPC

TreKCPS

C = Call option price P = Put option priceS = Current stock price K = Option strike pricer = Risk-free interest rate T = Time remaining until

option expiration

Buy the stock, buy a put, and write a call; the sum ofwhich equals the strike price discounted at the risk-free rate

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15-3

More Put-Call Parity

If the stock pays a dividend before option expiration:

Expiration Date PayoffsExpiration date stock price ST > K ST < KBuy stock ST ST

Sell one call option -(ST - K) 0Buy one put option 0 (K - ST)Total portfolio expiration date payoff K K

TreKCPS

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15-4

More Put-Call Parity

TreKDSPC

If the stock pays a dividend before option expiration:

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15-5

Black-Scholes-Merton Option Pricing Model

Value of a stock option is a function of 6 input factors:1. Current price of underlying stock.2. Dividend yield of the underlying stock.3. Strike price specified in the option contract.4. Risk-free interest rate over the life of the contract.5. Time remaining until the option contract expires.6. Price volatility of the underlying stock.

The price of a call option equals:

)d(NeK)d(NeSC 2Tr

1Ty

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15-6

B-S-M Option Pricing Model (cont’d)

Where the inputs are:S = Current stock pricey = Stock dividend yieldK = Option strike pricer = Risk-free interest rateT = Time remaining until option expiration = Sigma, representing stock price volatility

The price of a put option equals:

)d(NeK)d(NeSC 2Tr

1Ty

)d(NeS)d(NeKP 1Ty

2Tr

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15-7

B-S-M Option Pricing Model (cont’d)

)d(NeK)d(NeSC 2Tr

1Ty

)d(NeS)d(NeKP 1Ty

2Tr

Where d1 and d2 equal:

T

T2

yrKSln

d

2

1

Tdd 12

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15-8

B-S-M Option Pricing Model (cont’d)

Remembering put-call parity, the value of a put,given the value of a call equals:

TreKSCP Also, remember at expiration:

KSC

SKP

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15-9

B-S-M Option Pricing Model Example

Assume S = $50, K = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put option.

125.1$e45$50$32.8$P )50.0(10.0

32.8$)754.0(e45)812.0(e50C )50.0(10.0)5.0(0

884.0

50.028.0

50.0228.0

010.04550ln

d

2

1

686.050.028.0884.0d2

From a standard normal probability table, look upN(d1) = 0.812 and N(d2) = 0.754

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15-10

Varying the Option Input Values

Sign of inputeffect

Input Call Put GreekStock price (S) + - DeltaStrike price (K) - +Time until expiration (T) + + ThetaVolatility () + + VegaRisk-free rate ( r) + - RhoDividend yield (y) - +

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15-11

Varying Option Input Values (cont’d)

• Stock price:• Call: as stock price increases call option

price increases• Put: as stock price increases put option

price decreases• Strike price:• Call: as strike price increases call option

price decreases• Put: as strike price increases put option

price increases

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15-12

Varying Option Input Values (cont’d)

• Time until expiration:• Call & Put: as time to expiration

increases call and put option price increase

• Volatility:• Call & Put: as volatility increases call &

put value increase

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15-13

Varying Option Input Values (cont’d)

• Risk-free rate:• Call: as the risk-free rate increases call

option price increases• Put: as the risk-free rate increases put

option price decreases• Dividend yield:• Call: as the dividend yield increases call

option price decreases• Put: as the dividend yield increases put

option price increases

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15-14

Figure 15.1. Put and Call Option Prices

0

5

10

15

20

25

Stock Price ($)

Op

tio

n P

rice

($) Call PricePut Price

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15-15

Figure 15.2. Option Prices and Time to Expiration

0

5

10

15

20

25

30

35

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Time to Expiration (months)

Op

tio

n P

rice

($)

Call Price

Put Price

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15-16

Figure 15.3. Option Prices and Sigma

0

5

10

15

20

25

Sigma (%)

Op

tio

n P

rice

($)

Call Price

Put Price

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15-17

Figure 15.4. Options Prices and Interest Rates

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Interest Rate (%)

Op

tio

n P

rice

($)

Call Price

Put Price

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15-18

Measuring the Impact of Changes - Delta

0dNeDeltaoptionCall 1Ty

0dNeDeltaoptionPut 1Ty

• Delta measures the impact of a change in the stock price on the value of the option.

• A $1 change in stock price causes the option to change by delta dollars.

• Delta is positive for calls and negative for puts

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15-19

Measuring the Impact of Changes - Eta

1CSdNeEtaoptionCall 1

Ty

1PSdNeEtaoptionPut 1

Ty

• Eta measures the percentage impact of a change in the stock price on the value of the option.

• A 1% change in stock price causes the option to change by eta percent.

• Eta is positive for calls and negative for puts.

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15-20

Measuring the Impact of Changes - Vega

0TdneSVega 1Ty

• Vega measures the impact of a change in stock price volatility on the value of the option.

• A 1% change in sigma causes the option price to change by vega percent.

• Vega is positive for calls and puts• Where:

2

d

1

1

e2

1dn

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15-21

Measuring the Impact of Changes - Example

• From the previous example: P = $50, K = $45, T = 6 months, r = 10%, = 28%, N(d1) = 0.812, N(d2) = 0.754, C = $8.32, and P= $1.13.

• Call Delta = 0.812, so for every $1.00 the stock price increases, the call option increases by $0.81

• Call Eta = 0.812 x (50 / 8.32) = 0.812, so for a 1% increase in stock price, the call option increases by 4.88%

• Call Vega = $50 x 0.26998 x (.5)1/2 = 9.545, for a 1% increase in sigma, the call option will increase by 9.55 cents.

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15-22

Other Measures of Sensitivity

• Gamma: measure of delta sensitivity to a stock price change.

• Theta: measure of option price sensitivity to a change in time to expiration.

• Rho: measure of option price sensitivity to a change in the interest rate.

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15-23

Implied Standard Deviation

• Implied Standard Deviation (ISD)• Implied Volatility (IV)• Use current option price to compute an

estimate of the stock’s standard deviation.

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15-24

Implied Volatility Skews

• Volatility smiles• Relationship between implied volatility's

and strike prices• Steep negative slope between ISD’s and

strike prices for both calls & puts• Stochastic volatility• Black-Scholes-Merton option pricing

model assumes constant volatility• Use at-the-money options

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15-25

Figure 15.5. Volatility Skews for IBM Options

40

42

44

46

48

50

52

54

56

58

60

110 115 120 125 130 135 140 145

Strikes ($)

Imp

lied

Sta

nd

ard

Dev

iati

on

(%

)

Call ISDs

Put ISDs

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15-26

Hedging a Stock Portfolio with Options

• To calculate the number of index option contracts need to hedge an equity portfolio:

valuecontractOptionxdeltaOption

valuePortfolioxbetaPortfoliocontractsoptionof.No

• To maintain hedge, must rebalance portfolio

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15-27

Problem 15-6

A call option is currently selling for $10. It has a strike price of $80 and 3 months to maturity. What is the price of a put option with a $80 strike and 3 months to maturity? The current stock price is $85, and the risk-free rate is 6%.

Solution:

Using put-call parity:

TreKSCP

81.3$e80$85$10$P )25(.06.

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15-28

Problem 15-7

What is the value of a call option if the underlying stock price is $100, the strike price is $70, the underlying stock volatility is 30%, and the risk-free rate is 5%. Assume the option has 30 days to expiration.

Solution:

S = $100sigma = .30

K = $70 T = 30 days

r = .05

solving for C = $30.29

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15-29

Problem 15-19Suppose you have a stock market portfolio with a

beta of 1.4 that is currently worth $150 million. You wish to hedge against a decline using index option. Describe how you might do this with puts and calls. Suppose you decide to use SPX calls. Calculate the number of contracts needed if the contract you pick has a delta of .50, and S&P 500 index is at 1200.

Solution:

You can either buy puts or sell calls. In either case, gains or losses on your stock portfolio will be offset by gains or losses on your options contracts. [cont’d next slide]

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15-30

Problem 15-19 (cont’d)Solution:

valuecontractOptionxdeltaOption

valuePortfolioxbetaPortfoliocontractsoptionof.No

contracts500,3000,120$x50.0

million150$x4.1Number

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15-31

Problem 15-20

Using an options calculator, calculate the price and the following “greeks” for a call and a put option with 1 year to expiration: delta, gamma, rho, eta, vega, and theta. The stock price is $80, the strike price is $75, the volatility is 40%, the dividend yield is 3%, and the risk-free rate is 5%.

Solution:

S = $80 sigma = 0.40

K = $75 T = 365 days

r = .05 y = 3%

[cont’d next slide]

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15-32

Problem 15-20 (cont’d)

Solution:

Call Put

Value $15.21 $8.92

Delta 0.640 -0.330

Gamma 0.011 0.011

Rho 0.360 -0.353

Eta 3.366 -2.963

Vega 0.285 0.285

Theta 0.016 0.015