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© The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw-Hill 15-1 Chapter 15 Option Valuation Put-Call...
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Transcript of © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw-Hill 15-1 Chapter 15 Option Valuation Put-Call...
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-4
More Put-Call Parity
TreKDSPC
If the stock pays a dividend before option expiration:
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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$)754.0(e45)812.0(e50C )50.0(10.0)5.0(0
884.0
50.028.0
50.0228.0
010.04550ln
d
2
1
686.050.028.0884.0d2
From a standard normal probability table, look upN(d1) = 0.812 and N(d2) = 0.754
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-10
Varying the Option Input Values
Sign of inputeffect
Input Call Put GreekStock price (S) + - DeltaStrike price (K) - +Time until expiration (T) + + ThetaVolatility () + + VegaRisk-free rate ( r) + - RhoDividend yield (y) - +
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-11
Varying Option Input Values (cont’d)
• Stock price:• Call: as stock price increases call option
price increases• Put: as stock price increases put option
price decreases• Strike price:• Call: as strike price increases call option
price decreases• Put: as strike price increases put option
price increases
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-12
Varying Option Input Values (cont’d)
• Time until expiration:• Call & Put: as time to expiration
increases call and put option price increase
• Volatility:• Call & Put: as volatility increases call &
put value increase
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-13
Varying Option Input Values (cont’d)
• Risk-free rate:• Call: as the risk-free rate increases call
option price increases• Put: as the risk-free rate increases put
option price decreases• Dividend yield:• Call: as the dividend yield increases call
option price decreases• Put: as the dividend yield increases put
option price increases
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-14
Figure 15.1. Put and Call Option Prices
0
5
10
15
20
25
Stock Price ($)
Op
tio
n P
rice
($) Call PricePut Price
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-15
Figure 15.2. Option Prices and Time to Expiration
0
5
10
15
20
25
30
35
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
Time to Expiration (months)
Op
tio
n P
rice
($)
Call Price
Put Price
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-16
Figure 15.3. Option Prices and Sigma
0
5
10
15
20
25
Sigma (%)
Op
tio
n P
rice
($)
Call Price
Put Price
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-17
Figure 15.4. Options Prices and Interest Rates
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Interest Rate (%)
Op
tio
n P
rice
($)
Call Price
Put Price
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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.
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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 = $50 x 0.26998 x (.5)1/2 = 9.545, for a 1% increase in sigma, the call option will increase by 9.55 cents.
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-22
Other Measures of Sensitivity
• Gamma: measure of delta sensitivity to a stock price change.
• Theta: measure of option price sensitivity to a change in time to expiration.
• Rho: measure of option price sensitivity to a change in the interest rate.
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-23
Implied Standard Deviation
• Implied Standard Deviation (ISD)• Implied Volatility (IV)• Use current option price to compute an
estimate of the stock’s standard deviation.
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-24
Implied Volatility Skews
• Volatility smiles• Relationship between implied volatility's
and strike prices• Steep negative slope between ISD’s and
strike prices for both calls & puts• Stochastic volatility• Black-Scholes-Merton option pricing
model assumes constant volatility• Use at-the-money options
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-25
Figure 15.5. Volatility Skews for IBM Options
40
42
44
46
48
50
52
54
56
58
60
110 115 120 125 130 135 140 145
Strikes ($)
Imp
lied
Sta
nd
ard
Dev
iati
on
(%
)
Call ISDs
Put ISDs
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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.
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-29
Problem 15-19Suppose you have a stock market portfolio with a
beta of 1.4 that is currently worth $150 million. You wish to hedge against a decline using index option. Describe how you might do this with puts and calls. Suppose you decide to use SPX calls. Calculate the number of contracts needed if the contract you pick has a delta of .50, and S&P 500 index is at 1200.
Solution:
You can either buy puts or sell calls. In either case, gains or losses on your stock portfolio will be offset by gains or losses on your options contracts. [cont’d next slide]
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
15-30
Problem 15-19 (cont’d)Solution:
valuecontractOptionxdeltaOption
valuePortfolioxbetaPortfoliocontractsoptionof.No
contracts500,3000,120$x50.0
million150$x4.1Number
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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]
© The McGraw-Hill Companies, Inc., 2000Irwin/McGraw-Hill
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