Option Valuation

At expiration, an option is worth its intrinsic value.
Before expiration, put-call parity allows us to price
options. But,

 To calculate the price of a call, we need to know the put price.
 To calculate the price of a put, we need to know the call price.

So, what we want to know the value of a call option:
 Before expiration, and
 Without knowing the price of the put
The Black-Scholes option pricing model allows us to calculate
the price of a call option before maturity (put price is not
needed).

 Dates from the early 1970s
 Created by Professors Fischer Black and Myron Scholes
 Made option pricing much easier—The CBOE was launched soon after
the Black-Scholes model appeared.
Today, many finance professionals refer to an extended
version of the model

 The Black-Scholes-Merton option pricing model.
 Recognizing the important contributions by professor Robert Merton.
The Black-Scholes-Merton option pricing model says
the value of a stock option is determined by six
factors:

S, the current price of the underlying stock
y, the dividend yield of the underlying stock
K, the strike price specified in the option contract
r, the risk-free interest rate over the life of the option contract
T, the time remaining until the option contract expires
 , (sigma) which is the price volatility of the underlying stock





The price of a call option on a single share of
common stock is: C = Se–yTN(d1) – Ke–rTN(d2)

The price of a put option on a single share of
common stock is: P = Ke–rTN(–d2) – Se–yTN(–d1)

d1 and d2 are calculated using these two formulas:
d1 


ln S K   r  y  σ 2 2 T
d 2  d1  σ T
σ T
In the Black-Scholes-Merton formula, three common
functions are used to price call and put option prices:

 e-rt, or exp(-rt), is the natural exponent of the value of –rt (in common
terms, it is a discount factor for continuous time)
 ln(S/K) is the natural log of the "moneyness" term, S/K.
 N(d1) and N(d2) denotes the standard normal probability for the
values of d1 and d2.

In addition, the formula makes use of the fact that:
N(-d1) = 1 - N(d1)

Suppose you are given the following inputs:
S = $50
y = 2%
K = $45
T = 3 months (or 0.25 years)
 = 25% (stock volatility)
r = 6%
What is the price of a call option and a put option,
using the Black-Scholes-Merton option pricing
formula?

d1 


ln S K   r  y  σ 2 2 T
σ T




ln 50 45  0.06  0.02  0.252 2 0.25
0.25 0.25
0.10536  0.071258 0.25
0.125
 0.98538
d 2  d 1  σ T  0.98538  0.25 0.25  0.8604

If we use =NORMSDIST(0.98538), we obtain 0.837781.

If we use =NORMSDIST(0.8604), we obtain 0.805216.

Let’s make use of the fact N(-d1) = 1 - N(d1).
N(-0.98538) = 1 – N(0.89538) = 1 – 0.837781 = 0.162219.
N(-0.8604) = 1 – N(0.8604) = 1 – 0.805216= 0.194784
We now have all the information needed to price the
call and the put.

Values for N(d), given d: Rows give first decimal of d and columns give second and third decimals
d
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
-2.9
0.001866
0.001836
0.001807
0.001778
0.001750
0.001722
0.001695
0.001668
0.001641
-2.8
0.002555
0.002516
0.002477
0.002439
0.002401
0.002364
0.002327
0.002291
0.002256
-2.7
0.003467
0.003415
0.003364
0.003314
0.003264
0.003215
0.003167
0.003119
0.003072
-2.6
0.004661
0.004594
0.004527
0.004461
0.004396
0.004332
0.004269
0.004207
0.004145
-2.5
0.006210
0.006123
0.006037
0.005952
0.005868
0.005785
0.005703
0.005622
0.005543
-2.4
0.008198
0.008086
0.007976
0.007868
0.007760
0.007654
0.007549
0.007446
0.007344
-2.3
0.010724
0.010583
0.010444
0.010306
0.010170
0.010036
0.009903
0.009772
0.009642
-2.2
0.013903
0.013727
0.013553
0.013380
0.013209
0.013041
0.012874
0.012709
0.012545
-2.1
0.017864
0.017646
0.017429
0.017215
0.017003
0.016793
0.016586
0.016381
0.016177
-2.0
0.022750
0.022482
0.022216
0.021952
0.021692
0.021434
0.021178
0.020925
0.020675
-1.9
0.028717
0.028390
0.028067
0.027746
0.027429
0.027115
0.026803
0.026495
0.026190
-1.8
0.035930
0.035537
0.035148
0.034762
0.034380
0.034001
0.033625
0.033253
0.032884
-1.7
0.044565
0.044097
0.043633
0.043173
0.042716
0.042264
0.041815
0.041370
0.040930

Call Price = Se–yTN(d1) – Ke–rTN(d2)
= $50 x e-(0.02)(0.25) x 0. 837781 – 45 x e-(0.06)(0.25) x 0.805216
= 50 x 0.99501 x 0.837781 – 45 x 0.98511 x 0.805216
= $5.985.

Put Price = Ke–rTN(–d2) – Se–yTN(–d1)
= $45 x e-(0.06)(0.25) x 0.194784– 50 x e-(0.02)(0.25) x 0.162219
= 45 x 0.98511 x 0.194784 – 50 x 0.99501 x 0.162219
= $0.563.
Note: The options must have European-style exercise.
C  P  Se -yT  Ke  rT
$5.985  $0.563  50e (0.020.25)  45e (0.060.25)
$5.42  $49.75  $44.33
Stock Price:
Strike Price:
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
50.00
45.00
25.00
0.2500
6.00
2.00
Discounted Stock:
Discounted Strike:
49.75
44.33
d(1):
N(d1):
0.98538
0.83778
N(-d1):
0.16222
d(2):
N(d2):
0.86038
0.80521
N(-d2):
0.19479
Call Price:
$ 5.985
Put Price:
$ 0.565
Companies issuing stock options to employees must
report estimates of the value of these ESOs

The Black-Scholes-Merton formula is widely used for this
purpose.

For example, in December 2002, the Coca-Cola Company
granted ESOs with a stated life of 15 years.

However, to allow for the fact that ESOs are often
exercised before maturity, Coca-Cola also used a life of 6
years to value these ESOs.

Stock Price:
Discounted Stock:
44.55
35.10
Stock Price:
Discounted Stock:
44.55
40.23
Strike Price:
Discounted Strike:
44.66
19.13
Strike Price:
Discounted Strike:
44.66
36.41
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
25.53
15
5.65
1.59
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
30.20
6
3.40
1.70
d(1):
N(d1):
1.10792
0.86605
d(1):
N(d1):
0.50458
0.69307
d(2):
N(d2):
0.11915
0.54742
d(2):
N(d2):
-0.23517
0.40704
Call Price:
$ 13.06
Call Price:
$ 19.92

Changes in the stock price has a big effect on option prices.
Option traders must know how changes in input
prices affect the value of the options that are in their
portfolio.

Two inputs have the biggest effect over a time span
of a few days:

 Changes in the stock price (Greek name: Delta)
 Changes in the volatility of the stock price (Greek name: Vega)
Delta measures the dollar impact of a change in the
underlying stock price on the value of a stock option.

Call option delta
= e–yTN(d1) > 0
Put option delta
= –e–yTN(–d1) < 0
A $1 change in the stock price causes an option price
to change by approximately delta dollars.

Stock Price:
Strike Price:
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
50.00
45.00
25.00
0.2500
6.00
2.00
Discounted Stock:
Discounted Strike:
49.75
44.33
d(1):
N(d1):
Call Delta w/ Divy:
0.98538
0.83778
0.83360
N(-d1):
0.16222
d(2):
N(d2):
Put Delta w/ divy:
0.86038
0.80521
-0.16141
N(-d2):
0.19479
Call Price:
$
5.985
Put Price:
$
0.565
The call delta value of 0.8336 predicts that if the stock price
increases by $1, the call option price will increase by $0.83.

 If the stock price is $51, the call option value is $6.837—an actual
increase of about $0.85.
 How well does Delta predict if the stock price changes by $0.25?
The put delta value of -0.16141 predicts that if the stock price
increases by $1, the put option price will decrease by $0.16.

 If the stock price is $51, the put option value is $0.422—an actual
decrease of about $0.14.
 How well does Delta predict if the stock price changes by $0.25?
Vega measures the impact of a change in stock price
volatility on the value of stock options.


Vega is the same for both call and put options.
Vega = Se–yTn(d1)T > 0
n(d) represents a standard normal density, e-d/2/ 2p
If the stock price volatility changes by 100% (i.e., from
25% to 125%), option prices increase by about vega.

Stock Price:
Strike Price:
Volatility (%):
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
50.00
45.00
25.00
0.2500
6.00
2.00
d(1):
N(d1):
Call Delta:
0.98538
0.83778
0.83360
d(2):
N(d2):
Put Delta:
0.86038
0.80521
-0.19382
Call Price:
$ 5.985
Put Price:
$ 0.565
Discounted Stock:
Discounted Strike:
49.75
44.33
N(-d1):
n(d1):
0.16222
0.24375
N(-d2):
0.19479
Vega:
6.06325
The vega value of 6.063 predicts that if the stock price volatility increases
by 100% (i.e., from 25% to 125%), call and put option prices will increase by
$6.063.

Generally, traders divide vega by 100—that way the prediction is: if the
stock price volatility increases by 1% (25% to 26%), call and put option prices
will both increase by about $0.063.

If stock price volatility increases from 25% to 26%, you can use the
spreadsheet to see that the

 Call option price is now $6.047, an increase of $0.062.
 Put option price is now $0.627, an increase of $0.062.
Gamma measures delta sensitivity to a stock price
change.

 A $1 stock price change causes delta to change by approximately the
amount gamma.
Theta measures option price sensitivity to a change in
time remaining until option expiration.

 A one-day change causes the option price to change by approximately
the amount theta.
Rho measures option price sensitivity to a change in the
interest rate.

 A 1% interest rate change causes the option price to change by
approximately the amount rho.
Of the six input factors for the Black-Scholes-Merton
stock option pricing model, only the stock price volatility
is not directly observable.

A stock price volatility estimated from an option price is
called an implied standard deviation (ISD) or implied
volatility (IVOL).


Calculating an implied volatility requires:
 All other input factors, and
 Either a call or put option price
Sigma can be found by trial and error, or by using
the following formula.
 This simple formula yields accurate implied volatility
values as long as the stock price is not too far from
the strike price of the option contract.

2
2
2π T 

YX
YX
Y  X

σ
C
 C 


YX 
2
2 
π


Y  Se  yT
X  Ke -rT




Stock Price:
Strike Price:
Time (in years):
Riskless Rate (%):
Dividend Yield (%):
50.00
45.00
0.2500
6.00
2.00
Volatility (%):
39.25
Discounted Stock, Y:
Discounted Strike, X:
49.75
44.33
Estimated Volatility (Sigma, in %): 38.888
Percent Error: -0.92%
d(1):
N(d1):
0.68595
0.75363
(SQRT(2P/T))/Y+X
C- (Y-X)/2
d(2):
N(d2):
0.48970
0.68783
B-S-M Call Price:
$7.00
(C-(Y-X)/2)^2
((Y-X)^2)/p
Observed Call Price:
0.05
4.29
18.40
9.35
$7.00
< 1%,
not bad!
Chance/Brooks approximation:

C
(0.398)S 0 T

The CBOE publishes data for three implied volatility indexes:
 S&P 500 Index Option Volatility, ticker symbol VIX
 S&P 100 Index Option Volatility, ticker symbol VXO
 Nasdaq 100 Index Option Volatility, ticker symbol VXN
Each of these volatility indexes are calculating using IVs from eight
options:

 4 calls with two maturity dates:
 2 slightly out of the money
 2 slightly in the money
 4 puts with two maturity dates:
 2 slightly out of the money
 2 slightly in the money
The purpose of these indexes is to give investors information about
market volatility in the coming months.

You own 1,000 shares of XYZ stock AND you want
protection from a price decline.

Let’s use stock and option information from before—in
particular, the “delta prediction” to help us hedge.

Here you want changes in the value of your XYZ shares to be
offset by the value of your options position. That is:

Change in stock price  shares  Change in option price  number of options
Change in stock price  shares  Option Delta  number of options

Using a Delta of 0.8336 (slide 23) and a stock price decline of $1:
Change in stock price  shares  Option Delta  number of options
- 1 1,000  0.8336  number of options
Number of options  - 1,000 / 0.8336  - 1,199.62
- 1,199.62 / 100  - 12.
You should write 12 call options to hedge your stock.

XYZ Shares fall by $1—so, you lose $1,000.

What about the value of your option position?
 At the new XYZ stock price of $49, each call option is now worth
$5.17—a decrease of $.81 for each call ($81 per contract).
 Because you wrote 12 call option contracts, your call option gain was
$972.

Your call option gain nearly offsets your loss of $1,000.

Why is it not exact?
 Call Delta falls when the stock price falls.
 Therefore, you did not quite sell enough call options.

Using a Delta of -0.1614 (slide 23) and a stock price decline of $1:
Change in stock price  shares  Option Delta  number of options
- 1 1,000  - 0.1614  number of options
Number of options  - 1,000 / - 0.1614  6,195.79
6,195.79 / 100  62.
You should buy 62 put options to hedge your stock.

XYZ Shares fall by $1—so, you lose $1,000.

What about the value of your option position?
 At the new XYZ stock price of $49, each put option is now worth
$.75—an increase of $.19 for each put ($19 per contract).
 Because you bought 62 put option contracts, your put option gain
was $1,178.

Your put option gain more than offsets your loss of $1,000.

Why is it not exact?
 Put Delta also falls (gets more negative) when the stock price falls.
 Therefore, you bought too many put options—this error is more
severe the lower the value of the put delta.
 So, use a put with a strike closer to at-the-money.
Many institutional money managers use stock index options to hedge the
equity portfolios they manage.

To form an effective hedge, the number of option contracts needed can
be calculated with this formula:

Number of Option Contracts 
Portfolio Beta  Portfolio Value
Option Delta  Underlying Value  100
Note that regular rebalancing is needed to maintain an effective hedge
over time. Why? Well, over time:

 Underlying Value Changes
 Option Delta Changes
 Portfolio Value Changes
 Portfolio Beta Changes

Your $45,000,000 portfolio has a beta of 1.10.
You decide to hedge the value of this portfolio with the
purchase of put options.

 The put options have a delta of -0.31
 The value of the index is 1050.
Number of Option Contracts 
Portfolio Beta  Portfolio Value
Option Delta  Underlying Value  100

So, you buy 1,521 put options.
1.10  45,000,000
 1,520.74
0.31  1050  100










www.jeresearch.com (information on option formulas)
www.cboe.com (for a free option price calculator)
www.DerivativesModels.com (derivatives calculator)
www.numa.com (for “everything option”)
www.wsj.com/free (option price quotes)
www.aantix.com (for stock option reports)
www.ino.com (Web Center for Futures and Options)
www.optionetics.com (Optionetics)
www.pmpublishing.com (free daily volatility
summaries)
www.ivolatility.com (for applications of implied
volatility)

The Black-Scholes-Merton Option Pricing Model

Valuing Employee Stock Options

Varying the Option Price Input Values






Varying the Underlying Stock Price
Varying the Option’s Strike Price
Varying the Time Remaining until Option Expiration
Varying the Volatility of the Stock Price
Varying the Interest Rate
Varying the Dividend Yield
Measuring the Impact of Input Changes on Option
Prices









Interpreting Option Deltas
Interpreting Option Etas
Interpreting Option Vegas
Interpreting an Option’s Gamma, Theta, and Rho
Implied Standard Deviations
Hedging with Stock Options
Hedging a Stock Portfolio with Stock Index Options
Homework: 1, 4, 6, 10, 12, 15, 16, 17