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)

Tσdd

Tσ

T2σyrKSlnd

12

2

1

d1 and d2 are calculated using these two formulas:

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 = $50y = 2%K = $45T = 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?

0.86040.250.250.98538Tσdd0.98538

0.125 0.25 0.071258 0.10536 0.250.25 0.2520.250.020.064550lnTσ T2σyrKSlnd

12

221

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.

$44.33$49.75$5.4245e50e$0.563$5.985

KeSePC0.25)(0.060.25)(0.02

rT-yT

Note: The options must have European-style exercise.

Stock Price: 50.00 Discounted Stock: 49.75Strike Price: 45.00 Discounted Strike: 44.33

Volatility (%): 25.00Time (in years): 0.2500

Riskless Rate (%): 6.00Dividend Yield (%): 2.00

d(1): 0.98538N(d1): 0.83778 N(-d1): 0.16222

d(2): 0.86038N(d2): 0.80521 N(-d2): 0.19479

Call Price: 5.985$

Put Price: 0.565$

16-13

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: 44.55 Stock Price: 44.55Discounted Stock: 35.10 Discounted Stock: 40.23

Strike Price: 44.66 Strike Price: 44.66Discounted Strike: 19.13 Discounted Strike: 36.41

Volatility (%): 25.53 Volatility (%): 30.20Time (in years): 15 Time (in years): 6

Riskless Rate (%): 5.65 Riskless Rate (%): 3.40Dividend Yield (%): 1.59 Dividend Yield (%): 1.70

d(1): 1.10792 d(1): 0.50458N(d1): 0.86605 N(d1): 0.69307

d(2): 0.11915 d(2): -0.23517N(d2): 0.54742 N(d2): 0.40704

Call Price: 19.92$ Call Price: 13.06$

16-15

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: 50.00 Discounted Stock: 49.75Strike Price: 45.00 Discounted Strike: 44.33

Volatility (%): 25.00Time (in years): 0.2500

Riskless Rate (%): 6.00Dividend Yield (%): 2.00

d(1): 0.98538N(d1): 0.83778 N(-d1): 0.16222

Call Delta w/ Divy: 0.83360

d(2): 0.86038N(d2): 0.80521 N(-d2): 0.19479

Put Delta w/ divy: -0.16141

Call Price: 5.985$

Put Price: 0.565$

16-23

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/ 2

If the stock price volatility changes by 100% (i.e., from 25% to 125%), option prices increase by about vega.

Stock Price: 50.00 Discounted Stock: 49.75Strike Price: 45.00 Discounted Strike: 44.33

Volatility (%): 25.00Time (in years): 0.2500

Riskless Rate (%): 6.00Dividend Yield (%): 2.00

d(1): 0.98538N(d1): 0.83778 N(-d1): 0.16222

Call Delta: 0.83360 n(d1): 0.24375

d(2): 0.86038N(d2): 0.80521 N(-d2): 0.19479

Put Delta: -0.19382

Call Price: 5.985$ Vega: 6.06325

Put Price: 0.565$

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.

rT-yT

22

Ke X SeY

π

XY

2

XYC

2

XYC

XY

T2πσ

Stock Price: 50.00 Discounted Stock, Y: 49.75Strike Price: 45.00 Discounted Strike, X: 44.33

Time (in years): 0.2500Riskless Rate (%): 6.00

Dividend Yield (%): 2.00

Volatility (%): 39.25 Estimated Volatility (Sigma, in %): 38.888

Percent Error: -0.92%d(1): 0.68595

N(d1): 0.75363 (SQRT(2P/T))/Y+X 0.05C- (Y-X)/2 4.29

d(2): 0.48970 (C-(Y-X)/2)^2 18.40N(d2): 0.68783 ((Y-X)^2)/ 9.35

B-S-M Call Price: $7.00 Observed Call Price: $7.00

< 1%, not bad!

T(0.398)S

C

0

Chance/Brooks approximation:

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:

options ofnumber Delta Option shares price stock in Change

options ofnumber price option in Change shares price stock in Change

Using a Delta of 0.8336 (slide 23) and a stock price decline of $1:

12.- 100 / 1,199.62-

1,199.62- 0.8336 / 1,000- options of Number

options of number 0.8336 1,000 1-

options of number Delta Option shares price stock in Change

16-38

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:

62. 100 / 6,195.79

6,195.79 0.1614- / 1,000- options of Number

options of number 0.1614- 1,000 1-

options of number Delta Option shares price stock in Change

16-40

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.

16-41

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:

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

100 ValueUnderlyingDelta Option

ValuePortfolioBeta Portfolio Contracts Option of Number

16-42

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.

1,520.74 10010500.31

45,000,0001.10

100 ValueUnderlyingDelta Option

ValuePortfolioBeta Portfolio Contracts Option of Number

So, you buy 1,521 put options.

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