fin 545 prof. rogers spring 2011 option valuation

FIN 545PROF. ROGERSSPRING 2011

Option valuation

PRINCIPLES OF OPTION VALUATIONASSOCIATED AUDIO CONTENT = APPROX 20

MINUTES (15 SLIDES)

Segment 1

Basic Notation and Terminology

Symbols S0 (stock price) X (exercise price) T (time to expiration = (days until expiration)/365) r (risk-free rate) ST (stock price at expiration)

C(S0,T,X), P(S0,T,X)

Principles of Call Option Pricing

Minimum Value of a Call C(S0,T,X) 0 (for any call) For American calls:

Ca(S0,T,X) Max(0,S0 - X)

Concept of intrinsic value: Max(0,S0 - X) Concept of time value of option

C(S,T,X) – Max(0,S – X) For example, what is the time value of a call option

trading at $5 with exercise price of $20 when the the underlying asset is trading at $22.75?

Principles of Call Option Pricing (continued)

Maximum Value of a Call C(S0,T,X) S0

“Right” but not “obligation” can never be more valuable than underlying asset (and will typically be worth less).

Option values are always equal to a percentage of the underlying asset’s value!


Effect of Time to Expiration More time until expiration, higher option value!

Volatility is related to time (we’ll see this in binomial and Black-Scholes models).

Calls allow buyer to invest in other assets, thus a pure time value of money effect.


Effect of Exercise Price Lower exercise prices on call options with same

underlying and time to expiration always have higher values!


Lower Bound of a European Call Ce(S0,T,X) Max[0,S0 - X(1+r)-T] A call option can never be worth less than the

difference between the underlying’s value and the present value of the exercise price on the call (or zero, if this difference is negative).


American Call Versus European Call Ca(S0,T,X) Ce(S0,T,X) If there are no dividends on the stock, an American call

will never be exercised early (unless there are complicating factors…we’ll discuss employee options eventually).

Rather than exercise, better to sell the call in the market. Options are worth more alive than dead!

If no dividends, the value of the American call and identical European call should be equal.

If dividend is sufficiently large to invoke potential for early exercise, this “early exercise option” is a source of additional value for an American call (vs. the equivalent European).

Principles of Put Option Pricing

Minimum Value of a Put P(S0,T,X) 0 (for any put) For American puts:

Pa(S0,T,X) Max(0,X - S0)

Concept of intrinsic value: Max(0,X - S0)

Principles of Put Option Pricing (continued)

Maximum Value of a Put Pe(S0,T,X) X(1+r)-T

European put option value must be no more than the present value of the exercise price of the put option.

Pa(S0,T,X) X American put option value is bounded above by the

exercise price. No “present value effect” because of potential for

early exercise (more on this shortly).


The Effect of Time to Expiration Same effect as call options: more time, more value!


Effect of Exercise Price Raising exercise price of put options increases

value!


Lower Bound of a European Put Pe(S0,T,X) Max(0,X(1+r)-T - S0) The value of a put option cannot be less than the

difference between the present value of the put option’s exercise price and the underlying’s value (or zero if this difference is negative).


American Put Versus European Put Pa(S0,T,X) Pe(S0,T,X)

Early Exercise of American Puts There is typically the probability of a sufficiently low

stock price occurring that will make it optimal to exercise an American put early.

Dividends on the stock reduce the likelihood of early exercise.


Put-Call Parity Form portfolios A and B where the options are

European. Portfolio A: Buy share of stock; buy put option on stock with

exercise price X, and maturity date T Portfolio B: Buy call option on stock with exercise price X,

and maturity date T; buy risk-free bond with face value X and maturity date T

The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T

Equation illustrates “put-call parity.” Equation can be rearranged to offer various interpretations.

Put-Call Parity Example

Price of underlying asset (S) = $19.50Premium for call option on underlying asset with

exercise price = $20 and 3 months until expiration = $2.50

Premium for put option on underlying asset with exercise price = $20 and 3 months until expiration = $1.50

Risk-free rate = 5%Does put-call parity hold?Which option is overpriced?What would be the trading strategy?

VALUING OPTIONS WITH BINOMIAL MODELSASSOCIATED AUDIO CONTENT = APPROX 38

MINUTES (12 SLIDES)

Segment 2

One-Period Binomial Model

Conditions and assumptions One period, two outcomes (states) S = current stock price u = 1 + return if stock goes up d = 1 + return if stock goes down r = risk-free rate

Value of European call at expiration one period later Cu = Max(0,Su - X) or Cd = Max(0,Sd - X)

One-Period Binomial Model (continued)

This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.

The probabilities of the up and down moves are never specified. They are irrelevant to the option price.

d)-d)/(u-r(1=p

wherer1

p)C(1pCC du


An Illustrative Example Let S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 First find the values of Cu, Cd, h, and p:

Cu = Max(0,100(1.25) - 100) = Max(0,125 - 100) = 25

Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 h = (25 - 0)/(125 - 80) = 0.556 p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6

Then insert into the formula for C: 14.02

1.07

0.0).40((0.6)25C

Student exercises

Calculate the option values if the following changes are made to the prior example: S = 110 S = 90 u = 1.40, d = 0.70 u = 1.15, d = 0.90 r = 10% r = 4%

(Return to text slide)

Two-Period Binomial Model

We now let the stock go up another period so that it ends We now let the stock go up another period so that it ends up Suup Su22, Sud or Sd, Sud or Sd22..

The option expires after two periods with three possible The option expires after two periods with three possible values:values:

X]SdMax[0,C

X]SudMax[0,C

X]SuMax[0,C

2d

ud

2u

2

2

Two-Period Binomial Model (continued)

After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values

The price of the call today will be ????

r1

p)C(1pCC

or ,r1

p)C(1pCC

2

2

ddud

uduu


2d

2udu

2

du

r)(1

Cp)(1p)C2p(1CpC

:over time)constant are d"" and u" (if

following theas writtenbe also can whichr1

p)C(1pCC

22


An Illustrative Example Let S = 100, X = 100, u = 1.25, d = 0.80, r =

0.07 Su2 = 100(1.25)2 = 156.25 Sud = 100(1.25)(0.80) = 100 Sd2 = 100(0.80)2 = 64 The call option prices at maturity are as follows:

0.0100]Max[0,64X]SdMax[0,C

0.0100]Max[0,100X]SudMax[0,C

56.25100]25Max[0,156.X]SuMax[0,C

2d

ud

2u

2

2


The two values of the call at the end of the first period are

0.01.07

0.0).40((0.6)0.0

r1

p)C(1pCCor

31.541.07

(0.4)0.0+(0.6)56.25=

r1

p)C(1pCC

2

2

ddud

uduu


Therefore, the value of the call today is

17.691.07

0.0).40((0.6)31.54r1

p)C(1pCC du

Student exercises

Calculate the option values if the following changes are made to the prior example: S = 110 S = 90 u = 1.40, d = 0.70 u = 1.125, d = 0.90, r = 3.5%

ABSENCE OF ARBITRAGE AND OPTION VALUATIONASSOCIATED AUDIO CONTENT = APPROX 24

MINUTES (10 SLIDES)

Segment 3

The “no-arbitrage” concept

Important point: d < 1 + r < u to prevent arbitrage

We construct a hedge portfolio of h shares of stock and one short call. Current value of portfolio: V = hS - C

At expiration the hedge portfolio will be worth Vu = hSu - Cu

Vd = hSd - Cd

If we are hedged, these must be equal. Setting Vu = Vd and solving for h gives (see next page!)


These values are all known so h is easily computed. The variable, h, is called “hedge ratio.”

Since the portfolio is riskless, it should earn the risk-free rate. Thus V(1+r) = Vu (or Vd)

Substituting for V and Vu

(hS - C)(1+r) = hSu - Cu

And the theoretical value of the option is

SdSu

CCh du

No-arbitrage condition

C = hS – [(hSu – Cu)(1 + r)-1]Solving for C provides the same result as we

determined in our earlier example!Can alternatively substitute Sd and Cd into

equationIf the call is not priced “correctly”, then

investor could devise a risk-free trading strategy, but earn more than the risk-free rate….arbitrage profits!

1-period binomial model risk-free portfolio example

A Hedged Portfolio Short 1,000 calls and long 1000h = 1000(0.556) =

556 shares. Value of investment: V = 556($100) -

1,000($14.02) $41,580. (This is how much money you must put up.)

Stock goes to $125 Value of investment = 556($125) - 1,000($25)

= $44,500 Stock goes to $80

Value of investment = 556($80) - 1,000($0) = $44,480 (difference from 44,500 is due to rounding error)


An Overpriced Call Let the call be selling for $15.00 Your amount invested is 556($100) -

1,000($15.00) = $40,600

You will still end up with $44,500, which is a 9.6% return.

Everyone will take advantage of this, forcing the call price to fall to $14.02

You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.

An Underpriced Call Let the call be priced at $13 Sell short 556 shares at $100 and buy 1,000 calls

at $13. This will generate a cash inflow of $42,600.

At expiration, you will end up paying out $44,500. This is like a loan in which you borrowed $42,600

and paid back $44,500, a rate of 4.46%, which beats the risk-free borrowing rate.


2-period binomial model risk-free portfolio example

A Hedge Portfolio Call trades at its theoretical value of $17.69. Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) =

0.701 So

Buy 701 shares at $100 for $70,100 Sell 1,000 calls at $17.69 for $17,690 Net investment: $52,410


A Hedge Portfolio (continued) The hedge ratio then changes depending on

whether the stock goes up or down

What is the hedge ratio if “up”? What is the hedge ratio if “down”? Describe how you alter your portfolio in each circumstance.

In each case, you wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.

2dud

d2

uduu

du

SdSud

CCh ,

SudSu

CCh ,

SdSu

CCh

22


A Hedge Portfolio (continued) Your wealth then grows by 7% from the end of the

first period to the end of the second. Conclusion: If the option is correctly priced and you

maintain the appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.


A Mispriced Call in the Two-Period World If the call is underpriced, you buy it and short the

stock, maintaining the correct hedge over both periods. You end up borrowing at less than the risk-free rate.

If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.

EXTENSIONS OF THE BINOMIAL MODEL VALUATION PROCESS

ASSOCIATED AUDIO CONTENT = APPROX 33 MINUTES (16 SLIDES)

Segment 4

Extensions of the binomial model

Early exercise (American options)Put optionsCall options with dividendsReal option examples

Pricing Put Options

Same procedure as calls but use put payoff formula at expiration. Using our prior example, put prices at expiration are

36.064]Max[0,100]SdMax[XP

0.0100]Max[0,100Sud]Max[XP

0.0156.25]Max[0,100]SuXMax[0,P

2d

ud

2u

2

2

Pricing Put Options (continued)

The two values of the put at the end of the first period are

13.461.07

36).40((0.6)0.0

r1

p)P(1pPPor

0.0,1.07

(0.4)0.0+(0.6)0.0=

r1

p)P(1pPP

2

2

ddud

uduu


Therefore, the value of the put today is

5.031.07

13.46).40((0.6)0.0r1

p)P(1pPP du


Let us hedge a long position in stock by purchasing puts. The hedge ratio formula is the same except that we ignore the negative sign:

Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.

0.29980125

13.460h


Stock goes from 100 to 125. We now have 299 shares at $125 + 1,000 puts at $0.0 = $37,375 This is a 7% gain over $34,930. The new hedge ratio is

So sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.

0.000100156.25

0.00.0h


Stock goes from 100 to 80. We now have 299 shares at $80 + 1,000 puts at $13.46 = $37,380 This is a 7% gain over $34,930. The new hedge ratio is

So buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.

1.00064100

360h


Stock goes from 125 to 156.25. We now have Bond worth $37,375(1.07) = $39,991 This is a 7% gain.

Stock goes from 125 to 100. We now have Bond worth $37,375(1.07) = $39,991 This is a 7% gain.


Stock goes from 80 to 100. We now have 1,000 shares worth $100 each, 1,000 puts worth $0 each,

plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain

Stock goes from 80 to 64. We now have 1,000 shares worth $64 each, 1,000 puts worth $36 each,

plus a loan in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain

Early Exercise & American Puts

Now we must consider the possibility of exercising the put early. At time 1 the European put values were

Pu = 0.00 when the stock is at 125

Pd = 13.46 when the stock is at 80

When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at 7.48

1.07

20).40((0.6)0.0P

Call options and dividends

One way to incorporate dividends is to assume a constant yield, , per period. The stock moves up, then drops by the rate . Using the same parameters from our earlier 2-

period binomial example (now incorporate 10% dividend yield to be paid at the first “up” or “down” node).

The call prices at expiration are

0100)0Max(0,57.6C

0.0100)Max(0,90CC

40.625100)625Max(0,140.C

2

2

d

duud

u

Calls and dividends (continued)

The European call prices after one period are

The European call value at time 0 is

00.01.070.0).40( (0.6)0.0

C

22.781.07

0.0).40(5(0.6)40.62C

u

u

12.771.07

0.0).40((0.6)22.78C

Will an American call option be exercised early?

If there are no dividends, a call option should never be exercised early (NOTE: not true for employee options later!)

If the call is American, when the stock is at 125, it pays a dividend of $12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25.

Therefore, a rational investor would exercise early (to capture the “high” dividend! In the example we just completed, replace the binomial value of Cu = $22.78 with Cu = $25. Thus, if the the call is American, at time 0 its value is $14.02. The $12.77 value on the prior page would be the value of the European call option.

14.021.07

0.0).40((0.6)25C

Calls and dividends

Alternatively, we can specify that the stock pays a specific dollar dividend at time 1. Unfortunately, the tree no longer recombines. We can still calculate the option value but the tree grows large very fast.

Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.

Calls and dividends

Yet another alternative (and preferred) specification is to subtract the present value of the dividends from the stock price and let the adjusted stock price follow the binomial up and down factors.

The tree now recombines and we can easily calculate the option values following the same procedure as before.

Real options

An application of binomial option valuation methodology to corporate financial decision making.

Consider an oil exploration company Traditional NPV analysis assumes that decision to

operate is “binding” through the life of the project.

Real options analysis adds “flexibility” by allowing management to consider abandonment of project if oil prices drop too low.

If “option” adds value to the project, then Project value = NPV of project + value of real options

See spreadsheet example.

VALUING OPTIONS WITH BLACK-SCHOLES-MERTONASSOCIATED AUDIO CONTENT = APPROX 37

MINUTES (15 SLIDES)

Segment 5

Origins of the Black-Scholes-Merton Formula

Brownian motion and the works of Einstein, Bachelier, Wiener, Itô

Black, Scholes, Merton and the 1997 Nobel Prize

Black-Scholes-Merton Model as the Limit of the Binomial Model

Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.

Demonstrate (using spreadsheet model) that binomial valuation converges toward Black-Scholes valuation.

The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.

Assumptions of the Model

Stock prices behave randomly and evolve according to a lognormal distribution. A lognormal distribution means that the log

(continuously compounded) return is normally distributed.

The risk-free rate and volatility of the log return on the stock are constant throughout the option’s life

There are no taxes or transaction costsThe options are European

A Nobel Formula

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

The model is derived with complex mathematics but is easily understandable. The formula is (if no dividends on underlying stock):

Tσdd

Tσ

/2)Tσ(r/X)ln(Sd

where

)N(dXe)N(dSC

12

2c0

1

2Tr

10c

A Nobel Formula (continued)

where N(d1), N(d2) = cumulative normal probability = annualized standard deviation (volatility) of the

continuously compounded return on the stock rc = continuously compounded risk-free rate


A Digression on Using the Normal Distribution The familiar normal, bell-shaped curve

(Figure 5.5) See Table 5.1 for determining the normal probability

for d1 and d2. This gives you N(d1) and N(d2).


A Numerical Example S0 = 30, X = 30, r = 4%, T = 0.5, = 0.40. See spreadsheet for calculations. C = $3.65.


Characteristics of the Black-Scholes-Merton Formula Interpretation of the Formula

The concept of risk neutrality, risk neutral probability, and its role in pricing options

The option price is the discounted expected payoff, Max(0,ST - X). We need the expected value of ST - X for those cases where ST > X.


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

The first term of the formula is the expected value of the stock price given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.

The second term is the expected value of the payment of the exercise price at expiration.


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

Lower Bound of a European Call Recall that the lower bound would be

The Black-Scholes-Merton formula always exceeds this value as seen by letting S0 be very high and then let it approach zero.

)XeSMax(0, rT0


Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When T = 0

At expiration, the formula must converge to the intrinsic value.

It does but requires taking limits since otherwise it would be division by zero.

Must consider the separate cases of ST X and ST < X.


Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When S0 = 0

Here the company is bankrupt so the formula must converge to zero.

It requires taking the log of zero, but by taking limits we obtain the correct result.


Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When = 0

Again, this requires dividing by zero, but we can take limits and obtain the right answer

If the option is in-the-money as defined by the stock price exceeding the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.


Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When X = 0

Call price converges to the stock price. Here both N(d1) and N(d2) approach 1.0 so by

taking limits, the formula converges to S0.


Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When r = 0

A zero interest rate is not a special case and no special result is obtained.

“GREEKS” OF BLACK-SCHOLESASSOCIATED AUDIO CONTENT = APPROX 20

MINUTES (8 SLIDES)

Segment 6

Variables in the Black-Scholes-Merton Model

The Stock Price Let S then C . See Figure 5.6. This effect is called the delta, which is given by

N(d1). Measures the change in call price over the change

in stock price for a very small change in the stock price.

Delta ranges from zero to one. See Figure 5.7 for how delta varies with the stock price.

The delta changes throughout the option’s life. See Figure 5.8.

Variables in the Black-Scholes-Merton Model (continued)

The Stock Price (continued) Delta hedging/delta neutral: holding shares of stock

and selling calls to maintain a risk-free position The number of shares held per option sold is the delta,

N(d1). As the stock goes up/down by $1, the option goes

up/down by N(d1). By holding N(d1) shares per call, the effects offset.

The position must be adjusted as the delta changes.


The Stock Price (continued) Delta hedging works only for small stock price

changes. For larger changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:

T2σS

eGamma Call

0

/2d21


The Stock Price (continued) The larger is the gamma, the more sensitive is

the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.

See Figure 5.9 for gamma vs. the stock price See Figure 5.10 for gamma vs. time


The Exercise Price 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.


The Risk-Free Rate Let r then C See Figure 5.11. The effect is

called rho

See Figure 5.12 for rho vs. stock price.

)N(dTXe RhoCall 2rT


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

model because the option price is very sensitive to the volatility and it is the only unobservable variable.

Let , then C See Figure 5.13. This effect is known as vega.

See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.

2

eTS vegaCall

/2-d0

21


The Time to Expiration Calculated as (days to expiration)/365 Let T , then C . See Figure 5.15. This effect is

known as theta:

See Figure 5.16 for theta vs. the stock price Chance’s spreadsheet BSMbin7e.xls calculates the

delta, gamma, vega, theta, and rho for calls and puts (thus can use as useful check).

)N(dXer T22

eS- thetaCall 2

Trc

/2d0 c

21

EXTENSIONS TO BLACK-SCHOLESASSOCIATED AUDIO CONTENT = APPROX 27

MINUTES (8 SLIDES)

Segment 7

Black-Scholes-Merton Model When the Stock Pays Dividends

Known Discrete Dividends Assume a single dividend of Dt where the ex-dividend

date is time t during the option’s life. Subtract present value of dividends from stock price. Adjusted stock price, S, is inserted into the B-S-M

model:

See Table 5.3 for example. The Excel spreadsheet BSMbin7e.xls allows up to 50

discrete dividends.

rtt00 eDSS

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

Continuous Dividend Yield Assume the stock pays dividends continuously at the

rate of . Subtract present value of dividends from stock price.

Adjusted stock price, S, is inserted into the B-S model.

See Table 5.4 for example. This approach could also be used if the underlying is a

foreign currency, where the yield is replaced by the continuously compounded foreign risk-free rate.

The Excel spreadsheet BSMbin7e.xls permit you to enter a continuous dividend yield.

T00 eSS

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

Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life

The value obtained upon exercise is compared to the ex-dividend value of the option.

High dividends and low time value lead to early exercise.

Chance’s Excel spreadsheet will calculate the American call price using the binomial model.

Estimating the Volatility

Historical Volatility This is the volatility over a recent time

period. Collect daily, weekly, or monthly returns on

the stock. Convert each return to its continuously

compounded equivalent by taking ln(1 + return). Calculate variance.

Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12 (monthly returns). Take square root.

Estimating the Volatility (continued)

Implied Volatility This is the volatility implied when the market

price of the option is set to the model price. Figure 5.17 illustrates the procedure. Substitute estimates of the volatility into the

B-S-M formula until the market price converges to the model price.

A short-cut for at-the-money options is

T(0.398)S

C

0

Estimating the Volatility (continued)

Implied Volatility (continued) Interpreting the Implied Volatility

The relationship between the implied volatility and the time to expiration is called the term structure of implied volatility. See Figure 5.18.

The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.

The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.

Put Option Pricing Models

Restate put-call parity with continuous discounting

Substituting the B-S-M formula for C above gives the B-S-M put option pricing model

N(d1) and N(d2) are the same as in the call model.

rT00e0e XeSX)T,,(SC),,(P XTS

)]N(d[1S)]N(d[1XeP 102rT

Put Option Pricing Models (continued)

The Black-Scholes-Merton price does not reflect early exercise. A binomial model would be necessary to get an accurate price. effect of the input variables on the Black-Scholes-Merton put formula.

Chance’s spreadsheet also calculates put prices and Greeks.

(Return to text slide)

fin 545 prof. rogers spring 2011 option valuation

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