Dr. Hassan Mounir El-SadyChapter 6 1 Black-Scholes Option Pricing Model (BSOPM)

Dr. Hassan Mounir El-Sady Chapter 61

Chapter 6

Black-Scholes Option Pricing Model (BSOPM)

Chapter 6Dr. Hassan Mounir El-

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Introduction

The Black-Scholes option pricing model (BSOPM) has been one of the

most important developments in finance in the last 50 years

– Has provided a good understanding of what options should sell for

– Has made options more attractive to individual and institutional investors


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The Model

Tdd

T

TRXS

d

dNXedSNC RT

12

2

1

21

and

2ln

where

)()(


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The Model (cont’d)

Variable definitions:S = Current Stock PriceX = Option Strike Price

e = Base of Natural LogarithmsR = Riskless Interest Rate

T = Time Until Option Expiration = Standard Deviation (Sigma) of Returns

on the Underlying Securityln = Natural Logarithm

N(d1) and N(d2) = Cumulative Standard Normal Distribution Functions


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Determinants of the Option Premium

1.1. Striking PriceStriking Price– The lower the striking price for a given stock, the

more the option should be worth, because a call option lets you buy at a predetermined striking price

2.2. Time Until ExpirationTime Until Expiration– The longer the time until expiration, the more the

option is worth– The option premium increases for more distant

expirations for puts and calls

3.3. Stock PriceStock Price– The higher the stock price, the more a given call

option is worth


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4.4. VolatilityVolatility – The greater the price volatility, the more the

option is worth.– The volatility estimate sigma cannot be directly

observed and must be estimated.

5.5. DividendsDividends– A company that pays a large dividend will have a

smaller option premium than a company with a lower dividend, everything else being equal

(Capital Gain Effect).– Listed options do not adjust for cash dividends.

6.6. Risk-Free Interest RateRisk-Free Interest Rate– The higher the risk-free interest rate, the higher

the option premium, everything else being equal.

Determinants of the Option Premium (cont’d)


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Assumptions of the Black-Assumptions of the Black-Scholes ModelScholes Model

1. The stock pays no dividends during the option’s life

2. European exercise style3. Markets are efficient4. No transaction costs

5. Interest rates remain constant6. Prices are lognormally distributed


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Assumptions of the Black-Scholes Model:

1.1. The stock pays no dividends during The stock pays no dividends during the option’s life.the option’s life.

– If you apply the BSOPM to two securities, one with no dividends and

the other with a dividend yield, the model will predict the same call

premium– Developed BSOPM account for the

payment of dividends.


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The BSOPM if Stock Pays Dividends During the Option’s Life

The Robert Miller Option Pricing Model

Tdd

T

TdRXS

d

dNXedSNeC RTdT

*1

*2

2

*1

*2

*1

*

and

2ln

where

)()(


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Assumptions of the Black-Scholes Model (cont’d) :

2.2. European Exercise Style:European Exercise Style:– A European option can only be exercised on the expiration

date– American options are more valuable than European options

– Few options are exercised early due to time value

3.3. Informational EfficiencyInformational Efficiency– People cannot predict the direction of the market or of

an individual stock– Put/call parity implies that you and everyone else will

agree on the option premium, regardless of whether you are bullish or bearish

4.4. No Transaction CostsNo Transaction Costs– There are no commissions and bid-ask spreads

– This is not true, because transaction costs causes slightly different actual option prices for different

market participants


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Assumptions of the Black-Scholes Model (cont’d):

5.5. Interest Rates Remain ConstantInterest Rates Remain Constant:– There is no real “riskfree” interest rate

– Often the 30-day T-bill rate is used– Must look for ways to value options when the

parameters of the traditional BSOPM are unknown or dynamic

6.6. Prices Are Lognormally Distributed:Prices Are Lognormally Distributed:– The logarithms of the underlying security prices

are normally distributed– A reasonable assumption for most assets on

which options are available


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Intuition Into the Black-Scholes Model

The valuation equation has two parts– One gives a “pseudo-probability” weighted

expected stock price (an inflow)– One gives the time-value of money adjusted

expected payment at exercise (an outflow)

)( 1dSNC )( 2dNKe RT

Cash Inflow Cash Outflow


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Intuition Into the Black-Scholes Model (cont’d)

The value of a call option is the difference between the expected benefit from

acquiring the stock outright and paying the exercise price on expiration day.


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Calculating Black-Scholes Prices from Historical Data

To calculate the theoretical value of a call option using the BSOPM, we

need:– The stock price

– The option striking price– The time until expiration– The riskless interest rate– The volatility of the stock


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Valuing a Microsoft Call ExampleValuing a Microsoft Call Example We would like to value a MSFT OCT 70 call in the year

2000. Microsoft closed at $70.75 on August 23 (58 days before option expiration). Microsoft pays no dividends.

We need the interest rate and the stock volatility to value the call.

Consulting the “Money Rate” section of the Wall Street Journal, we find a T-bill rate with about 58 days to

maturity to be 6.10%.

To determine the volatility of returns, we need to take the logarithm of returns and determine their volatility. Assume we find the annual standard deviation of MSFT

returns to be 0.5671.


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Valuing a Microsoft Call Example Valuing a Microsoft Call Example (cont’d)(cont’d)

Using the BSOPM:

2032.1589.5671.

1589.02

5671.0610.

7075.70

ln

2ln

2

2

1

T

TRXS

d


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Valuing a Microsoft Call Example (cont’d)Valuing a Microsoft Call Example (cont’d)

Using the BSOPM (cont’d):Using the BSOPM (cont’d):

Using normal probability tables, we find:Using normal probability tables, we find:

The value of the MSFT OCT 70 call is:The value of the MSFT OCT 70 call is:

4909.)0029.(

5805.)2032(.

N

N

0229.2261.2032.12

Tdd

04.7$

)4909(.70)5805(.75.70

)()()1589)(.0610(.

21

e

dNXedSNC RT


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Using Black-Scholes to Solve for the Put Premium

Can combine the BSOPM with put/call parity:

)()( 12 dSNdNXeP RT


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Implied Volatility

Introduction Calculating implied volatility

An implied volatility heuristic Historical versus implied volatility

Pricing in volatility units Volatility smiles


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Introduction

Instead of solving for the call premium, assume the market-

determined call premium is correct– Then solve for the volatility that makes

the equation hold– This value is called the implied volatility


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Calculating Implied Volatility

Sigma cannot be conveniently isolated in the BSOPM

– We must solve for sigma using trial and error

Valuing a Microsoft Call Example

The implied volatility for the MSFT OCT 70 call is 35.75%, which is much lower than

the 57% value calculated from the monthly returns over the last two years.


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An Implied Volatility Heuristic

For an exactly at-the-money call, the correct value of implied volatility is:

TRK

TPC

)1/(

/2)(5.0implied


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Historical Versus Implied Volatility

The volatility from a past series of prices is historical volatility

Implied volatility gives an estimate of what the market thinks about

likely volatility in the future


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Pricing in Volatility Units

You cannot directly compare the dollar cost of two different options

because– Options have different degrees of

“moneyness”– A more distant expiration means more

time value– The levels of the stock prices are

different


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Volatility Smiles

Volatility smiles are in contradiction to the BSOPM, which assumes

constant volatility across all strike prices

– When you plot implied volatility against striking prices, the resulting graph often

looks like a smile


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Volatility Smiles (cont’d)Volatility Smile

Microsoft August 2000

0

10

20

30

40

50

60

40 45 50 55 60 65 70 75 80 85 90 95 100 105

Striking Price

Imp

lie

d V

ola

tili

ty (

%)

Current Stock Price


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Problems Using the Black-Scholes Model

Does not work well with options that are deep-in-the-money or substantially out-of-

the-money Produces biased values for very low or

very high volatility stocks– Increases as the time until expiration increases May yield unreasonable values when an

option has only a few days of life remaining

Documents

Dr. Hassan Mounir El-SadyChapter 6 1 Black-Scholes Option Pricing Model (BSOPM)