blackscholes(1)

1

THE BLACK-SCHOLES AND BINOMIAL OPTION PRICING MODEL

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

2

THE BLACK-SCHOLES OPTION PRICING MODEL

• The model

• Development and assumptions of the model

• Determinants of the option premium

• Assumptions of the Black-Scholes model

• Intuition into the Black-Scholes model

3

THE MODEL

Tdd

T

TRKS

d

dNKedSNC RT

12

2

1

21

and

2ln

where

)()(

4

THE MODEL (CONT’D)

• Variable definitions:S = current stock priceK = option strike pricee = base of natural logarithmsR = riskless interest rateT = 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

5

DEVELOPMENT AND ASSUMPTIONS OF THE MODEL

• Derivation from:

• Physics

• Mathematical short cuts

• Arbitrage arguments

• Fischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM

6

DETERMINANTS OF THE OPTION PREMIUM

• Striking price

• Time until expiration

• Stock price

• Volatility

• Dividends

• Risk-free interest rate

7

STRIKING 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

8

TIME 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

9

STOCK PRICE

• The higher the stock price, the more a given call option is worth

• A call option holder benefits from a rise in the stock price

10

VOLATILITY

• The greater the price volatility, the more the option is worth

• The volatility estimate sigma cannot be directly observed and must be estimated

• Volatility plays a major role in determining time value

11

DIVIDENDS

• A company that pays a large dividend will have a smaller option premium than a company with a lower dividend, everything else being equal

12

RISK-FREE INTEREST RATE

• The higher the risk-free interest rate, the higher the option premium, everything else being equal

13

ASSUMPTIONS OF THE BLACK-SCHOLES MODEL

• The stock pays no dividends during the option’s life

• European exercise style

• Markets are efficient

• No transaction costs

• Interest rates remain constant

• Prices are lognormally distributed

14

THE STOCK PAYS NO DIVIDENDS DURING 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

• Robert Merton developed a simple extension to the BSOPM to account for the payment of dividends

15

The Robert Miller Option Pricing Model

16

Tdd

T

TdRKS

d

dNKedSNeC RTdT

*1

*2

2

*1

*2

*1

*

and

2ln

where

)()(

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

17

MARKETS ARE EFFICIENT

• The BSOPM assumes informational 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.

18

NO TRANSACTION COSTS

• There are no commissions and bid-ask spreads

• Not true

• Causes slightly different actual option prices for different market participants

19

INTEREST 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

20

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

21

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)

22

INTUITION INTO THE BLACK-SCHOLES MODEL (CONT’D)

23

)( 1dSNC )( 2dNKe RT

Cash Inflow Cash Outflow

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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BINOMIAL OPTION PRICING MODEL

Assumptions:

A single period

Two dates: time t=0 and time t=1 (expiration)

The future (time 1) stock price has only two possible values

The price can go up or down

The perfect market assumptions

No transactions costs, borrowing and lending at the risk free interest rate, no taxes…

26

BINOMIAL OPTION PRICING MODELEXAMPLE

The stock price Assume S= $50,

u= 10% and d= (-3%)

S

Su=S·(1+u)

Sd=S·(1+d)

S=$50

Su=$55

Sd=$48.5

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The call option price Assume X= $50,

T= 1 year (expiration)

C

Cu= Max{Su-X,0}

Cd= Max{Sd-X,0}

C

Cu= $5 = Max{55-50,0}

Cd= $0 = Max{48.5-50,0}

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The bond price Assume r= 6%

1

(1+r)

(1+r)

$1

$1.06

$1.06

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REPLICATING PORTFOLIO

At time t=0, we can create a portfolio of N shares of the stock and an investment of B dollars in the risk-free bond. The payoff of the portfolio will replicate the t=1 payoffs of the call option:

N·$55 + B·$1.06 = $5

N·$48.5 + B·$1.06 = $0

Obviously, this portfolio should also have the same price as the call option at t=0:

N·$50 + B·$1 = C

We get N=0.7692, B=(-35.1959) and the call option price is C=$3.2656.

30

A DIFFERENT REPLICATION

The price of $1 in the “up” state:

The price of $1 in the “down” state:

qu

$1

$0

qd

$0

$1

31

REPLICATING PORTFOLIOS USING THE STATE PRICES

We can replicate the t=1 payoffs of the stock and the bond using the state prices:

qu·$55 + qd·$48.5 = $50

qu·$1.06 + qd·$1.06 = $1

once we solve for the two state prices we can price any other asset in that economy. In particular, the call option:

qu·$5 + qd·$0 = C

We get qu=0.6531, qd =0.2903 and the call option price is C=$3.2656.

32


The put option price Assume X= $50,

T= 1 year (expiration)

P

Pu= Max{X-Su,0}

Pd= Max{X-Sd,0}

P

Pu= $0 = Max{50-55,0}

Pd= $1.5 = Max{50-48.5,0}

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REPLICATING PORTFOLIOS USING THE STATE PRICES


qu·$55 + qd·$48.5 = $50

qu·$1.06 + qd·$1.06 = $1

But the assets are exactly the same and so are the state prices. The put option price is:

qu·$0 + qd·$1.5 = P

We get qu=0.6531, qd =0.2903 and the put option price is P=$0.4354.

34

TWO PERIOD EXAMPLE

• Assume that the current stock price is $50, and it can either go up 10% or down 3% in each period.

• The one period risk-free interest rate is 6%.

• What is the price of a European call option on that stock, with an exercise price of $50 and expiration in two periods?

35

THE STOCK PRICE

S= $50, u= 10% and d= (-3%)

S=$50

Su=$55

Sd=$48.5

Suu=$60.5

Sud=Sdu=$53.35

Sdd=$47.05

36

THE BOND PRICE

r= 6% (for each period)

$1

$1.06

$1.06

$1.1236

$1.1236

$1.1236

37

THE CALL OPTION PRICE

X= $50 and T= 2 periods

C

Cu

Cd

Cuu=Max{60.5-50,0}=$10.5

Cud=Max{53.35-50,0}=$3.35

Cdd=Max{47.05-50,0}=$0

38

STATE PRICES IN THE TWO PERIOD TREE


qu·$55 + qd·$48.5 = $50

qu·$1.06 + qd·$1.06 = $1

Note that if u, d and r are the same, our solution for the state prices will not change (regardless of the price levels of the stock and the bond):

qu·S·(1+u) + qd·S ·(1+d) = S

qu ·(1+r)t + qd ·(1+r)t = (1+r)(t-1)

Therefore, we can use the same state-prices in every part of the tree.

39

THE CALL OPTION PRICE

qu= 0.6531 and qd= 0.2903

C

Cu

Cd

Cuu=$10.5

Cud=$3.35

Cdd=$0

Cu = 0.6531*$10.5 + 0.2903*$3.35 = $7.83

Cd = 0.6531*$3.35 + 0.2903*$0.00 = $2.19

C = 0.6531*$7.83 + 0.2903*$2.19 = $5.75 40

TWO PERIOD EXAMPLE

• What is the price of a European put option on that stock, with an exercise price of $50 and expiration in two periods?

• What is the price of an American call option on that stock, with an exercise price of $50 and expiration in two periods?

• What is the price of an American put option on that stock, with an exercise price of $50 and expiration in two periods?

41

TWO PERIOD EXAMPLE

European put option - use the tree or the put-call parity

What is the price of an American call option - if there are no dividends …

American put option – use the tree

42

THE EUROPEAN PUT OPTION PRICE

qu= 0.6531 and qd= 0.2903

P

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

Pu = 0.6531*$0 + 0.2903*$0 = $0

Pd = 0.6531*$0 + 0.2903*$2.955 = $0.858

PEU = 0.6531*$0 + 0.2903*$0.858 = $0.24943

THE AMERICAN PUT OPTION PRICE

qu= 0.6531 and qd= 0.2903

PAm

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

44

AMERICAN PUT OPTION

Note that at time t=1 the option buyer will decide whether to exercise the option or keep it till expiration.

If the payoff from immediate exercise is higher than the option value the optimal strategy is to exercise:

If Max{ X-Su,0 } > Pu(European) => Exercise

45

THE AMERICAN PUT OPTION PRICE

qu= 0.6531 and qd= 0.2903

P

Pu

Pd

Puu=$0

Pud=$0

Pdd=$2.955

Pu = Max{ 0.6531*0 + 0.2903*0 , 50-55 } = $0

Pd = Max{ 0.6531*0 + 0.2903*2.955 , 50-48.5 } = 50-48.5 =$1.5

PAm = Max{ 0.6531*0 + 0.2903*1.5 , 50-50 } = $0.4354 > $0.2490 = PEu

46

DETERMINANTS OF THE VALUESOF CALL AND PUT OPTIONS

VariableVariable C – Call ValueC – Call Value P – Put ValueP – Put Value

SS – – stock pricestock price IncreaseIncrease DecreaseDecrease

XX – – exercise priceexercise price DecreaseDecrease IncreaseIncrease

σσ – – stock price volatilitystock price volatility IncreaseIncrease IncreaseIncrease

TT – – time to expirationtime to expiration IncreaseIncrease IncreaseIncrease

rr – – risk-free interest raterisk-free interest rate IncreaseIncrease DecreaseDecrease

DivDiv – – dividend payoutsdividend payouts DecreaseDecrease IncreaseIncrease47

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