bs option pricing basics.pdf

7/29/2019 BS Option Pricing Basics.pdf

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The Black-Scholes Option Pricing

Model


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Outline Introduction

The Black-Scholes option pricing model Calculating Black-Scholes prices from

historical data

Implied volatility Using Black-Scholes to solve for the put

premium

Problems using the Black-Scholes model


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Introduction The Black-Scholes option pricing model

(BSOPM) has been one of the mostimportant developments in finance in thelast 50 years

Has provided a good understanding ofwhat options should sell for

Has made options more attractive to

individual and institutional investors


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The Black-Scholes Option

Pricing Model The model

Development and assumptions of themodel

Determinants of the option premium

Assumptions of the Black-Scholes model

Intuition into the Black-Scholes model


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The Model

Tdd

T

TRK

S

d

dNKedSNCRT

=

++

=

=

12

2

1

21

and

2ln

where)()(


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The Model (contd) Variable definitions:

S = current stock priceK = option strike price

e = base of natural logarithms

R = riskless interest rate

T = time until option expiration = standard deviation (sigma) of returns on

the underlying security

ln = natural logarithm

N(d1) and

N(d2) = cumulative standard normal distribution

functions


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Determinants of the Option

Premium Striking price

Time until expiration

Stock price

Volatility

Dividends

Risk-free interest rate


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Striking Price The lower the striking price for a given

stock, the more the option should beworth

Because a call option lets you buy at a

predetermined striking price


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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


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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


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Volatility The greater the price volatility, the more

the option is worth The volatility estimate sigmacannot be

directly observed and must be estimated

Volatility plays a major role in determiningtime value


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Dividends A company that pays a large dividend will

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

Listed options do not adjust for cashdividends

The stock price falls on the ex-dividend

date


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Risk-Free Interest Rate The higher the risk-free interest rate, the

higher the option premium, everythingelse being equal

A higher discount rate means that the

call premium must rise for the put/callparity equation to hold


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Assumptions of the Black-

Scholes Model The stock pays no dividends during the

options life European exercise style

Markets are efficient

No transaction costs

Interest rates remain constant

Prices are lognormally distributed


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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 totime value


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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 andeveryone else will agree on the optionpremium, regardless of whether you arebullish or bearish


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No Transaction Costs There are no commissions and bid-ask

spreads Not true

Causes slightly different actual option

prices for different market participants


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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


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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 weightedexpected stock price (an inflow)

One gives the time-value of money

adjusted expected payment at exercise (anoutflow)


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Model (contd)

)( 1dSNC= )( 2dNKeRT

Cash Inflow Cash Outflow


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Model (contd) The value of a call option is the difference

between the expected benefit fromacquiring the stock outright and payingthe 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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Prices from Historical Data

Valuing 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.


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Valuing a Microsoft Call Example (contd)

Consulting the Money Rate section of the WallStreet Journal, we find a T-bill rate with about 58days to maturity to be 6.10%.

To determine the volatility of returns, we need totake the logarithm of returns and determine theirvolatility. Assume we find the annual standard

deviation of MSFT returns to be 0.5671.


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Using the BSOPM:

2032.1589.5671.

1589.02

5671.0610.

70

75.70ln

2ln

2

2

1

=

++

=

++

=

T

TRK

S

d


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Using the BSOPM (contd):

0229.2261.2032.12

==

= Tdd


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Using normal probability tables, we find:

4909.)0029.(

5805.)2032(.

=

=

N

N


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The value of the MSFT OCT 70 call is:

04.7$

)4909(.70)5805(.75.70

)()(

)1589)(.0610(.

21

=

=

=

e

dNKedSNCRT


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The call actually sold for $4.88.

The only thing that could be wrong in ourcalculation is the volatility estimate. This isbecause we need the volatility estimate over theoptions life, which we cannot observe.


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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 callpremium 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


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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 inthe 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 timevalue

The levels of the stock prices are different


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Volatility Smiles Volatility smilesare in contradiction to the

BSOPM, which assumes constantvolatility across all strike prices

When you plot implied volatility against

striking prices, the resulting graph oftenlooks like a smile


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Volatility Smiles (contd)

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

ImpliedVolatility(%)

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 anoption has only a few days of liferemaining

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