black scholes pricing methodology

7/27/2019 Black Scholes Pricing Methodology

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Black Scholes Pricing

MethodologyDr A Vinay Kumar


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The What's?

What is Brownian Motion and Why is itimportant for Option Pricing?

What is Ito Calculus and How was it used?

Noble formula, how does it work?

Whats the implied volatility puzzle?


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Categorization of StochasticProcesses

Discrete time; discrete variable

Discrete time; continuous variable Continuous time; discrete variable

Continuous time; continuous variable


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Modeling Stock Prices

We can use any of the four types ofstochastic processes to model stock

prices The continuous time, continuous

variable process proves to be the mostuseful for the purposes of valuingderivatives


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Markov Processes (See pages 216-7)

In a Markov process future movementsin a variable depend only on where weare, not the history of how we got

where we are

We assume that stock prices followMarkov processes


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A Wiener Process (See pages 218)

We consider a variablez whose valuechanges continuously

The change in a small interval of time dt isdz

The variable follows a Wiener process if

1.2. The values ofdz for any 2 different (non-overlapping) periods of time are independent

(0,1)fromdrawingrandomaiswhere dd tz


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Properties of a WienerProcess

Mean of [z(T)z(0)] is 0

Variance of [z(T)z(0)] is T

Standard deviation of [z(T)z(0)] is

T


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Generalized Wiener Processes(See page 220-2)

A Wiener process has a drift rate (i.e.average change per unit time) of 0and a variance rate of 1

In a generalized Wiener process thedrift rate and the variance rate

can be set equal to any chosen

constants


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Generalized Wiener Processes(continued)

Mean change inx in time Tis aT

Variance of change inx in timeTis b2T

Standard deviation of change inxintime Tis

tbtax ddd

b T


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The Example Revisited

A stock price starts at 40 and has a probabilitydistribution of(40,10) at the end of the year

If we assume the stochastic process is Markovwith no drift then the process is

dS = 10dz

If the stock price were expected to grow by $8on average during the year, so that the year-

end distribution is (48,10), the process isdS = 8dt + 10dz


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Why a Generalized WienerProcess

is not Appropriate for Stocks For a stock price we can conjecture that its

expected percentage change in a short period

of time remains constant, not its expectedabsolute change in a short period of time

We can also conjecture that our uncertainty asto the size of future stock price movements is

proportional to the level of the stock price


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Ito Process (See pages 222)

In an Ito process the drift rate and thevariance rate are functions of time

dx=a(x,t)dt+b(x,t)dz

The discrete time equivalent

is only true in the limit as dttends to

zero

ttxbttxax ddd ),(),(


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An Ito Process for Stock Prices(See pages 222-3)

where is the expected return isthe volatility.

The discrete time equivalent is

dS Sdt Sdz

tStSS ddd


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Monte Carlo Simulation

We can sample random paths for thestock price by sampling values for

Suppose = 0.14, = 0.20, and dt= 0.01,then

d SSS 02.00014.0


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Monte Carlo Simulation One Path(See Table 11.1)

PeriodStock Price atStart of Period

Random

Sample forChange in Stock

Price, S

0 20.000 0.52 0.23

1 20.236 1.44 0.611

2 20.847 -0.86 -0.32

3 20.518 1.46 0.62

4 21.146 -0.69 -0.26


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Itos Lemma (See pages 226-227)

If we know the stochastic processfollowed byx, Itos lemma tells us thestochastic process followed by somefunction G (x, t)

Since a derivative security is a function ofthe price of the underlying and time, Itos

lemma plays an important part in theanalysis of derivative securities


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The Stock Price Assumption

Consider a stock whose price is S

In a short period of time of length dt, thereturn on the stock is normally distributed:

where is expected return and is volatility

ttS

Sdd

d,


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The Lognormal Property(Equations 12.2 and 12.3, page 235)

It follows from this assumption that

Since the logarithm ofST is normal, ST islognormally distributed

ln ln ,

ln ln ,

S S T T

S S T T

T

T

0

2

0

2

2

2

or


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The Lognormal Distribution

E S S e

S S e e

T

T

T

T T

( )

( ) ( )

0

0

2 2 2 1

var


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Continuously Compounded

Return, h Equations 12.6 and 12.7), page 236)S S e

T

S

S

T

T

T

T

0

0

1

2

or

=

or

2

h

h

h

ln

,


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The Expected Return

The expected value of the stock price isS0e

T

The expected return on the stock is

2/2

)/(ln

2/)/ln(

0

20

SSESSE

T

T


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The Volatility

The volatility of an asset is the standarddeviation of the continuouslycompounded rate of return in 1 year

As an approximation it is the standarddeviation of the percentage change in theasset price in 1 year


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Estimating Volatility from

Historical Data (page 239-41)

1. Take observations S0, S1, . . . , Sn atintervals of years

2. Calculate the continuously compounded

return in each interval as:

3. Calculate the standard deviation,s , ofthe uis

4. The historical volatility estimate is:

uS

Si

i

i

ln1

s


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The Concepts Underlying

Black-Scholes The option price and the stock price depend on the

same underlying source of uncertainty

We can form a portfolio consisting of the stock andthe option which eliminates this source of uncertainty

The portfolio is instantaneously riskless and mustinstantaneously earn the risk-free rate

This leads to the Black-Scholes differential equation


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Assumptions

Stock Prices follow GBM Short selling of securities with full use of

proceeds is permitted No transaction costs, taxes, All securities are perfectly divisible There are no dividends No Arbitrage

Security trading is continuous Risk free rate is constant and the same for allmaturities.


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The Derivation of the Black-Scholes Differential Equation

shares:

-

derivative:1

ofconsistingportfolioaupseteW

22

2

2

S

zSStSStSS

zStSS

d

d

d

ddd


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steptimetheduringchangenordiddeltathatNotice

bygivenisin timevalueitsinchangeThe

bygivenisportfoliotheofvalueThe

SS

t

S

S

d

dd

d

The Derivation of the Black-Scholes

Differential Equation continued


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The Derivation of the Black-Scholes

Differential Equation continued

:equationaldifferentiScholes-Blackthegetto

equationstheseinandforsubstituteWe

Hencerate.

free-riskthebemustportfoliotheonreturnThe

rS

SS

rSt

Str

dddd

2

2

22


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Derivation

istherforeportfolioThe

haveweItoFrom

22

2

2

22

2

2

S

S

tS

S

t

t

S

S

tSS

tt

SS

d

d

d

d

d

d

d

d

d


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The deterministic terms andrandom terms

All dt terms are deterministic and all dsterms are random

To almost eliminate the randomness youneed choose

That what is delta hedging obviously it hasto be dynamic because asset price is

dynamic

S


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Principle of No arbitrage

0

arbitragecreatetogoingisrateotherany

soratefreeriskonlyearntoneedsThis

betoremainsportfolioThe

222

2

22

2

2

rfS

rSSSt

tr

tSS

tt

dd

d

d

d


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The Differential Equation

Any security whose price is dependent on thestock price satisfies the differential equation

The particular security being valued is determined

by the boundary conditions of the differentialequation

In a forward contract the boundary condition is = SK when t =T

The solution to the equation is

= SKer(Tt)


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Risk-Neutral Valuation

The variable does not appear in the Black-Scholes equation

The equation is independent of all variables

affected by risk preference The solution to the differential equation is

therefore the same in a risk-free world as itis in the real world

This leads to the principle of risk-neutralvaluation


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Applying Risk-Neutral

Valuation1. Assume that the expected

return from the stock price isthe risk-free rate

2. Calculate the expected payofffrom the option

3. Discount at the risk-free rate


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The Black-Scholes Formulas(See pages 246-248)

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNSc

rT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where


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Implied Volatility

The implied volatility of an option is thevolatility for which the Black-Scholes priceequals the market price

The is a one-to-one correspondencebetween prices and implied volatilities

Traders and brokers often quote implied

volatilities rather than dollar prices


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Estimating the Volatility(continued)

Implied Volatility This is the volatility implied when the

market price of the option is set to themodel price.

Substitute estimates of the volatility into theB-S 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


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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 impliedvolatility.

The relationship between the implied volatility and theexercise price is called the volatility smile or volatilityskew. These volatilities are actually supposed to be the

same. This effect is puzzling and has not beenadequately explained.

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


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Causes of Volatility

Volatility is usually much greater when themarket is open (i.e. the asset is trading)than when it is closed

For this reason time is usually measuredin trading days not calendar days when

options are valued


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Dividends

European options on dividend-payingstocks are valued by substituting the stockprice less the present value of dividends

into Black-Scholes Only dividends with ex-dividend dates

during life of option should be included

The dividend should be the expectedreduction in the stock price expected


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Equity Options

Implied

Volatility

Strike

Price

The Volatility smirk forEquity Options Possible biases in

BS Model.

If the BS model isright we should havehad same IVS for all

moneyness andmaturity.

Low X:ITM calls or

OTM puts

High X:ITM puts or

OTM calls

The flat line is the BS

IV as it should be

The dark line is the IV

estimated from the market

data


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ITM calls or

OTM puts

ITM puts or

OTM calls

The left tail is fatter andthe right tail is thinnerthan the lognormaldistribution

Implied Risk neutral

distribution (IRND) ismore negatively skewedand leptokurtic.


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Biases in BS prices

Implied

Volatility

Strike

Price

The Volatility smirk for Equity Options

Low X:ITM calls or OTM puts.BS model under prices

High X:ITM puts or OTM calls.

BS model over prices

The flat line is the BS

IV as it should be

The dark line is the IV

estimated from the market

data


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Low X:ITM calls orOTM puts

High X:ITM puts orOTM calls

How about ForeignCurrency Options ?

Pricing Biases in B-S model


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Pricing Biases in B-S modelfor equity options

Moneyness:

It under prices ITM ( OTM)calls (puts) and over pricesdeep OTM (ITM) calls (puts) .

Maturity:

Volatility smile becomes lesspronounced as option maturityincreases

It under (over) prices long

(short) maturity options Market prices are higher

(lower) for long (short)maturity options

Volatility It under prices low volatility

stocks

It over prices high volatilitystocks


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Further evidence of pricing biases( Fleming, Dumas and Whaley JOF, 1997)

St h ti V l tilit


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Stochastic VolatilityHeston (RFS, 1993)

Consider the following return and volatility riskneutral processes:

The volatility process is the square root process of Cox,Ingersoll &Ross (1985)

Price of volatility risk =

dzVdtVdV

dwSdtSdS

dV,dC/CCovtVS aversionriskrelative,,


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Skewness and Kurtosis

Correlation controls the skewess of the riskneutral distributions - important for pricing ITMVs. OTM options.

Volatility of Volatility controls the kurtosis of therisk neutral distributions- important for pricing

ATM Vs. deep OTM options.

Ch t i ti F ti


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Characteristic FunctionApproach

The new PDE involves bothdelta, vega ( first order andsecond order effects and crosseffects) and volatility riskpremium term .

Extra boundary conditions.

The risk neutral probabilityfunction which is solution toPDE does not have a closedform.

However characteristicfunctions exist that satisfy thesame PDE.

One should invert the

characteristic function to obtainthe desired probabilities(numerical integrals).

Ch t i ti F ti


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Characteristic FunctionApproach

Eq 10,17, 18 give the soln for European calls

C l ti i H t SV d l


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Correlation in Hestons SV model

Nandi (JBF, 1998)

In order to estimate SV model besides the BSinputs we need

processSVofmeantermlong:

premiumriskvolatility:processSVforreversionmean:

yvolatilitofvolatility:

parameterncorrelatio:

,,,,:parametersmodel

t

v


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IV Function Explanation

Shape of IV function:

Hedging Pressure

Net demand pressure

y are ose op ons sm ng


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y are ose op ons sm ngEderington and Guan (JOD,

2000) Two sources of the smile: With respect to the underlying risk neutral

dynamics of the underlying asset

Violation of other assumptions such ascontinuous trading and frictionless markets.


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If BS Model is biased, trades based onIV smile ( long options at the bottom ofthe smile and short options at the top of

the smile ) would yield no profits, evenbefore transaction costs.

If BS Model is unbiased such tradesbased on IV smile should yield profits.

IV smile


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Trading on the smile

Follow the following strategy:

long options at the bottom of the smile andshort options at the top of the smile

Value of long portfolio/ Value of shortportfolio = 1

Portfolio is self financing

Delta and gamma hedge the portfolio


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Data

S & P 500 index futures and futures options

IV ( Blacks model) based on near the money options(OTM options Exclude).

No non-synchronous problem between two markets Futures options:

Better liquidity and easy to trade than the spot.

Better arbitrage, hedging and market linkages

Lower transaction costs

Futures option pricing is independent of future dividends


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Results

Trades based on IV smile ( long options at thebottom of the smile and short options at the topof the smile ) yield substantial profits, beforetransaction costs.

So the BS Model is not completely biased. i.e. BS formula is not completely wrong

i.e. Smile is not wholly caused by errors in the BSformula

However that does not mean that the market isinefficient.


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Results

However the profits are spurious. Theprofits drop once transaction costs areconsidered.

The residual profits are also related toimperfect delta-gamma hedging.


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Results

If the Smile is not wholly caused by errorsin the BS formula, what causes it?:

Hedging in OTM puts (and hence ITM calls

through put-call parity)

The authors find support for the hedginghypothesis:

IVs on low X options are relatively unrelatedto actual ex-post volatility than compared tohigh X options

Net Buying Pressure and shape if IV


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y g pfunction

Bollen and Whaley (JOF, 2004) Provide an explanation based on demand

and supply for different option series ( i.ec(X,T) and p(X,T) ).

Under BS model, the supply curve foreach option series is a horizontal line

Option price is unaffected by demand

shifts for options


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Market Maker

A market maker in option markets will not standready to sell an unlimited number of contracts ina particular option series.

As her position grows large and imbalanced, herhedging costs and/or volatility risk exposure alsoincreases and he is forced to charge a higherprice.

With an upward sloping supply curve, differentlyshaped IVF can be expected

Market Maker and upward sloping


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Market Maker and upward slopingSupply curve

If investor net (or excess) public demand for a particularoption series is to buy (sell) the option prices will be high(low) and hence IV will be accordingly high (low).

In the case of S & P 500 index OTM puts, excessdemand can cause higher option prices.

The market makers need higher compensation towards higher hedging costs and

exposure to volatility risk.


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Net buying Pressure:

Defined as difference between: buyer motivated contracts and

(trades executed at prices above prevailing bid-askprices)

seller motivated contacts (trades executed at prices below prevailing bid-ask

prices)

traded each day.

The difference is multiplied by absolute value ofoption delta to express demand in stock / indexequivalent units


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Index Vs Stock options

Index markets:

Most of the trading is in puts

Buying pressure on Index put options leads to

the IVF being more negatively sloped.

Stock markets:

Most of the trading is in calls

Buying pressure on Equity calls options leadsto the IVF being less negatively sloped.


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Delta Hedging

Delta hedging OTM Index puts leads tosignificant profits compared to deltahedging equity options.

Once the vega risk is factored in, theexcess profits are no longer significant.

black scholes pricing methodology

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