the stock price assumptionsuraj.lums.edu.pk/~adnan.khan/casmfin2018/day4-volatility-azmat.pdf ·...

VOLATILITY MODELING

The Stock Price Assumption

Consider a stock whose price is S

In a short period of time of length ∆119905 the return on the stock is normally distributed

∆119878

119878sim 119873(120583∆119905 1205902∆119905)

where 120583 is expected return and 120590 is volatility

The Lognormal Property

It follows from this assumption that

Since the logarithm of 119878119879 is normal 119878119879 is lognormally distributed

2

or

2

22

0

22

0

TTSS

TTSS

T

T

lnln

lnln

The Black-Scholes-Merton Formulas for Options

119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892

119875 = 119870119890minus119903119879119873 minus1198892 minus 1198780119873 minus1198891

where

1198891 =ln

1198780119870

+ 119903+1205902

2119879

120590radic(119879)

1198892 =ln

1198780119870

+ 119903minus1205902

2119879

120590radic(119879)= 1198891 minus 120590 119879

The N(x) Function

N(x) is the CDF of a standard normal random variable that is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x

5

Inputs of BS

Inputs

Current price 1198780

Strike price 119870

Risk-free rate 119903

Time to expiration 119879

Volatility 120590

Output

Price of the option

The Volatility

Volatility is a measure of the dispersion of returns ie a simple measure of asset return volatility is its variance over time

More formally volatility is standard deviation of the rate of return in 1 year

The standard deviation of the return in a short time period time Dt is approximately tD

Example Google stock prices last one year

In the BS model what value should be chosen for 120590 when pricing an option

Variance does not capture volatility clustering (periods of turbulence periods of calm impacts of economics events etc)

ndash It is a measure of unconditional variance

ndash It is a single number of a given sample

ndash It does not incorporate the past information

Volatility

Implied Volatility

The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price

There is a one-to-one correspondence between prices and implied volatilities

Traders and brokers often quote implied volatilities rather than dollar prices

11

Implied Volatility

Consider a call option

119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873(1198892)119891 120590 = 119862 minus (1198780119873 1198891 minus 119870119890minus119903119879119873(1198892))

Find the zeros of 119891(120590)

Example

Example

Implied Volatility (IV)-Task

Why is IV important

Stocks vs Options You can theoretically hold a stock forever-no timeline Whereas when dealing options you have to deal with time component T and the implied volatility (IV how far do traders expect the stock to move)

ATM How much would you willing to pay for an ATM option

OTM How much would you be willing to pay for an OTM option

Market participantsrsquo belief determines how far would the stock move


When IV is high people are willing to pay more money for an option contract

When IV is low people are willing to not pay a lot of money for an option contract

Suppose a stock with an IV of 20 is trading at $100 The expected range is $80-$120

68 of the time the stock will trade between $80 and $120

IV How do we know if IV is high or low

Consider two stocks Google and FedEx with IVs equal to 15

How do we determine if 15 is a high or a low volatility for GoogleFedEx

Suppose from the historical data for Google the high was 40 and the low was 10 Whereas for FedEx the high was 25 and the low was 5 Which stock would you trade

Lookup the historical highs and lows for each stock and determine the relative ranking-percentile

Auto-correlated Heteroscedasticity

Heteroscedasticity or unequal variabilityvariance

Auto-correlated heteroscedasticity

hellip is called the ARCH effect

Estimating Volatility

Autoregressive conditional heteroscedasticity (ARCH)

Exponentially weight moving average (EWMA)

Generalized autoregressive conditional heteroscedasticity

(GARCH)


Define n as the volatility per day between day n-1 and day n as estimated at end of day n-1

Define Si as the value of market variable at end of day i

Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )


Define 119906119894 as the percentage change in the market variable between the end of day 119894 minus 1 and end of day 119894

Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1

Assume that the mean value of ui is zero

Replace mminus1 by m

This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme

Instead of assigning equal weights to the observations we can assign more weight to recent data

n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model

In an ARCH(m) model we also assign some weight to the long-run variance rate VL

m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model

In an exponentially weighted moving average model the weights assigned to the u2 decline exponentially as we move back through time

This leads to

25

2

1

2

1

2 )1( nnn u

EWMA Weights Decline Exponentially

1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12

= 120582 [120582 120590119899minus22 + 1 minus 120582 119906119899minus2

2 ] + 1 minus 120582 119906119899minus12

= 1 minus 120582 119906119899minus12 + 1 minus 120582 119906119899minus2

2 + 1205822 120590119899minus22

Continue substituting for 120590119899minus22 120590119899minus3

2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1

2 + 120582 1 minus 120582 119906119899minus22 + 1205822 1 minus 120582 119906119899minus3

2 +⋯+ 120582119898minus1 1 minus 120582 119906119899minus1198982 +

120582119898120590119899minus1198982

Weights decline at the rate of 120582

Attractions of EWMA

Relatively little data needs to be stored

We need only remember the current estimate of the variance rate and the most recent observation on the market variable

Tracks volatility changes

094 is a popular choice for

27

GARCH (11)

In GARCH (11) we assign some weight to the long-run average variance rate

1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)

Let 120596 = 120574119881119871 then GARCH(11) becomes

1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596

1minus120572minus120573

GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u

Maximum Likelihood Methods

How do we estimate the parameters in the models discussed above

Maximum likelihood methods deal with choosing the parameters that maximize the likelihood (the chance) of data occurring

Example

Suppose that we sample 10 stocks at random on a certain day and find that the price of one of them declined on that day and the prices of the other nine either remained the same or increased What is the best estimate of the probability of a randomly chosen stockrsquos price declining on the day

Itrsquos 01

Maximum likelihood method Let 119901 be the probability of price decline The probability that one stock declines and others do not is 119901 1 minus 119901 9

We maximize 119891(119901) to obtain a maximum likelihood estimate

Itrsquos p = 01

32

Example Estimating a constant variance

Considers the problem of estimating the variance of a variable 119883 from 119898observations on 119883 when the underlying distribution is normal with zero mean

Assume that the observations are 1199061 1199062 ⋯ 119906119898

Denote the variance by 120584

The likelihood of 119906119894 being observed is defined as the probability density function for

119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898

The likelihood of m observations occurring in the order in which they are observed

is 119891 1199061 times 119891 1199062 times⋯times 119891 119906119898

Example

Estimate of 120584 ∶

m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result

maximizing to equivalent is this logarithms Taking

Maximize

Application to GARCH

We choose parameters that maximize

35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References

Options Futures and Other Derivatives (10th Edition) 10th Edition

by John C Hull

Handbook of Volatility Models and Their Application

by Luc Bauwens Christian M Hafner Sebastien Laurent

Option Volatility amp Pricing Advanced Trading Strategies and Techniques Updated Edition

by Sheldon Natenberg

The Stock Price Assumption

Consider a stock whose price is S

In a short period of time of length ∆119905 the return on the stock is normally distributed

∆119878

119878sim 119873(120583∆119905 1205902∆119905)

where 120583 is expected return and 120590 is volatility




2

or

2

22

0

22

0

TTSS

TTSS

T

T

lnln

lnln


119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892


where

1198891 =ln

1198780119870

+ 119903+1205902

2119879

120590radic(119879)

1198892 =ln

1198780119870

+ 119903minus1205902

2119879

120590radic(119879)= 1198891 minus 120590 119879

The N(x) Function


5

Inputs of BS

Inputs


Strike price 119870



Volatility 120590

Output

Price of the option

The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull








2

or

2

22

0

22

0

TTSS

TTSS

T

T

lnln

lnln


119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892


where

1198891 =ln

1198780119870

+ 119903+1205902

2119879

120590radic(119879)

1198892 =ln

1198780119870

+ 119903minus1205902

2119879

120590radic(119879)= 1198891 minus 120590 119879

The N(x) Function


5

Inputs of BS

Inputs


Strike price 119870



Volatility 120590

Output

Price of the option

The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull






119862 = 1198780119873 1198891 minus 119870119890minus119903119879119873 1198892


where

1198891 =ln

1198780119870

+ 119903+1205902

2119879

120590radic(119879)

1198892 =ln

1198780119870

+ 119903minus1205902

2119879

120590radic(119879)= 1198891 minus 120590 119879

The N(x) Function


5

Inputs of BS

Inputs


Strike price 119870



Volatility 120590

Output

Price of the option

The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





The N(x) Function


5

Inputs of BS

Inputs


Strike price 119870



Volatility 120590

Output

Price of the option

The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Inputs of BS

Inputs


Strike price 119870



Volatility 120590

Output

Price of the option

The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





The Volatility










Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull











Volatility

Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Implied Volatility




11

Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Implied Volatility




Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Example

Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Example


Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull






Why is IV important























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull























(GARCH)




Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull








Define 119906119894 = ln(119878119894

119878119894minus1)

20

n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )



Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull







Set 119906119894 =119878119894minus119878119894minus1

119878119894minus1



This gives

21

n n ii

m

mu2 2

1

1

Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Example

Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Weighting Scheme


n i n ii

m

ii

m

u2 2

1

1

1

where

ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





ARCH(m) Model


m

i

i

m

i iniLn uV

1

1

22

1

where

EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





EWMA Model


This leads to

25

2

1

2

1

2 )1( nnn u


1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull






1205901198992 = 120582 120590119899minus1

2 + 1 minus 120582 119906119899minus12


2 ] + 1 minus 120582 119906119899minus12


2 + 1205822 120590119899minus22


2 and so on we have

1205901198992 = 1 minus 120582 119906119899minus1



120582119898120590119899minus1198982


Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Attractions of EWMA





27

GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





GARCH (11)


1205901198992 = 120574119881119871 + 120572119906119899minus1

2 + 120573120590119899minus12

such that

120574 + 120572 + 120573 = 1

GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





GARCH (11)


1205901198992 = 120596 + 120572119906119899minus1

2 + 120573120590119899minus12

Where

119881119871 =120596


GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





GARCH (pq)

30

2

1 1

22

jn

p

i

q

j

jinin u




Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull








Example


Itrsquos 01



Itrsquos p = 01

32






119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull










119883 when 119883 = 119906119894 That is 119891 119906119894 =1

2120587120584119890minus

1199061198942

2120584 for 119894 = 12⋯ 119898



Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Example


m

i

i

m

i

i

m

i

i

um

v

v

uv

v

u

v

1

2

1

2

1

2

1

)ln(

2exp

2

1

Result


Maximize



35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull







35

m

i i

ii

i

im

i i

v

uv

v

u

v

1

2

2

1

)ln(

2exp

2

1

or

Example

References


by John C Hull





Example

References


by John C Hull





References


by John C Hull





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