CH12- WIENER PROCESSES AND IT

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CH12- WIENER PROCESSES AND ITÔ'S LEMMA
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OUTLINE
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the value of the variable changes only at certain
fixed time point
only limited values are possible for the variable
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12.1 THE MARKOV PROPERTY

• A Markov process is a particular type of
  stochastic process .
• The past history of the variable and the way that
  the present has emerged from the past are
  irrelevant.
• A Markov process for stock prices is consistent
  with weak-form market efficiency.

where only the present value of a variable is
relevant for predicting the future.
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12.2CONTINUOUS-TIME STOCHASTIC PROCESSES

• Suppose 10(now), change in its value during 1
  year is f(0,1).
• What is the probability distribution of the
  stock price at the end of 2 years? f(0,2)
• 6 months? f(0,0.5)
• 3 months? f(0,0.25)
• Dt years? f(0, Dt)

• In Markov processes changes in successive periods
  of time are independent
• This means that variances are additive.
• Standard deviations are not additive.

N(µ0, s1)
N(µ0, s2)
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A WIENER PROCESS (1/3)

• It is a particular type of Markov stochastic
  process with a mean change of zero and a variance
  rate of 1.0 per year.
• A variable z follows a Wiener Process if it has
  the following two properties
• (Property 1.)
• The change ?z during a small period of time
  ?t is

Wiener Process
?znormal distribution
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A WIENER PROCESS (2/3)

• (Property 2.)
• The values of ?z for any two different short
  intervals of time, ?t, are independent.
• Mean of Dz is 0
• Variance of Dz is Dt
• Standard deviation of Dz is

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A WIENER PROCESS (3/3)
Mean of z (T ) z (0) is 0 Variance of z
(T ) z (0) is T Standard deviation of z (T
) z (0) is

• Consider the change in the value of z during a
  relatively long period of time, T. This can be
  denoted by z(T)z(0).
• It can be regarded as the sum of the changes in z
  in N small time intervals of length Dt, where

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EXAMPLE12.1(WIENER PROCESS)

• ExInitially 25 and time is measured in years.
  Mean25, Standard deviation 1. At the end of 5
  years, what is mean and Standard deviation?
• Our uncertainty about the value of the variable
  at a certain time in the future, as measured by
  its standard deviation, increases as the square
  root of how far we are looking ahead.

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GENERALIZED WIENER PROCESSES(1/3)

• A Wiener process, dz, that has been developed so
  far has a drift rate (i.e. average change per
  unit time) of 0 and a variance rate of 1
• DR0 means that the expected value of z at any
  future time is equal to its current value.
• VR1 means that the variance of the change in z
  in a time interval of length T equals T.

Drift rate ?DR , variance rate ?VR
DR0 , VR1
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GENERALIZED WIENER PROCESSES (2/3)

• A generalized Wiener process for a variable x
  can be defined in terms of dz as
• dx a dt b dz

DR
VR
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GENERALIZED WIENER PROCESSES(3/3)

• In a small time interval ?t, the change ?x in the
  value of x is given by equations

Mean of ?x is Variance of ?x is Standard
deviation of ?x is
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EXAMPLE 12.2

• Follow a generalized Wiener process
• DR20 (year) VR900(year)
• Initially , the cash position is 50.
• At the end of 1 year the cash position will have
  a normal distribution with a mean of ?? and
  standard deviation of ??
• ANS??70, ??30

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ITÔ PROCESS

• Itô Process is a generalized Wiener process in
  which the parameters a and b are functions of the
  value of the underlying variable x and time t.
• dxa(x,t) dtb(x,t) dz
• The discrete time equivalent
• is only true in the limit as Dt tends to zero

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12.3 THE PROCESS FOR STOCKS

• The assumption of constant expected drift rate is
  inappropriate and needs to be replaced by
  assumption that the expected reture is constant.
• This means that in a short interval of time,?t,
  the expected increase in S is µS?t.
• A stock price does exhibit volatility.

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An Ito Process for Stock Prices

• where m is the expected return and s is the
  volatility.
• The discrete time equivalent is

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EXAMPLE 12.3

• Suppose m 0.15, s 0.30, then
• Consider a time interval of 1 week(0.0192)year,
  so that Dt 0.0192
• ?S0.00288 S 0.0416 S

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MONTE CARLO SIMULATION

• MCS of a stochastic process is a procedure for
  sampling random outcome for the process.
• Suppose m 0.14, s 0.2, and Dt 0.01 then
• The first time period(S20 0.52 )
• DS0.001420 0.02200.520.236
• The second time period
• DS'0.001420.236 0.0220.2361.440.611

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MONTE CARLO SIMULATION ONE PATH

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12.4 THE PARAMETERS

• µ?s
• We do not have to concern ourselves with the
  determinants of µin any detail because the value
  of a derivative dependent on a stock is, in
  general, independent of µ.
• We will discuss procedures for estimating s in
  Chaper 13

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12.5 ITÔ'S LEMMA

• If we know the stochastic process followed by x,
  Itô's lemma tells us the stochastic process
  followed by some function G (x, t )
• dxa(x,t)dtb(x,t)dz
• Itô's lemma shows that a functions G of x and t
  follows the process

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DERIVATION OF ITÔ'S LEMMA(1/2)

• IfDx is a small change in x and
• D G is the resulting small change in G

Taylor series
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DERIVATION OF ITÔ'S LEMMA(2/2)

• A Taylor's series expansion of G (x, t) gives

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IGNORING TERMS OF HIGHER ORDER THAN DT
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SUBSTITUTING FOR ?X
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THE E2?T TERM
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APPLICATION OF ITO'S LEMMA TO A STOCK PRICE
PROCESS
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APPLICATION TO FORWARD CONTRACTS
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THE LOGNORMAL PROPERTY

• We define

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THE LOGNORMAL PROPERTY

• The standard deviation of the logarithm of the
  stock price is

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