Elementary s to Chas Tic Calculus

7/30/2019 Elementary s to Chas Tic Calculus

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Elementary Stochastic Calculus1 In this lecture. . .

G all the stochastic calculus you need to know, and no moreG

the meaning of Markov and martingale

GBrownian motion

G

stochastic integrationG

stochastic differential equations

GIts lemma in one and more dimensions


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3 A motivating example

Toss a coin. Every time you throw a head you receive $1, every

time you throw a tail you pay out $1. In the experiment below the

sequence was THHTHT, and you finished even.

-2

-1

0

1

2

0 1 2 3 4 5 6 7

1.The outcome of a coin tossing experiment.


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G 5

L

denotes the random amount, either $1 or -$1, you make on the

L th toss:

( > 5

L

@ ( > 5

L

@ and ( > 5L

5

M

@

In this example it doesnt matter whether or not these expectations

are conditional on the past. In other words, if you threw five headsin a row it does not affect the outcome of the sixth toss.

GIntroduce

6

L

to mean the total amount of money you have won up

to and including theL

th toss so that

6

L

L

;

M

5

M

Later on it will be useful if we have 6

, i.e., you start with nomoney.


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If we now calculate expectations of 6L

it does matter what infor-

mation we have. If we calculate expectations of future events beforethe experiment has even begun then

( > 6

L

@ and ( > 6 L

@ ( > 5

5

5

? ? ? @ L

On the other hand, suppose there have been five tosses already, canwe use this information and what can we say about expectations for

the sixth toss?

GThis is the conditional expectation.

The expectation of 6

conditional upon the previous five tosses

gives

( > 6

M 5

5

( 6


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4 The Markov property

GThe expected value of the random variable

6

L

conditional upon all

of the past events only depends on the previous value $L >

. This

is the Markov property.

We say that the random walk has no memory beyond where it is

now. Note that it doesnt have to be the case that the expected value

of the random variable$

L

is the same as the previous value.

This can be generalized to say that given information about$

M

for

some values of T M L then the only information that is of use to

us in estimating 6L

is the value of 6M

for the largest M for which we

have information.


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5 The martingale property

The coin tossing experiment possesses another property that can be

important in finance. You know how much money you have won af-

ter the fifth toss. Your expected winnings after the sixth toss, and in-

deed after any number of tosses if we keep playing, is just the amountyou already hold.

G The conditional expectation of your winnings at any time in the

future is just the amount you already hold:

( > $

L

M $

M

M L @ $

M

This is called the martingale property.


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6 Quadratic variation

The quadratic variation of the random walk is defined by

L

;

M

$

M

> $

M >

Because you either win or lose an amount $1 after each toss,M $

M

b

$

M >

M . Thus the quadratic variation is always L :

L

;

M

$

M

> $

M >

L

We are going to use the coin-tossing experiment for one more demon-

stration. And that will lead us to a continuous-time random walk.


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7 Brownian motion

Change the rules of the coin-tossing experiment slightly. First of all

restrict the time allowed for the six tosses to a periodW

, so each toss

will take a timeW . Second, the size of the bet will not be $1 but

5

9 .This new experiment clearly still possesses both the Markov and

martingale properties, and its quadratic variation measured over the

whole experiment is

;

M

$

M

> $

M >

@

U

7

9

9


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Change the rules again, to speed up the game.

GWe will have

Qtosses in the allowed time

W, with an amount

5

9 3

riding on each throw.

Again, the Markov and martingale properties are retained and the

quadratic variation is stillQ

;

M

$

M

> $

M >

3 @

U

7

9

3

9

Now make Q larger and larger. This speeds up the game, decreasingthe time between tosses, with a smaller amount for each bet. But

the new scalings have been chosen very carefully, the timestep is

decreasing likeQ

> but the bet size only decreases byQ

> .


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-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

2.A series of coin tossing experiments.


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As we go to the limit3 , the resulting random walk stays

finite. It has an expectation, conditional on a starting value of zero,of

> $ 9 @

and a variance

> $ 9

@ 9

We use6 W

to denote the amount you have won or the value of the

random variable after a time W .

G The limiting process for this random walk as the timesteps go tozero is called Brownian motion, and we will denote it by ; W .


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The important properties of Brownian motion are as follows.

G Finiteness : Any other scaling of the bet size or increments with

timestep would have resulted in either a random walk going to

infinity in a finite time, or a limit in which there was no motion

at all. It is important that the increment scales with the square root

of the timestep.

G Continuity : The paths are continuous, there are no discontinuities.

Brownian motion is the continuous-time limit of our discrete time

random walk.G Markov : The conditional distribution of ; W given information

up until W depends only on ; .

G Martingale : Given information up until W the conditional

expectation of; W is ; .


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G Quadratic variation : If we divide up the time to W in a partition

with Q partition points W L L W Q thenQ

;

M

; W

M

> ; W

M >

9 (Technically almost surely.)

G Normality : Over finite time increments W L > to 9 L , ; W L > ; W L >

is Normally distributed with mean zero and variance9

L

> 9

L >

.


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Having built up the idea and properties of Brownian motion from

a series of experiments, we can discard the experiments, to leave theBrownian motion that is defined by its properties. These properties

will be very important for our financial models.


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8 Stochastic integration

Define a stochastic integral by

: W

=

9

1 G ; O L P

3

3

;

M

1 9

M >

; W

M

> ; W

M >

with

9

M

M W

Q

G Note that the function I W is evaluated in the summation at theleft-hand point W

M >

.

It will be crucially important that each function evaluation does not

know about the random increment that multiplies it, i.e. the integra-

tion is non-anticipatory.In financial terms, we will see that we take some action such as

choosing a portfolio and only then does the stock price move. This

choice of integration is natural in finance, ensuring that we use no

information about the future in our current actions.


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9 Stochastic differential equations

Stochastic integrals are important for any theory of stochastic cal-

culus since they can be meaningfully defined. However, it is very

common to use a shorthand notation for expressions such as

: W

=

9

1 G ; (1)

That shorthand comes from differentiating (1) and is

G : I W G ;

GThink of G ; as being an increment in ; , i.e. a Normal random

variable with mean zero and standard deviation G W .


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Pursuing this idea further, imagine what might be meant by

G : J W G W I W G ; (2)

This is simply shorthand for

: W

=

9

J G

=

9

1 G ;

GEquations like (2) are called stochastic differential equations.

Their precise meaning comes, however, from the technically more

accurate equivalent stochastic integral.


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10 The mean square limit

This is useful in the precise definition of stochastic integration.

Examine the quantity

(

3

;

M

; W

M

> ; W

M >

> 9

where

9

M

M W

Q


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This can be expanded as

(

3

;

M

; W

M

> ; W

M >

3

;

L

;

M L

; W

L

> ; W

L >

; W

M

> ; W

M >

> 9

3

;

M

; W

M

> ; W

M >

9

Since ; WM

> ; W

M >

is Normally distributed with mean zero and

variance 9 3 we have

A

; W

M

> ; W

M >

B

9

Q

and

(

A

; W

M

> ; W

M >

B

9

3


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Thus the required expectation becomes

Q

W

Q

Q Q >

W

Q

> 9 3

9

3

9

M

3

N

AsQ this tends to zero. We therefore say that

3

;

M

; W

M

> ; W

M >

9

in the mean square limit. This is often written, for obvious reasons,

as=

9

G ;

W

Whenever we talk about equality in the following proof we

mean equality in the mean square sense.


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11 Functions of stochastic variables and

Its lemma

Now well see the idea of a function of a stochastic variable. Below

is shown a realization of a Brownian motion ; W and the function

) ; ;

.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5

Brownian motion

Square of Brownian motion

3.A realization of a random walk and its square.


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If) ; is it true that G ) ; G ; ? No.

GThe ordinary rules of calculus do not generally hold in a stochastic

environment.

Then what are the rules of calculus?

We now derive the most important rule of stochastic calculus,Its lemma.


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In this derivation we will need various timescales. The first timescale

is very, very small. Denote it bys W

Q

K

This timescale is so small that the function ) ; W K can be ap-

proximated by a Taylor series:

) ; W K > ) ; W ; W K > ; W

G )

G ;

; W

; W K > ; W

G

)

G ;

; W ? ? ?


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From this it follows that

) ; W K > ) ; W ) ; W K > ) ; W K ? ? ?

) ; W Q K > ) ; W Q > K

3

;

M

; W M K > ; W M > K

G )

G ;

; W M > K

G

)

G ;

; W

3

;

M

; W M K > ; W M > K

? ? ?

This has used the approximation

G

)

G ;

; W M > K

G

)

G ;

; 9

This is consistent with the order of accuracy we require.

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The first line in this becomes simply

) ; W Q K > ) ; W ) ; W s W > ) ; W

The second is just the definition of=

9 I 9

9

G )

G ;

G ;

and the last is

G

)

G ;

; W s W

in the mean square sense. Thus we have

) ; W s W > ) ; W

=

9 I 9

9

G )

G ;

; G ;

=

9 I 9

9

G

)

G ;

; G

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We can now extend this result over longer timescales, from zero up

to W , over which ) does vary substantially to get

) ; W ) ;

=

9

G )

G ;

; G ;

=

9

/

/ ;

; G

This is the integral version ofIts lemma, which is usually written

as

G )

G )

G ;

G ;

G

)

G ;

G W

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We can now answer the question, If ) ; what stochastic dif-

ferential equation does ) satisfy? In this example

G )

G ;

; andG

)

G ;

Therefore Its lemma tells us that

G ) ; G ; G W

This is not what we would get if; were a deterministic variable. In

integrated form

;

) ; )

=

9

; G ;

=

9

/ Q

=

9

; G ; W

Therefore=

9

; G ;

;

>

9

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12 It and Taylor

If we were to do a naive Taylor series expansion of ) , completely

disregarding the nature of ; , and treating G ; as a small increment

in;

, we would get

) ; G ; ) ;

G )

G ;

G ;

G

)

G ;

G ;

ignoring higher-order terms. We could argue that ) ; G ; >

) ; was just the change in ) and so

G )

G )

G ;

G ;

G

)

G ;

G ;

This is very similar to the proper It result with the only difference

being that there is a G ; instead of a G W . However, since in a sense#

9

G ;

W we could perhaps write G ;

G W

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This lacks any rigor but it does give the correct result.

GYou can, with little risk of error, use Taylor series with the rule of

thumb

G ;

G W

and in practice you will get the right result.

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Suppose our stochastic differential equation is

G 6 D 6 G W E 6 G ; (3)

say, for some functions D 6 and E 6 . Here G ; is the usual Brown-

ian increment. Now if we have a function of6 , 9 6 , what stochas-

tic differential equation does it satisfy? The answer is

G 9

G 9

G 6

G 6

E

G

9

G 6

G W

We could derive this properly or just cheat by using Taylor series

with G ;

G W . We could, if we wanted, substitute for G 6 from (3)to get an equation for

G 9in terms of the pure Brownian motion

;:

G 9

M

, $

/ '

/ $

- $

/

'

/ $

N

G W E 6 G ;

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13 It in higher dimensions

In financial problems we often have functions of one stochastic vari-

able 6 and a determinisic variable W , time: 9 6 W . If

G 6 D 6 W G W E 6 W G ;

then the increment G 9 is given by

G 9

# 9

# W

G W

# 9

# 6

G 6

E

#

9

# 6

G W

Again, this is shorthand notation for the correct integrated form.

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Occasionally, we have a function of two, or more, random vari-

ables, and time as well: 9 6 6 W . Lets write the behaviour of6 and 6

in the general form

G 6

D

6

6

W G W E

6

6

W G ;

and

G 6

D

6

6

W G W E

6

6

W G ;

Note that we have two Brownian increments G ;

and G ;

. We can

think of these as being Normally distributed with variance G W , but

they are correlated. The correlation between these two random vari-ables we will call . This can also be a function of6

, 6

and W but

must satisfy

> T T

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G The rules of thumb can readily be imagined:

G ;

G W G ;

G W and G ;

G ;

G W

Its lemma becomes

G 9

# 9

# W

G W

# 9

# 6

G 6

# 9

# 6

G 6

E

#

9

# 6

G W E

E

#

9

# 6

# 6

G W

E

#

9

# 6

G W

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14 Some pertinent examples

The first example is like the simple Brownian motion but with a drift:

G 6 { G W G ;

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

4.A realization ofG 6 { G W G ; .

In this realization 6 has gone negative.

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This stochastic differential equation G 6 { G W G ; can be

integrated exactly to give6 W 6 { W ; W > ;

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Our second example is similar to the above but the drift and ran-

domness scale with 6 :G 6 { 6 G W 6 G ;

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1

5.A realization of G 6 { 6 G W 6 G ; .

If 6 starts out positive it can never go negative; the closer that 6

gets to zero the smaller the increments G 6 .

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This property of this random walk is clearly seen if we examine

the function ) 6 O R J 6 using Its lemma. From It we have

G )

G )

G 6

G 6

6

G

)

G 6

G W

6

{ 6 G W 6 G ; >

P

/ 9

?

{ >

P

@

/ 9 G ;

This shows us that O R J 6 can range between minus and plus infinity

but cannot reach these limits in a finite time, therefore 6 cannot reach

zero or infinity in a finite time.

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The integral form of this stochastic differential equation follows

simply from the stochastic differential equation for O R J 6 :

6 W 6 H

{ >

W ; W > ;

The stochastic differential equationG 6 { 6 G W 6 G ; will be

particularly important in the modeling of many asset classes. If wehave some function 9 6 W then from It it follows that

G 9

# 9

# W

G W

# 9

# 6

G 6

6

#

)

# 6

G W

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The third example is

G 6 | > { 6 G W G ;

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

6.A realization ofG 6 | > { 6 G W G ;

.

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The random walk

G 6 | > { 6 G W G ;

is an example of a mean-reverting random walk. If 6 is large, the

negative coefficient in front of G W means that 6 will move down on

average, if 6 is small it rises on average. There is still no incentive

for 6 to stay positive in this random walk. With U instead of 6 this

random walk is the Vasicek model for the short-term interest rate.

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The final example is similar to the third but we are going to adjust

the random term slightly:G 6 | > { 6 G W 6

G ;

Now if6 ever gets close to zero the randomness decreases, perhaps

this will stop 6 from going negative? Lets play around with this

example for a while. And well see It in practice.

Write ) 6 . What stochastic differential equation does )

satisfy? Since

G )

G 6

6

>

and/

/ $

>

$

>

we have

/

M

| >

)

>

{ )

N

/ 9

G ;

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We have just turned the original stochastic differential equation

with a variable coefficient in front of the random term into a sto-chastic differential equation with a constant random term. In so do-

ing we have made the drift term nastier. In particular, the drift is now

singular at ) 6 .

Continuing with this example, instead of choosing ) 6

, canwe find a function ) 6 such that its stochastic differential equation

has a zero drift term? For this we will need

| > { 6

G )

G 6

6

G

)

G 6

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This can be integrated once to give

G )

G 6

$ 6

>

|

H

{ 6

(4)

for any constant $ .

If |

M

we cannot integrate (4) at$ . This makes the origin non-

attainable. In other words, if the parameter|

is sufficiently large

it forces the random walk to stay away from zero.This particular stochastic differential equation for

6will be impor-

tant later on, it is the Cox, Ingersoll & Ross model for the short-term

interest rate.

Elementary s to Chas Tic Calculus

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