Chapter 4 3- Lec 6 Ito processes PDF

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DA 3470 –Stochastic Finance

Section 4.3: Ito Calculus 2 Suppose we had some function f of Brownian motion, say f (Wt )  Wt . What is the

stochastic differential dft ?

t

It can be proven that

  dWs 

2

 t or in differential form dWt   dt 2

0 Ito’s formula Consider a Stochastic differential equation in its general form dX t   t dWt   t dt and consider a transformation of the form Yt  f ( X t ) with a smooth function f . We can calculate the differential equation for Yt by considering the following Taylor expansion:

dYt  f ( Xt   X )  f ( Xt ) 2

 f '( X t ) dX t  12 f ''( X t )(dX t )  ... 2  f '( X t )( tdt   tdWt )  12 f ''( X t )( tdt   tdWt )  ...

 f '( X t )( tdt   tdWt )  12 f ''( X t )( t2 dt2   t2 (dWt )2  2 t tdtdW t ) Since (dWt )2  dt and since we consider terms onlyupto dt we have dYt  df (X t )  f '(X t )(tdt   tdW t )  12 f ''(X t ) t 2dt

Thus the Ito’s formula is given by

dYt  df (X t )   t f '(X t )dWt  (t f '(X t )  1  t2 f ''(X t ))dt 2

Special case Consider the special case, where X t  Wt and Yt  f Wt  . Then we have t  0 and

 t  1 and we see that dYt  df (X t )  f '(Wt )dWt  1 f ''(Wt )dt 2 1

Copyright ©2020, S.D. Perera

DA 3470 –Stochastic Finance Applying the above formula, we can find:

 

d Wt 2 . 

Here f'(Wt )=2Wt and f''(Wt )  2

 

Thus d Wt 2  2Wt dWt  dt Similarly we can find:   

  d  Wt 3  3Wt 2dWt  3Wt dt d  Wt 6  6Wt 5dWt 15Wt 4dt d Wt 2  t  2Wt dWt

In particular we can see that the solution of the stochastic differential equation dYt  2Yt dYt (together with the initial condition Y0  0 ) is given by Yt  Wt2  t . SDEs from processes Ito’s most immediate use is to generate SDE’s from a functional expression for a process. ( Wt   t) Consider the exponential Brownian motion, X t  e . Write the stochastic differential equation for X. Suppose Yt   Wt   t and f ( x)  e x . Then dYt   dWt   dt . X t  f (Yt )  eYt . So Ito’s formula give us dX t   t f '(Yt )dWt  ( t f '(Yt )  21  t2 f ''(Yt ))dt

Since f (Yt )  f '(Yt )  f "(Yt )  Xt , dX t  X t ( dWt  (   1  2 ) dt ) . 2

The variable  is called the log-volatility of the process and  as log-drift.

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DA 3470 –Stochastic Finance Processes from SDE’s Here we convert SDE’s to processes. (Or in other words, we solve them). In general we can’t solve SDE’s. What we can do is guess a solution and then prove that the proposed solution is indeed an actual solution via Ito. Suppose we are asked to solve the SDE dX t   X t dWt . We need an inspired guess. We notice that the stochastic term, Xt dWt is the same as the SDE we generated via ito earlier by choosing    12  2 . Thus we guess that ( Wt  12 2 t)

Xt  e

. By using Ito’s formula we can prove that dX t   X tdWt .

( dXt  X t ( dWt  (   12  2 )dt ) and since    1  2 , we have, dX t   X t dWt .) 2

Product Rule

d ( X t Yt )  X t dYt  Yt dX t  dX t dYt

If

dX t   t dWt   t dt dYt  t dWt  t dt

Then d ( X t Yt )  X t dYt  Yt dX t   t  t dt If X t and Yt are two stochastic processes adapted to two different and independent Brownian motions such as,

dX t   t dWt   t dt dYt  t dW Then d ( X t Yt )  X t dYt  Yt dX t Where  t and  t are the respective volatilities of X and Y, t and

 t are their drifts and W

and W are two independent Brownian motions. Exercise: Show that if Bt is a zero volatility process and X t is any stochastic process, then

d ( X t Bt )  X t dBt  Bt dX t 3

Copyright ©2020, S.D. Perera

DA 3470 –Stochastic Finance

Section 4.4: Martingales A stochastic process Mt is a martingale with respect to a measure ℙ if and only if i)

𝐸ℙ(𝑀𝑡 ) < ∞

ii)

𝐸ℙ(𝑀𝑡 |ℱ𝑠 ) = 𝑀𝑠 ∀𝑠 ≤ 𝑡

∀𝑡

Examples 1.

The constant process 𝑆𝑡 = 𝑐 ∀𝑡 is a martingale with respect to any measure: 𝐸ℙ(𝑆𝑡 |ℱ𝑠 ) = 𝑐 = 𝑆𝑠 ∀𝑠 ≤ 𝑡 and for any measure ℙ.

2.

The ℙ-Brownian motion is a ℙ-martingale. Proof: For Brownian motion to be a ℙ-martingale, we need𝐸ℙ(𝑊𝑡 |ℱ𝑠 ) = 𝑊𝑠 . The increment 𝑊𝑡 − 𝑊𝑠 is independent of ℱ𝑠 (property 4) and 𝑊𝑡 − 𝑊𝑠 ~𝑁(0, 𝑡 − 𝑠) 𝐸ℙ(𝑊𝑡 |ℱ𝑠 ) = 𝐸ℙ(𝑊𝑡 − 𝑊𝑠 + 𝑊𝑠 |ℱ𝑠 ) = 𝐸ℙ (𝑊𝑠 |ℱ𝑠 ) + 𝐸ℙ (𝑊𝑡 − 𝑊𝑠 |ℱ𝑠 )=𝐸ℙ(𝑊𝑠 |ℱ𝑠 ) = 𝑊𝑠 In the last step, we used the (trivial) fact that 𝐸ℙ(𝑊𝑠 |ℱ𝑠 ) = 𝑊𝑠 and that 𝐸ℙ(𝑊𝑡 − 𝑊𝑠 |ℱ𝑠 ) = 0 since we know that 𝑊𝑡 − 𝑊𝑠 ~𝑁(0, 𝑡 − 𝑠).

3.

For any claim X depending only on events up to time t, the process 𝑁𝑡 = 𝐸ℙ(𝑋|ℱ𝑡 ) is a ℙ-martingale. For 𝑁𝑡 to be a ℙ-martingale, we require𝐸ℙ (𝑁𝑡 |ℱ𝑠 ) = 𝑁𝑠 , but for this we merely need to be satisfied that 𝐸ℙ(𝐸ℙ(𝑋|ℱ𝑡 )|ℱ𝑠 ) = 𝐸ℙ (𝑋|ℱ𝑠 ). This means conditioning firstly up to time t and then on information up to time s is just the same as conditioning up to time s.

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DA 3470 –Stochastic Finance 4.

Show that 𝑌𝑡 = 𝑊𝑡 2 -t is a ℙ-martingale. Proof: First we will show that 𝐸ℙ(𝑊𝑡 2 − 𝑊𝑠 2 |ℱ𝑠 ) = 𝑡 − 𝑠 2 𝐸ℙ(𝑊𝑡 2 − 𝑊𝑠 |ℱ𝑠 ) = 𝐸ℙ((𝑊𝑡 − 𝑊𝑠 )2 + 2𝑊𝑠 (𝑊𝑡 − 𝑊𝑠 )|ℱ𝑠 ) 2 = 𝐸ℙ((𝑊𝑡 − 𝑊 𝑠 ) |ℱ𝑠 ) + 2𝑊𝑠 𝐸ℙ (𝑊𝑡 − 𝑊𝑠 )|ℱ𝑠 )

=𝑡−𝑠 𝐸ℙ(𝑊𝑡 2 − 𝑡|ℱ𝑠 ) = 𝐸ℙ(𝑊𝑡 2 − 𝑊𝑠 2 + 𝑊𝑠 2 − 𝑡|ℱ𝑠 ) 2 2 =𝐸ℙ (𝑊𝑡 2 − 𝑊𝑠 |ℱ𝑠 ) + 𝐸ℙ (𝑊𝑠 − 𝑡|ℱ𝑠 ) 2 2 =𝑡−𝑠+𝑊 𝑠 − 𝑡 = 𝑊𝑠 − 𝑠

Using Ito’s lemma to identify martingales Aside from solving stochastic differential equations, we can also use Ito’s lemma to identify martingales. To see why, let’s prove the following (trivial) statement. Consider a stochastic process X t given by Xt  Wt   t . Then X t is a martingale if and only if

  0.

Proof: Assume X t is a martingale for t>s. Therefore 𝐸ℙ(𝑋𝑡 |ℱ𝑠 ) = 𝑋𝑠 . As X s  Ws  s and using the fact that 𝑊𝑡 is a ℙ-martingale, 𝐸ℙ (𝑋𝑡 |ℱ𝑠 ) = 𝐸ℙ (𝜎𝑊𝑡 + 𝜇𝑡|ℱ𝑠 ) = 𝐸ℙ(𝜎𝑊𝑡 |ℱ𝑠 ) + 𝜇𝑡 =𝜎𝑊𝑠 + 𝜇𝑡 But since X t is a martingale, 𝐸ℙ(𝑋𝑡 |ℱ𝑠 ) = 𝑋𝑠 = 𝜎𝑊𝑠 + 𝜇𝑠 Thus we have 𝜎𝑊𝑠 + 𝜇𝑡= 𝜎𝑊𝑠 + 𝜇𝑠. Thus 𝜇 = 0. The other direction is trivial.

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DA 3470 –Stochastic Finance The important interpretation of this statement is that for a stochastic process, we need the drift to vanish in order for the process to be a martingale. This statement is generalized by the following theorem. Theorem Assume that a stochastic process X t is the solution of the stochastic differential equation

T    dX t  t dWt  t dt and that the condition  (   s2ds    0 

1/2

)   is satisfied. Then X t is a

martingale if and only if X t is driftless (hence t 0 ).

Examples:





 The stochastic process Yt  Wt 2  t is a martingale as there is no drift present in

dYt  2Wt dWt .  Yt  Wt2

is not a martingale since the stochastic differential equation is

dYt  2Wt dWt  dt with the drift term present.  Yt  e

( Wt  12 2 t )

is a martingale as there is no drift tem in the corresponding stochastic

differential equation dYt   Yt dWt

Exercise: Show that the process Xt  Wt   t where 𝑊𝑡 is a ℙ-Brownian motion, is a ℙmartingale if and only if 𝛾 = 0.

In the discrete world, we had Binomial representation theorem where if Mt and Nt are both ℙmartingales, we could represent the changes in Nt by scaled changes in the other Mt ℙmartingale. Thus Nt itself can be represented by the scaled sum of these changes.

In the continuous world:

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DA 3470 –Stochastic Finance Martingale Representation Theorem Suppose Mt is a

martingale process with P(t  0) 1 . If Nt is any other

T  exists a ℱ -previsible process  such that P   t 2t 2dt     1,   0  t

N t  N 0   sdM s 0

where  t is the ratio of their respective volatilities.

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martingale, there

and N can be written as...