1. Brownian Motion & Itô Formula
Chapter 4
Brownian Motion
& Itô Formula

2. Stochastic Process
The price movement of an underlying asset is a stochastic process.
The French mathematician Louis Bachelier was the first one to describe the stock share price movement as a Brownian motion in his 1900 doctoral thesis.
introduction to the Brownian motion
derive the continuous model of option pricing
giving the definition and relevant properties Brownian motion
derive stochastic calculus based on the Brownian motion including the Ito integral & Ito formula.
All of the description and discussion emphasize clarity rather than mathematical rigor.

3. Coin-tossing Problem Define a random variable
It is easy to show that it has the following properties:
& are independent

4. Random Variable
With the random variable, define a random variable and a random sequence

5. Random Walk
Consider a time period [0,T], which can be divided into N equal intervals. Let Δ=T\ N, t_n=nΔ ,(n=0,1,\cdots,N), then
A random walk is defined in [0,T]:
is called the path of the random walk.

6. Distribution of the Path
Let T=1,N=4,Δ=1/4,

7. Form of Path
the path formed by linear interpolation between the above random points. For
Δ=1/4 case, there are 2^4=16 paths. S t 1

8. Properties of the Path

9. Central Limit Theorem For any random sequence
where the random variable X~ N(0,1), i.e. the random variable X obeys the standard normal distribution:
E(X)=0,Var(X)=1.

10. Application of Central Limit Them.
Consider limit as Δ→ 0.

11. Definition of Winner Process (Brownian Motion)
1) Continuity of path: W(0)=0,W(t) is a continuous function of t.
2) Normal increments: For any t>0,W(t)~ N(0,t), and for 0 < s < t, W(t)-W(s) is normally distributed with mean 0 and variance t-s, i.e.,
3) Independence of increments: for any choice of in [0,T] with the increments
are independent.

12. Continuous Models of Asset Price Movement
Introduce the discounted value
of an underlying asset as follows:
in time interval [t,t+Δt], the BTM can be written as

13. Lemma If ud=1, σis the volatility, letting
then under the martingale measure Q,

14. Proof of the Lemma
According to the definition of martingale measure Q, on [t,t+Δt],
thus by straightforward computation,

15. Proof of the Lemma
Moreover, since

16. Proof of the Lemma cont. by the assumption of the lemma,
input these values to the ori. equation.
This completes the proof of the lemma.

17. Geometric Brownian Motion
By Taylor expansion
neglecting the higher order terms of Δt, we have

18. Geometric Brownian Motion cont.
By definition
therefore after partitioning [0,T], at each instant ,
i.e.

19. Geometric Brownian Motion cont.-

20. Geometric Brownian Motion cont.--
This means the underlying asset price movement as a continuous stochastic process, its logarithmic function is described by the Brownian motion. The underlying asset price S(t) is said to fit geometric Brownian motion.
This means: Corresponding to the discrete BTM of the underlying asset price in a risk-neutral world (i.e. under the martingale measure), its continuous model obeys the geometric Brownian motion .

21. Definition of Quadratic Variation
Let function f(t) be given in [0,T], and Π be a partition of the interval [0,T]:
the quadratic variation of f(t) is defined by

22. Quadratic Variation for classical function

23. Theorem 4.1
Let Π be any partition of the interval [0,T], then the quadratic variation of a Brownian motion has a limit as follows:

24. Path of a Brownian motion
For any let be an
arbitrary partition of the interval and be the quadratic variation of the Brownian motion corresponding to the partition , then by Theorem 4.1,
Referring to the conclusion regarding the differentiable function, we have:
The path of a Brownian motion W_t as a random walk of a particle is continuous everywhere but differentiable nowhere.

25. Remark If dt 0 (i.e. Δ 0), let denote the limit
of then by Theorem 4.1,
Hence neglecting the higher order terms of dt,
i.e. neglecting higher order terms, the square of the random variable is a definitive infinitesimal of the order of dt.

26. An Example
A company invests in a risky asset, whose price movement is given by
Let f(t) be the investment strategy, with f(t)>0(<0) denoting the number of shares bought (sold) at time t. For a chosen investment strategy, what is the total profit at t=T?

27. An Example cont. Partition [0,T] by:
If the transactions are executed at time
only, then the investment strategy can only be adjusted on trading days, and the gain (loss) at the time interval is
Therefore the total profit in [0,T] is

28. Definition of Itô Integral
If f(t) is a non-anticipating stochastic process, such that the limit
exists, and is independent of the partition, then the limit is called the Itô Integral of f(t), denoted as

29. Remark of Itô Integral
Def. of the Ito Integral ≠ one of the Riemann integral.
- the Riemann sum under a particular partition.
However, f(t) - non-anticipating,
Hence in the value of f must be taken at the left endpoint of the interval, not at an arbitrary point inΔ.
Based on the quadratic variance Them. 4.1 that the value of the limit of the Riemann sum of a Wiener process depends on the choice of the interpoints.
So, for a Wiener process, if the Riemann sum is calculated over arbitrarily point in Δ, the Riemann sum has no limit.

30. Remark of Itô Integral 2 In the above proof process : since the
quadratic variation of a Brownian motion is nonzero, the result of an Ito integral is not the same as the result of an ormal integral.

31. Ito Differential Formula
This indicates a corresponding change in the differentiation rule for the composite function.

32. Itô Formula Let , where is a stochastic process. We want to know
This is the Ito formula to be discussed in this section. The Ito formula is the Chain Rule in stochastic calculus.

33. Composite Function of a Stochastic Process
The differential of a function is the linear principal part of its increment. Due to the quadratic variation theorem of the Brownian motion, a composite function of a stochastic process will have new components in its linear principal part. Let us begin with a few examples.

34. Expansion By the Taylor expansion ,
Then neglecting the higher order terms,

35. Example 1

36. Differential of Risky Asset
In a risk-neutral world, the price movement of a risky asset can be expressed by,
We want to find dS(t)=?

37. Differential of Risky Asset cont.

38. Stochastic Differential Equation
In a risk-neutral world, the underlying asset satisfies the stochastic differential equation
where is the return of over a time interval dt, rdt is the expected growth of the return of , and is the stochastic component of the return, with variance σ is called volatility.

39. Theorem 4.2 (Ito Formula)
V is differentiable ~ both variables. If satisfies SDE
then

40. Proof of Theorem 4.2
By the Taylor expansion
But

41. Proof of Theorem 4.2 cont. Substituting it into ori. Equ., we get
Thus Ito formula is true.

42. Theorem 4.3
If are stochastic processes satisfying respectively the following SDE
then

43. Proof of Theorem 4.3
By the Ito formula,

44. Proof of Theorem 4.3 cont. Substituting them into above formula
Thus the Theorem 4.3 is proved.

45. Theorem 4.4
If are stochastic processes satisfying the above SDE, then

46. Proof of Theorem 4.4
By Ito formula

47. Proof of Theorem 4.4 cont. Thus by Theorem 4.3, we have
Theorem is proved.

48. Remark Theorems 4.3--4.4 tell us:
Due to the change in the Chain Rule for differentiating composite function of the Wiener process, the product rule and quotient rule for differentiating functions of the Wiener process are also changed.
All these results remind us that stochastic calculus operations are different from the normal calculus operations!

49. Multidimensional Itô formula
Let be independent standard Brownian motions,
where Cov denotes the covariance:

50. Multidimensional Equations
Let be stochastic processes satisfying the following SDEs
where are known functions.

51. Theorem 4.5
Let be a differentiable function of n+1 variables, are stochastic processes , then
where

52. Summary 1
The definition of the Brownian motion is the central concept of this chapter. Based on the quadratic variation theorem of the Brownian motion, we have established the basic rules of stochastic differential calculus operations, in particular the Chain Rule for differentiating composite function------the Ito formula, which is the basis for modeling and pricing various types of options.

53. Summary 2
By the picture of the Brownian motion, we have established the relation between the discrete model (BTM) and the continuous model (stochastic differential equation) of the risky asset
price movement. This sets the ground for further study of the BTM for option pricing (such as convergence proof).