Computational Finance

1
Computational Finance

• Lecture 4
• Part IV
• Black-Scholes Formula

2
Random Movements of Stocks

• Binomial trees only give approximation for stock
  price movements.
• In principle, the stock price could be any number
  larger than zero. Thus, we need a model with the
  property that price can range over a continuum.

3
Random Movements of Stocks

• Google stock (Jan. 19, 2005 to Feb. 25, 2008)
• The daily returns of Google stock price
  demonstrate randomness.
• Statistical tools needed.

4
Random Movements of Stocks

• Main procedure of statistical analysis
• Daily Return
• Mean and standard deviation (SDV)
• Mean
• Standard deviation

5
Random Movements of Stocks

• Main procedure of statistical analysis
• Scaled return
• From scaled returns, we can infer probability
  density function. It looks like a normal random
  number.

6
An Approximation Normal Distribution

• A random number, , is called standard normal
  distributed if it has the following frequency
  distribution (or probability density)
• Mean
• Standard deviation

7
An Approximation Normal Distribution

• For a normal distribution Y with mean and
  standard deviation , it can be represented by
• with the density function

8
Probability Density Function

• Once we have normal approximation, then we can
  answer questions regarding the probability of
  stock returns
• Probabilities

9
Leptokurtic Phenomenon

• One drawback hinders the application of normal
  distribution in financial modeling Leptokurtic
  Phenomenon
• Higher peak in empirical distribution than normal
• Heavier tails in empirical distribution than
  normal

10
Timescales

• Daily return, weekly return and monthly return
• The mean of returns is proportional to the length
  of time period. The standard deviation is
  proportional to the square root of the length of
  time period.

11
Timescales

• Usually we take one year as the basic unit of
  time. Let denote the mean of return in a
  year (drift), denote the standard deviation
  of return in a year (volatility).
• Then, the mean of daily return should be
  and the standard deviation of daily
  return should be

12
Timescales

• In general,
• The mean of return over a period of should
  be
• The standard deviation of return over a period of
  should be

13
Stock Price Dynamics

• Therefore, the scaled return of a stock should be
• i.e.,
• or

14
Stock Price Dynamics

• Thus, if we consider a time period with length
  , then

15
Stock Price Dynamics

• The following is a widely accepted model for
  stock prices
• where is a normal distributed random
  number

16
One Caution

• Heavy mathematics ahead!

17
Review of CalculusDifferentiation

• Differentiation
• Suppose that is a function of . Then
  the derivative of is defined as

18
Review of CalculusDifferentiation

• Differentiation as a limit

19
Review of CalculusDifferentiation

• When is very small, approximately, we have

20
Review of CalculusDifferentiation

• Higher order differentiation
• We can view as another function of
  and define its derivative
• This is called the second order derivative of
  .
• And the third, fourth

21
Review of CalculusTaylor Series Expansion

• Taylor series
• Given a function , it can be approximated
  by the following series
• This series is known as the Taylor series
  expansion.

22
Review of CalculusTaylor Series Expansion

• Taylor series of exponential function
• Consider exponential function
• Its every order derivative at 0 is 1. Then,

23
Review of CalculusPartial Derivatives

• Partial derivative
• Sometimes we need to consider a function with
  more than one variable
• . Therefore, when we take derivative,
  we should specify which variable it is with
  respect to

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Review of CalculusPartial Derivatives

• Second order partial derivative
• If we continue to take derivatives on
• and , we can have second
  order partial derivatives
• , ,

25
Review of CalculusPartial Derivatives

• Taylor series for functions with two variables

26
Itos Lemma

• Suppose that the price of one stock is given by
• We care about a function of and

27
Itos Lemma

• By Taylor series approximation of two variable
  function

28
Itos Lemma

• We have
• This formula is known as the Itos formula, which
  names after a Japanese mathematician Kiyoshi Ito.
  

29
Black-Scholes Formula for Option Pricing

• The objective is to set up a formula to calculate
  a European option price under Black-Scholes model
  of stocks.
• Suppose that indicates the price of an
  option at time when the stock price is .

30
Black-Scholes Formula for Option Pricing

• Idea Replication!
• Over the small time period (t, tdt ), we may
  replicate the change of option value by select
  proper shares of underlying stock and amount of
  cash in a bank
• stock cash

31
Black-Scholes Formula for Option Pricing

• Replication arguments
• At time tdt, stockcash portfolio
• Option

32
Black-Scholes Formula for Option Pricing

• Replication arguments (continued)
• For a successful replication,

33
Black-Scholes Formula for Option Pricing

• Duplication arguments imply that
• When
• We want to know the value of

34
Black-Scholes Formula for Option Pricing

• Partial differential equation (PDE) and boundary
  condition
• Two US economists, Fischer Black (1938-1995) and
  Myron Scholes (1941-), discovered this equation
  to price options.
• They, together with Robert Merton, were awarded
  the Nobel Prize in Economics in 1997 because of
  this work.

35
Black-Scholes Formula for Option Pricing

• They found the solution to that PDE is given by
• European call
• European put
• where

36
What is N?

• N is the cumulative probability function of a
  standard normal distribution. In other words,
• No analytical expression for N. But people have
  already calculated a table of this function and
  Excel provides a function to calculate it.

37
What is N?

• Illustration of function N