Test 2 Stock Option Pricing

1
Test 2Stock Option Pricing

• Math 115a

2
Formulas
3
Compound Interest

• P dollars invested at an annual rate r,
  compounded n times per year, has a value of F
  dollars after t years.
• Discrete Compounding

4
Compound Interest (cont.)

• P dollars invested at an annual rate r,
  compounded continuously, has a value of F dollars
  after t years.
• Continuous Compounding

5
Compound Interest (cont.)

• Investments with different rates and
  frequencies of compounding can be compared by
  looking at the values of P at the end of one
  year, and then computing the annual rates that
  would produce these amounts, without compounding.
  Such a rate is called an effective annual yield,
  annual percentage yield, or just the yield.

6
Compound Interest (cont.)

• Discrete
• Interest at an annual rate r, compounded n times
  per year has yield y.
• Continuous
• Interest at an annual rate r, compounded
  continuously has yield y.

7
Compound Interest (cont.)

• Under continuous compounding, the ratio, R, of
  future to present value is given by
• This allows us to convert the interest rate
  for a given period to a ratio of future to
  present value for the same period.

8
Excel Functions

• Sort (under Data)
• Histogram (under Tools Data Analysis)
• MIN/MAX (Statistical)
• COUNT (Statistical)
• BINOMDIST (Statistical)
• RANDBETWEEN (Math Trig)
• VLOOKUP (Lookup Reference)
• COUNTIF (Statistical)
• RAND (Math Trig)
• IF (Logical)

9
Probability Distribution

• Finite Random Variables
• Take on only finitely many values
• Ex. X0, 1, 2, or 3 (you can list all values)
• Probability Mass Function (p.m.f.), fX(x), where
  fX(x)P(Xx)area of the xs rectangle.
• The p.m.f. is a histogram.
• The area under the p.m.f. is equal to one.
• 0 ? fX(x) ? 1 for all x.

10
Probability Distribution

• Finite Random Variables (cont.)
• Cumulative Distribution Function (c.d.f.),
  FX(x), where FX(x)P(X ? x).
• 0 ? FX(x) ? 1 for all x.
• FX(x) 1 at the largest value of X.
• Expected Value

11
Probability Distribution

• Special Finite Random Variable
• Binomial Random Variable
• Must satisfy the following
• You have n repeated trials of an experiment.
• On a single trial, there are two possible
  outcomes, success or failure.
• The probability of success, p, is the same from
  trial to trial.
• The outcome of each trial is independent.
• Expected value E(X)np

12
Probability Distribution

• Continuous Random Variables
• The possible values of X form an interval
• Ex. Xany real number (you cannot list all
  values)
• Probability Density Function (p.d.f.), fX(x),
  where fX(x) ? P(Xx)
• The height of the function at x does NOT give the
  probability at x.
• P(Xx) 0 for all continuous random variables.
• The area under the p.d.f. is equal to one.
• fX(x) ? 0 for all x.

13
Probability Distribution

• Continuous Random Variables
• Cumulative Distribution Function (c.d.f.),
  FX(x), where FX(x)P(X ? x).
• 0 ? FX(x) ? 1 for all x.
• FX(x) 1 at the largest value of X, or
  approaches 1 in the case of an exponential R.V.
• Expected Value
• There is no general form of E(X) for all
  continuous random variables. However, for some
  forms of R.V.s, it is known.

14
Probability Distribution

• NOTE
• The fact that P(Xx)0 is not implying that X
  cannot take on the value x. In reality, there
  are an infinite number of choices for X. The
  chance it equals one particular value is just
  very very smallessentially zero.

15
Probability Distribution

• ALSO NOTE
• Because there is no practical interpretation of
  fX(x), other than the height of the function at
  x, it does not mean that fX(x) is not important.
  It not only shows the relative frequencies of
  values of X occurring, but it also gives the form
  for which FX(x) may find the area under.

16
Probability Distribution

• Special Continuous Random Variables
• Exponential Random Variables
• Usually describe the waiting time between
  consecutive events

17
Probability Distribution

• Special Continuous Random Variables
• Uniform Random Variables
• X is uniform on the interval (0,u)

18
Random Samples

• In the real world, most R.V.s for practical
  applications are continuous, and have no
  generalized form for fX(x) and FX(x).
• We may approximate the density functions by
  taking a random sample, with a large enough
  sample size, n, and plot the relative frequencies
  within the sample.
• Then,

19
Random Samples

• The whole idea behind random sampling is to
• let a part represent the whole.

20
Simulations

• We do repeated experiments to get a large enough
  random sample, such that the sample mean may
  approximate E(X), is not always practical. (Try
  flipping a coin 80 times, right?)
• Computers may be used to simulate these
  experiments, making it much more time- efficient
  to create the random sample.