Test 2
Stock Option Pricing
Math 115a
Formulas
 r
P  F  1  
 n
 nt
 1
e  lim 1  
m 
 m
 r
F  P  1  
 n
nt
n
 r
y  1    1
 n
1

r  n   1  y n  1


F  P  e r t
y  er 1
m
P  F  e  r t
E ( X )   x  f X ( x)
E( X )  n  p
E( X )  
all x
0 if x  0

f X ( x)   1  x / 
if 0  x
  e
0 if x  0
FX ( x)  
 x /
1

e
if 0  x

Compound Interest
P dollars invested at an annual rate r,
compounded n times per year, has a value of
F dollars after t years.
 r
F  P  1  
 n
nt
or
 r
P  F  1  
 n
 nt
Discrete Compounding
3
Compound Interest (cont.)
P dollars invested at an annual rate r,
compounded continuously, has a value of F
dollars after t years.
F  Pe
r t
or
P  F e
 r t
Continuous Compounding
4
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.
5
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.
n
 r
y  1    1
 n
y  e 1
r
6
Compound Interest (cont.)
Under continuous compounding, the ratio, R,
of future to present value is given by
F Per t
R 
 e r t
P
P
This allows us to convert the interest rate for a
given period to a ratio of future to present
value for the same period.
7
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)
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Probability Distribution
• Finite Random Variables
– Take on only finitely many values
• Ex.: X=0, 1, 2, or 3 (you can list all values)
– Probability Mass Function (p.m.f.), fX(x), where
fX(x)=P(X=x)=area of the x’s 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.
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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
E ( X )   x  P( X  x)
all x
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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
11
Probability Distribution
• Continuous Random Variables
– The possible values of X form an interval
• Ex.: X=any real number (you cannot list all values)
– Probability Density Function (p.d.f.), fX(x),
where fX(x)  P(X=x)
• The height of the function at x does NOT give the
probability at x.
• P(X=x) = 0 for all continuous random variables.
• The area under the p.d.f. is equal to one.
• fX(x)  0 for all x.
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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.
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Probability Distribution
• NOTE:
The fact that P(X=x)=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 small—essentially
zero.
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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.
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Probability Distribution
• Special Continuous Random Variables
– Exponential Random Variables
• Usually describe the waiting time between consecutive
events
 p.d.f. :
0 if x  0

f X ( x)   1  x / 
if 0  x
  e
 c.d.f. :
0 if x  0
FX ( x)  
 x /
if 0  x
1  e
 E( X )  
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Probability Distribution
• Special Continuous Random Variables
– Uniform Random Variables
• X is uniform on the interval (0,u)
 p.d.f. :
 c.d.f. :
0 if x  0
1

f X ( x)   if 0  x  u
u
0 if x  u
 E( X ) 
0 if x  0
x

FX ( x)   if 0  x  u
u
0 if x  u
u
2
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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,
1 n
x    xi  E ( X )
n i 1
18
Random Samples
• The whole idea behind random sampling
is to
let a part represent the
whole.
19
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 timeefficient to create the random sample.
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