Chapter 18 Lognormal Distribution

Question 1

pdf of a Normal Distribution

Answer 1

(1/(sigma * sqrt(2pi))) * exp(-(1/2) * (((x - u) / (sigma))^2))

Question 2

What is symmetry of a distribution and does the normal distribution have this property

Answer 2

it is phi(mu - x) = phi(mu + x) for the same sigma and mu and yes it has it

Question 3

N(-a) =

Answer 3

1 - (N(a))

Question 4

aX1 + bX2 where x1 and x2 are normally distributed

Answer 4

N(amu1 + bmu2, sigma1 + sigma2 + 2abcovsigma1*sigma2)

Question 5

Central Limit Theorem

Answer 5

As number of observations grow very large, it resembles a normal ditribution.

Question 6

A variable y is log normal if

Answer 6

ln(y) is normal

Question 7

why are stock prices lognormal distributed.

Answer 7

The returns are normally distributed, and ln(st/s0) = r(0,t), then st = e(r(0,t)) * s0

Question 8

the product of lognormal distributions is

Answer 8

lognormal

e^(x) * e^(y) = e^(x+y) and x + y is normal

Question 9

pdf of lognormal

Answer 9

(1/(y * sqrt(sigma^2 * 2pi))) * exp(-(1/2) * ((ln(y) - mu)/sigma)^2)

Question 10

Expected value of a lognormal distribution

Answer 10

e^(mu + (1/2) * sigma^2)

Question 11

Using the lognormal model for stocks, we find St =

Answer 11

s0 * exp(((a-delta-1/2(sigma^2))t) + sigma * sqrt(t) * Z)

Question 12

In lognormal model, E(St) =

Answer 12

s0 * exp((a - delta) * t)

Question 13

Median price for ST in the lognormal model

Answer 13

Set Z = 0

s0exp((a - delta - 1/2sigma^2)*t)

Question 14

using log-normal pricing P(S < K)

Answer 14

N(-d2) with A (actual rate) replacing r risk free rate

Question 15

using log-normal pricing P(S > K)

Answer 15

N(d2) with A (actual rate) replacing r risk free rate

Question 16

getting intervals using the log-normal model is done by

Answer 16

finding mean and SD of St, then plug in p like always and exponentiate and divide by S0

Question 17

E(st | st < K) i.e. a put

Answer 17

s0 * (exp(a - delta)*t) * (n(-d1) / n(-d2))

For a call there are no negatives on the ds

Question 18

for mean adding more frequent periods of data

Answer 18

does nothing

Question 19

for SD adding more frequent periods of data

Answer 19

helps improve accuracy

Question 20

the lognormal model predicts that returns on stocks will be normal, but in practice, they

Answer 20

have leptokurtosis (peak around 0) and fat tails
one explanation is normal but mean an sd change over time.

Question 21

order statistics

Answer 21

is the data in order

i.e. 1,4,5,7,10…

Question 22

normal probability plot

Answer 22

takes the value of the quantiles graphed agains the coresponding Normal quantile. So for 3, 4, 5, 7, 11, 3 is the 10% quantiles, so N-1(0.1) = -1.282 so, the first point is 3, -1.282. The straight line connects the 25 and 75 quantile and if the data is also a stright line then the plot is noral.