The lognormal distribution usage is very common for market values because it results from the assumptions of independent periodical returns following a normal distribution over small intervals. A lognormal distribution is obtained when the Neperian logarithm (In) of a random variable follows a normal distribution.

Hence, Xat some horizon H is lognormal. This implies that ln(X) ~ N(l, o) where In is the Neperian logarithm and N([l, o) the PDF of the normal distribution. It can be shown that the expectation and variance of a lognormal variable are:

THE POISSON DISTRIBUTION

The Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and are independent of each other.

Definition

The Poisson distribution serves for modeling the distribution of events having a preset time intensity. The random variable X is the count of a number of discrete occurrences (sometimes called "arrivals") that take place during a time-interval of given length. If the expected number of occurrences in this interval is 1, then the probability of exactly X— k occurrences (k being a non-negative integer, k= 1,2 ...) is equal to:

The parameter 1 is a positive real number, equal to the expected number of occurrences that occur during a given interval. It is also called an intensity or hazard rate. The time intensity is the number of events occurring per unit of time. The Poisson distribution serves, for instance, for assigning a probability to the number of typos when the rate of typos per unit of time is given.

Financial Applications

The Poisson distribution is the law of rare events when used in finance. It serves for modeling the behavior of prices, for assigning a probability to "jumps," or large price deviations, during a given time interval. The Poisson distribution also serves for modeling the number of claims in insurance.

For defaults, the intensity is analogous to a default probability. For example, if we say that the annual default probability is 1 % for a portfolio of 1000 borrowers, the default intensity for one year is 1% x 1000 =10. The default intensity is the number of defaults per unit of time, of the portfolio. It is equivalent to a default probability measured over a time interval such as 1 year. However, the Poisson distribution requires defaults to be independent, as for the binomial distribution, when using the same default intensity for a portfolio of borrowers.

The Poisson distribution can be derived as a limiting case of the binomial distribution as the number of trials goes to infinity and the expected fraction of successes remains fixed. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05. For sufficiently large values of n (say n > 1000), the normal distribution is an approximation to the Poisson distribution. The hazard rate is X — n x p and is a constant. For example, X — 1 for n — 100 and p — 1%, or n -1000 and/? = 0.1%.

The mean of a Poisson distribution is identical to its variance, and to the square of its standard deviation.

The sum of Poisson variables is also a Poisson variable of which Poisson parameter is the summation of all Poisson parameters. This property is used in portfolio models of credit risk.

Number of Default Events over a Given Horizon

The Poisson distribution is notably used in the modeling of time to defaults and in the modeling the number of defaults over a given horizon.

If the number of yearly defaults is 3 out of 100 exposures, this corresponds to a yearly default intensity of 3% and a Poisson parameter equal to 3 per year. The probability of observing 8 defaults is:

The mean of a Poisson distribution is the Poisson parameter, 3 in this case, and its volatility is its square root or V3 = 1.732. Table 10.1 shows the probabilities of the numbers of defaults

TABLE 10.1 The Poisson probability function, Poisson parameter = 3

Number of defaults, k

PDF

CDF

0

4.979%

4.979%

1

14.936%

19.915%

2

22.404%

42.319%

3

22.404%

64.723%

4

16.803%

81.526%

5

10.082%

91.608%

6

5.041%

96.649%

7

2.160%

98.810%

8

0.810%

99.620%

9

0.270%

99.890%

10

0.081%

99.971%

11

0.022%

99.993%

12

0.006%

99.998%

13

0.001%

100.000%

FIGURE 10.6 Poisson distribution (hazard rate: 3 per year)

k with a mean of 3 defaults per year, and Figure 10.6 charts the probability density and the cumulative density functions.

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