13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

14 Visualization of the risk process 161

Pawel Mista, Rafal Weron

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

15 Approximation of ruin probability 163

Krzysztof Burnecki, Pawel Mista, Aleksander Weron

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

16 Deductibles 165

Krzysztof Burnecki, Joanna Nowicka-Zagrajek, Aleksander Weron, A. Wyloma´ska

n

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

17 Premium calculation 167

Krzysztof Burnecki, Joanna Nowicka-Zagrajek, W. Otto

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

18 Premium calculation when independency and normality assumptions

are relaxed 169

W. Otto

18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8 Contents

19 Joint decisions on premiums, capital invested in insurance company,

rate of return on that capital and reinsurance 171

W. Otto

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

20 Stable Levy motion approximation in collective risk theory 173

Hansjoerg Furrer, Zbigniew Michna, Aleksander Weron

20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

21 Di¬usion approximations in risk theory 175

Zbigniew Michna

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Part I

Finance

1 Stable distributions in ¬nance

Szymon Borak, Wolfgang H¨rdle, Rafal Weron

a

1.1 Introduction

Stable laws “ also called ±-stable or Levy-stable “ are a rich family of probabil-

ity distributions that allow skewness and heavy tails and have many interesting

mathematical properties. They appear in the context of the Generalized Cen-

tral Limit Theorem which states that the only possible non-trivial limit of

normalized sums of independent identically distributed variables is ±-stable.

The Standard Central Limit Theorem states that the limit of normalized sums

of independent identically distributed terms with ¬nite variance is Gaussian

(±-stable with ± = 2).

It is often argued that ¬nancial asset returns are the cumulative outcome of

a vast number of pieces of information and individual decisions arriving almost

continuously in time (McCulloch, 1996; Rachev and Mittnik, 2000). Hence, it

is natural to consider stable distributions as approximations. The Gaussian

law is by far the most well known and analytically tractable stable distribution

and for these and practical reasons it has been routinely postulated to govern

asset returns. However, ¬nancial asset returns are usually much more leptokur-

tic, i.e. have much heavier tails. This leads to considering the non-Gaussian

(± < 2) stable laws, as ¬rst postulated by Benoit Mandelbrot in the early 1960s

(Mandelbrot, 1997).

Apart from empirical ¬ndings, in some cases there are solid theoretical reasons

for expecting a non-Gaussian ±-stable model. For example, emission of particles

from a point source of radiation yields the Cauchy distribution (± = 1), hitting

times for Brownian motion yield the Levy distribution (± = 0.5, β = 1), the

gravitational ¬eld of stars yields the Holtsmark distribution (± = 1.5), for

a review see Janicki and Weron (1994) or Uchaikin and Zolotarev (1999).

12 1 Stable distributions in ¬nance

Dependence on alpha

-2

-4

log(PDF(x))

-6

-8

-10

-10 -5 0 5 10

x

Figure 1.1: A semilog plot of symmetric (β = µ = 0) ±-stable probability

density functions for ± = 2 (thin black), 1.8 (red), 1.5 (thin, dashed

blue) and 1 (dashed green). The Gaussian (± = 2) density forms

a parabola and is the only ±-stable density with exponential tails.

STFstab01.xpl

1.2 ±-stable distributions

Stable laws were introduced by Paul Levy during his investigations of the be-

havior of sums of independent random variables in the early 1920s (Levy, 1925).

A sum of two independent random variables having an ±-stable distribution

with index ± is again ±-stable with the same index ±. This invariance property

does not hold for di¬erent ±™s, i.e. a sum of two independent stable random

variables with di¬erent ±™s is not ±-stable. However, it is ful¬lled for a more

general class of in¬nitely divisible distributions, which are the limiting laws for

sums of independent (but not identically distributed) variables.

1.2 ±-stable distributions 13

Dependence on beta

0.3

0.25

0.2

PDF(x)

0.15

0.1

0.05

-5 0 5

x

Figure 1.2: ±-stable probability density functions for ± = 1.2 and β = 0 (thin

black), 0.5 (red), 0.8 (thin, dashed blue) and 1 (dashed green).

STFstab02.xpl

The ±-stable distribution requires four parameters for complete description:

an index of stability ± ∈ (0, 2] also called the tail index, tail exponent or

characteristic exponent, a skewness parameter β ∈ [’1, 1], a scale parameter

σ > 0 and a location parameter µ ∈ R. The tail exponent ± determines the

rate at which the tails of the distribution taper o¬, see Figure 1.1. When ± = 2,

a Gaussian distribution results. When ± < 2, the variance is in¬nite. When