Rosazza Gianin Emanuela - Mathematical Finance: From Binomial Model to Risk Measures

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Emanuela Rosazza Gianin and Carlo Sgarra UNITEXT Mathematical Finance: Theory Review and Exercises From Binomial Model to Risk Measures 10.1007/978-3-319-01357-2_1

Springer International Publishing Switzerland 2013

1. Short review of Probability and of Stochastic Processes

Emanuela Rosazza Gianin 1 and Carlo Sgarra 2

(1)

Dipartimento di Statistica e Metodi Quantitativi Milano-Bicocca, Milan, Italy

(2)

Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

Abstract

Given a probability space (, , P ), where denotes a non-empty set, a -algebra and P a probability measure on :

• A random variable (r.v.) is a function X : such that { : X () A } F for any Borel set A in .
• A stochastic process is a family ( X t ) t 0 of random variables defined on (, , P ).
  The stochastic process ( X t ) t 0 is said to be a discrete-time stochastic process if t takes values in and a continuous-time stochastic process if t takes values in +.
• A filtration on is a family of -algebras on such that for any u v .

1.1 1.1 Review of Theory

Given a probability space (, , P ), where denotes a non-empty set, a -algebra and P a probability measure on :

• A random variable (r.v.) is a function X : such that { : X () A } F for any Borel set A in .
• A stochastic process is a family ( X t ) t 0 of random variables defined on (, , P ).
  The stochastic process ( X t ) t 0 is said to be a discrete-time stochastic process if t takes values in and a continuous-time stochastic process if t takes values in +.
• A filtration on is a family of -algebras on such that for any u v .

A stochastic process ( X t ) t 0 is called adapted to the filtration if, for any s 0, X s is -measurable.

Given a stochastic process ( X t ) t 0, a filtration is said to be generated by ( X t )t=0 if, for any s 0, is the smallest -algebra that makes X s measurable.

A filtration can be interpreted as the evolution of the information available up to a given time.

For a more detailed and exhaustive treatment of the notions recalled here, we refer to the books of Mikosch [28] and Ross [34] .

The following families of random variables and stochastic processes are widely used in Mathematical Finance.

Bernoulli random variable

A random variable X has a Bernoulli distribution if it assumes only two values (typically 1 and 0) with probability p and (1 p ) respectively. In such a case, we write X B(p) .

As a consequence, the expected value and the variance of X are equal, respectively, to E[X] = p and V (X) = p(1 p) .

Binomial random variable and Binomial process

A binomial random variable Y n counts the number of successes in a series of n independent trials, where p is the probability of success in any one trial. In such a case, Y n Bin(n; p) .

Y n can be written as a sum Y n = i =1 n X i where ( X i ) i =1,..., n are independent and identically distributed (i.i.d.), X i B(p) and X i = 1 denotes a success in trial i . The expected value and the variance of a binomial random variable Y n Bin(n; p) are, respectively, E [ Y n [ = np and V ( Y n ) = np (1 p ).

The sequence is called a binomial process.

Poisson random variable and Poisson process

A random variable Z has a Poisson distribution with parameter > 0 (in symbols, Z Poi ()) if it takes values in and its probability mass function is given by

for any n . Consequently, the expected value and the variance of Z are equal, respectively, to E [ Z ] = and to V ( Z ) = .

Furthermore, recall that a Poisson random variable can be obtained by taking the limit of a sequence of binomial random variables as p 0, n + and with pn = .

Setting = t (for t 0), can be understood as the rate or average number of arrivals per unit of time.

A process ( Z t ) t =0 is said to be a Poisson process of rate if Z 0 = 0 and if all the increments Z t Z s (for 0 s t ) are independent and identically distributed as a Poisson with parameter ( t s ).

If ( Z t ) t 0 is a Poisson process of rate counting the number of arrivals and T denotes the time between two arrivals, then T has an exponential distribution with parameter 0 ( T Exp ()). The density function of T is given by

The expected value and the variance of T are then equal to and , respectively.

Pareto random variable

A random variable X has a Pareto distribution with parameters x 0 0 and a > 0 if its cumulative distribution function is given by

For a 1, X has finite expected value

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