The second feature is the volatility process This model includes the Vasicek model when γ= 0. The Vasicek model is particularly convenient because it leads to closed-form solutions for many fixed-income products With γ = 1, this is the lognormal model. Ignoring the trend, this gives Δrt =σrtΔzt, or
F ( x)
x

f (u)du c

value at risk (VAR) can be interpreted as the cutoff point such that a loss will not happen with probability greater than p = 95%, say. If f(u) is the distribution of profit and losses on the portfolio, VAR is defined from

u drawn from U(0, 1) Next, we compute x sucht hat u = N(x), or x = N−1(u)
EXCEL NORMSINV()


mean reversion
rt ( rt )t rt z

‣ First, it displays mean reversion to a long-run value of θ. ‣ The parameter κ governs the speed of mean reversion. ‣ When the current interest rate is high (i.e., rt > θ), the model creates a negative drift κ(θ − rt) toward θ. ‣ Conversely, low current rates create a positive drift toward θ.
Δrt/rt = σΔzt

With γ = 0.5, this is the Cox, Ingersoll, and Ross (CIR) model, a good fit to the data


The binomial model can be viewed as a discrete equivalent to the geometric Brownian motion. we subdivide the horizon T into n intervals Δt = T/n. At each node, the price is assumed to go either up with probability p or down with
P( X t 1 | X t ,..., X1 ) P( X t 1 | X t )


a variable Δz, whose change is measured over the interval Δt such that its mean change is
zero and variance proportional to Δt
Quantitative Analysis Financial Risk Manager Handbook （sixth edition） Part two

Fundamentals of Probability Fundamentals of Statistics Monte Carlo Methods Modeling Risk Factors
The distribution can also be described by its quantile, which is the cutoff point x with an associated probability c
E ( X ) uf (u)du


Define this quantile as Q(X, c). The 50% quantile is known as the median.


EXCEL RAND() uniform distribution aX+b Normal distribution the cumulative p.d.f. N(x) is always between 0 and 1 First, we generate a uniform random variable
2

V(ε) = E(εε‘)= R the matrix R is a symmetric real matrix, it can be decomposed into its so-called Cholesky
factors (Cholesky factorization) construct the transformed variable ε = Tη
the underlying variable and time
x a( xt , t )t b( xt , t )z

A particular example of Ito process
S S t S z S / S t z
dS/S =dln(S), S follows a lognormal distribution
Markov process Wiener process Generalized Wiener process Martingale Ito process

a particular stochastic process independent of its past history; the entire distribution of the future price relies on the current price only
VAR(c) E( X ) Q( X , c)
SIMULATIONS WITH ONE RANDOM VARIABLE IMPLEMENTING SIMULATIONS MULTIPLE SOURCES OF RISK IMPORTANT FORMULAS


F ( x) f (u)du 1 p

x

VAR can be defined as minus the quantile itself, or alternatively, the deviation between the expected value and the quantile,

Univariate Distribution Functions A random variable X is characterized by a
distribution function
F ( x ) P( X x )
cumulative distribution function(C.D.F)

When the variable is continuous, the distribution is given by
F ( x) f (u)du

x
The density can be obtained from the distribution using
f ( x) dF ( x) / dx

When the variable X takes discrete values, this
distribution is obtained by summing the step values less than or equal to x.
F ( x)
xj x
f (x )
j
where the function f(x) is called the frequency function or the probability density function(p.d.f.)
2

a zero-drift stochastic process, a = 0,which leads to E(Δx) = 0
E(x | xt ) 0 E( xt t | xt ) xt

a generalized Wiener process, whose trend and volatility depend on the current value of
How to get frequency function if X takes discrete values?
F ()
j xj


f (u)du 1
f (x ) 1

the expected value for x, or mean, is given by
the integral

many sources of financial risk
S j ,t S j ,t 1 j t S j ,t 1 j j ,t t
1 1 2 1 1 2
2
(1 ) cov(1 , 2 ) V ( ) 2 ( 2 ) cov(1 , 2 )


the expected value on the target date discounted into the present risk-neutral approach risk-free rate path-dependent