Financial and Accounting Writing Memo
1
SUMMARY
Be able to write clear and understandable document, as accountant is really important. Companies today
are looking for best writers to hire. According to the University of Montana, a lot of characteristics n
1. Prepare forecasted financial statements for Year 7 (cash flow statement, income
statement, and balance sheet). Prepare a brief justification for each assumption that you
make when forecasting a specific value.
- New Deposit
Data given
The change is cal
2005b). The absence of reserves is not mandatory, policy holders are
free to decide to leave part of any surplus in the waqf (Ismail n.d.). It
has been argued by fi qh scholars that the commercial risks for the
insurer under commercial forms of insurance
i. Br., 2000. [Sey06] R. Seydel. Tools for Computational Finance.
Springer, Berlin, third edition, 2006. 88
Continuous Time Finance Lecture Notes by Ulrich Horst The objective of
this course is to give an introduction to the probabilistic techniques
requi
function v : [0, 1] R and integration, Equation (6.42) turns into Z 1 0 u
00(x)v(x)dx = Z 1 0 f(x)v(x)dx. (6.44) If (6.44) holds for all functions v
from a suffciently large space, any solution u to (6.44) solves (6.42) as
well. We focus on test functions
famous Black-Scholes option pricing formula. Armed with the necessary
results from stochastic calculus we then discuss the riskneutral
approach to pricing derivatives. In particular, we prove that the market
is free of arbitrage if and only if there exist
First of all the Black-Scholes model provides only a very rough
approximation to real markets. This is more or less true for any model.
Moreover, parameters are subject to estimation error. In the finite
difference method, discretisation of the derivative
Altogether, the Monte Carlo method provides a universal tool which can
be implemented relatively easily. Variance reduction and similar
refinements may be necessary to achieve a reasonable accuracy. The
computation of American options is not obvious in th
Therefore X = log(1 U)/ is exponentially distributed with parameter
if U has uniform law on [0, 1]. One can in fact also choose X =
log(U)/ because 1 U has the same law as U. Acceptance/rejection
method Unfortunately, the pseudo inverse is not easily av
large N, where (x) = 1 2 R x 0 exp(z 2/2)dz denotes once more the
cumulative distribution function of the standard normal law. Suppose
that we want the probability of an error greater than to be smaller
than e.g. 0.01. In principle, we can now compute how
numerical methods relying on e.g. solving PDEs or using integral
transforms run into severe difficulties. In these cases Monte Carlo
simulation may help although it does not seem very efficient on first
glance. Recall that prices of European options with
det(Dh1 (x) on h(A). As an example consider a pair U1, U2 of
independent random variables with uniform distribution on [0, 1]. Now
set Z1 = p 2 log(U1) cos(2U2), Z2 = p 2 log(U1) sin(2U2). 5.3.
VARIANCE REDUCTION 53 Then Z1, Z2 is a pair of independent st
underlying is known in closed form. However, the approach cannot be
applied immediately to American options. A way out is to approximate
the latter by Bermudan options, which can be exercised only at finitely
many instances. These can be priced by a recur
(1999); further aspects of discrete and continuous time are discussed in
the books by Shreve (2005a, 2005b); students with an interest in
economics are encouraged to also consult Duffie (1996) and Hull (2000).
The second part of the course provides an int
for every t 0. The processes are called indistinguishable if almost all
their sample paths agree, i.e., P[Xt = Yt for all t 0] = 1. Evidently, if two
processes are indistinguishable, then they are modifications of each
other. The following lemma establish
this algorithm are m1 pairs of initial stock prices and the
corresponding option prices. Stability The approximation error in (6.4)
resp. (6.5) is small if the grid is very fine. On the other hand, choosing a
fine grid means repeating (6.7) many times. It
this can be done. Inversion Suppose that we want to simulate a random
variable with arbitrary cumulative distribution function F. We denote
the inverse function of F as F 1 , which can be defined even in the case
that F is not invertible, namely as pseudo
space, e.g. the same space V . In other words, the discrete weak
solution w V is defined here as the solution to (w, v) = hf, vi, v V.
If the dimension n of the subspace tends to , we can hope for
convergence of this discrete weak solution to the solution
X N n=1 f(Xn) (5.1) as an approximation for V . Why does this make
sense? Unbiasedness Note that VN is random because it is computed
from random numbers. But at least its expectation equals E(VN ) = 1 N
X N n=1 E(f(Xn) = 1 N X N n=1 E(f(X) = E(f(X) = V, 4
Black-Scholes case. Implicit and Crank-Nicolson schemes can be
deduced as well. However, since the respective matrices are no longer
tridiagonal, numerical solution of the corresponding linear equations is
more involved. As a way out we discuss the so-cal
xe 2 )/4 0 . . . 0 ej (1 + xe 2 )/4 1 ej/2 . We do not
discuss the incorporation of boundary conditions a this point, which
may be based on limit considerations as in Section 6.2. Likewise, the ADI
scheme can be modified for the solution of linear comple
. Hence we obtain the following algorithm for simulating a standard
normal random variable X: 1. Simulate an exponential random variable
Y with parameter 1, i.e. simulate a uniform random variable U1 on [0, 1]
and set Y = log U1. 2. Simulate another unifo
variables and hence dimensions. This curse of dimensionality partly
explains the popularity of Monte Carlo methods in Finance. Asset price
models with jumps lead to partial integro-differential equations (PIDE)
rather than PDEs. They contain integral and
small , where denotes some constant. If we run the numerical
schemes with two different parameters 1, 2, we obtain 1 = (1)
+ p 1 2 = (2) + p 2 , (6.30) which leads to 1 2
p 1 p 2 . For 2 = 1/2 we have p 2 1 2 2 p 1 , (6.31)
which yields 2 2 1 2 3 for p
unbiased and consistent estimator for the variance 2 (f(X) = Var(f(X).
Using Slutskys theorem it is not hard to show that N(VN V ) p 2
N (f(X) (5.6) converges to a standard Gaussian random variable as well.
This means that we can just replace the unknown
(Wti1 Wti2 ) = Wti Wti2 of two neighbouring increments
corresponds to one increment on the grid with double mesh size 2t. In
the above approaches, the increments Wti1 Wti2 , Wti Wti1 are
simulated as tZi1 resp. tZi with independent standard normal
random