Part 1: Group Development
Forming Stage
Member in this stage will ask a number of questions as they are still in the process of getting to
know one another. The questions asked can vary. Group member may raise the following
questions:
1. What can I contri
Pretest:
Sophomore
Never
16 to 21African American
N/A
1.False
2.False
3.False
4.False
5.True
6.True
7.True
8.True
9.False
10.False
11.False
12.False
13.False
Yes becuase it could help get rid of stereotypes and inform us how to become better people
Postte
for low r. In addition, we expect the solution of (7.1-7.2) for the zerocoupon bond is of a simple form, for example Z(r, t; T) = e A(t;T)rB(t;T) .
Some named models are given as follows: 7.1. SHORT-TERM INTEREST
RATE MODELING 83 Vasicek This is the first
, F2 = F. Now consider a stochastic process (Xt)tcfw_0,1,2 that is
adapted to the filtration (Ft)tcfw_0,1,2. Intuitively, the value of the
random variable Xt is known once after t tosses of the coin. For
instance, X0 must be a constant, X0() = a for all ,
martingale deflators and 1FTP When does the above optimal
investment problem (*) have a solution? To answer the question, we
introduce a crucial definition: Definition. An arbitrage is a portfolio H
R n such that H P0 0 H P1 almost surely, and P(H P0 = 0
already has the n assets, with total supply of asset i given by S i and S =
(S 1 , . . . , Sn ). The agents trade with each other until each arrive at an
optimal allocation H j and collectively determine an initial price P
0 . To formalise this, we have
P1 a.s. So it remains to rule out the case that the sequence (Hk)k is
unbounded. Suppose that it was unbounded. Then we can pass to a
subsequence that kHkk . Again, let H k = Hk kHkk and pass to a
subsequence such that H k H . Note that we have that H V a
promise of future wealth is worth something now: it cannot have zero
or negative value. Furthermore, except in extreme circumstances, the
amount we pay initially will be smaller than the amount we receive at
maturity. A coupon-bearing bond is similar to t
particular, if there is an arbitrage then there cannot be an optimal
investment strategy to the utility maximisation problem. 2.1.
Motivation: Langrangian duality. As usual in a constrained optimisation
problem, we apply the Lagrangian method. Recall that
deflator (also called a state price density), we try to keep it simple. Let T
> 0 be some non-random time horizon, and let U(c) = Eu(c0, . . . , cT ),
3The term predictable is used in the US, while the synonym previsible
is more common in the UK. I am Ame
maturity. This is exactly same as what we have discussed about the
option pricing on non-traded assets in Section 2.5.2. We set up a
portfolio containing two bonds with different maturities T1 and T2. The
bond with maturity T1 has price V1(r, t; T1) and t
at a rate r(t) : dV = r(t)V dt. The solution of this equation is V (t; T) = e
R T t r()d . As a matter of fact, the short rate interest rate can be thing
of as the interest rate from a money market account, which is usually
unpredictable. To price intere
bonds options, swap, swaptions, and so on. Note that interest rates are
used for discounting as well as defining the payoff for some interest rate
derivative products. The values of these derivatives depend sensibly on
the level of interest rates. In the
probability space (, F, P) is a collection of sigma-fields such that Fs Ft
F for all 0 s t. Definition. A process X = (Xt)t0 is adapted to F iff
the random variable Xt is Ftmeasurable for all t 0. To gain some
intuition about these definitions, consider
Given the market (Pt)tcfw_0,1, the payout of the claim 1, the initial
wealth x and the utility function U, the utility indifference price 0 is
defined to any solution to max H E[U(x 0 H P0, 1 + H P1)] =
max K E[U(x K P0, K P1)]. assuming right-hand side i
could scale it to larger and larger proportions and consume more and
more. Eventually, the assumption that we are price takers (meaning we
are so small relative to the market that we can trade with no price
impact) becomes unrealistic. Aside. There is a f
P. 542-543). Ho & Lee The risk-neutral process of the short-term
interest rate is dr = (t)dt + cdW, where (t) and c are parameters. The
value of zero-coupon bonds is given by Z(r, t; T) = e A(t;T)rB(t;T) where
B = T t 84 CHAPTER 7. INTEREST RATE DERIVATIV
Collecting all V1 terms on the left-hand side and all V2 terms on the
right-hand side we find that V1 t + 1 2 2 2V1 r2 rV1 V1 r =
V2 t + 1 2 2 2V2 r2 rV2 V2 r . 82 CHAPTER 7. INTEREST RATE
DERIVATIVES The left-hand side is a function of T1 but not T2 and
claim is replicable, then the interval collapses into a single price, which
can be calculated by computing the expected value of 1Y1/Y0 for any
martingale deflator Y . Since attainable claims have unique no-arbitrage
prices, we single out the markets for
Balanced Scorecard
Aspect of
Company
Performance
Factors to be
Considered
Organizational
Goal
Financial
Quarterly Profit
Results
Return on Capital
Employed
Customer
Customer
Satisfaction Rate
Customer
Recommendation
95%
Rate (rate of new
80%
business gene
1
Strategy and Planning
MGT 521
05/30/2016
Dr. Minaudo
2
Strategy and Planning
A goal defined as a purpose do something significant within a given time. The value of
goal setting and knowing how to set goals is essential and the foundation to achieve grea
number of assets We now consider the seemingly unrealistic situation
where the market is allowed to have an infinite number of assets.
Rather than being an exercise for mathematicians to generalise
needlessly, we will see shortly that this modelling frame
(5.21) Letting U1, U2, . . . denote a sequence of i.i.d. random variables,
uniformly distributed on [0, 1], and recalling (1.5), in terms of the
language of the present section, we set Yi := f Ui for each i N. (5.22)
To obtain a control, we start with a f
[KP99, Th. 14.2.4] which, as remarked before, is proved as a special case
of [KP99, Th. 14.5.2]. Example 6.29. The commutativity condition is
satisfied for the LIBOR market model of Ex. 6.18(b) and Ex. 6.23: For
each k = 1, . . . , d, each l, = 1, . . . ,
case of R d -valued (Xt)t[0,T] and R m-valued (Wt)t[0,T] , d, m N.
For k = 1, . . . , d and l = 1, . . . , m, let (Xt)k, ak, and bkl denote the
components of the functions Xt , a, and b, respectively. One proceeds
analogous to the 1-dimensional case: For
:= (Wti+h)l (Wti )l , (6.33a) i=0,.,N1, ,l=1,.,m I (i) l := Z ti+h ti
(Wu) (Wt) d(Wl)u , (6.33b) and reinstating possible explicit time
depencencies of a and b, the multi-dimensional Milstein scheme can be
written as X(0) := Xinit, (6.34a) i=0,.,N1, k=1,.
monotonicity properties are present. In simple situations, the following
Th. 5.7 can be applied. Many related, more general, theorems can be
found in the literature. Theorem 5.7. Let U be a real-valued random
variable with range R(U). If h, k : R(U) R are
rigorous sense. So the goal is to compare discrete processes (X 0, X
h, . . . , XNh), N := T/h, given by (6.15), (6.28), or (6.34), at least for
sufficiently small h, with the continuous process (Xt)t[0,T] ,
constituting the strong solution to (6.1). 6 SI
to be a 1-dimensional geometric Brownian motion with drift R and
variance 2 . Using obvious notation, we can summarize the above as
(St)t0 is GBM(, 2 ) (Xt)t0 is BM( 2 /2, 2 ). (4.32) Remark
4.14. The relation (4.32) allows to exploit methods for simulati
on the last integral in (6.21) a bit further: Z t+h t (Wu Wt) dWu = Z t+h
t Wu dWu Wt Z t+h t dWu = Z t+h 0 Wu dWu Z t 0 Wu dWu Wt
Wt+h Wt = Yt+h Yt Wt Wt+h Wt (6.22) with random variables Yt
:= Z t 0 Wu dWu . (6.23) Noting that the stochastic process (Yt
of the notation from (4.18), we have [1] = s, [2] = u t , [11] =
2 s, [12] = 2 (u, s), (4.24a) [21] = 2 u s , [22] = 2 u u u t .
(4.24b) For u > 0, [22] is invertible, where 1 [22] = 1 2 (ut u 2 ) t
u u u = 1 2 (t u) t/u 1 1 1 . (4.24c) For u = 0, [22] =
X = E(f Y ), (5.90) showing the importance sampling estimator is
unbiased. (b) The variance of the importance sampling estimator (5.87)
is, for n N, V (Y n,g) = 1 n V f(X) g(X) = 1 n E f 2 g 2 X E(f Y ) 2
= 1 n Z S f(x) 2 g(x) 2 dPX(x) E(f Y ) 2 = 1 n Z
convergence > 0 if, and only if, c>0, >0 0<h0, >0 0<h< .
Thus, in general, conditions (6.37) and (6.38) are just different, and it
does not always make sense to try to compare them. Theorem 6.9.
Assume the hypotheses of Th. 6.3. The following statements a
equilibrium then no agent can believe there is an arbitrage. We now
return to our market model P0, P1. Now we explore heuristically the
implication of having a maximiser to an optimal investment problem.
Consider the function F(H) = E U x H P0, H P1 , and