Elements of Investments:
Introduction and Asset
Classes
Investment Theory FIN 320
Roberto Robatto
Wisconsin School of Business, University of Wisconsin-Madison
References:
Bodie, Kane, Marcus Essentials of Investments, 9th Edition
Chapters 1 and 2; some r
Elements of Investments:
Markets
Investment Theory FIN 320
Roberto Robatto
Wisconsin School of Business, University of Wisconsin-Madison
References:
Bodie, Kane, Marcus Essentials of Investments, 9 th Edition
Chapters 3
Extra readings (not required):
Hans
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FIN 320, Investment Theory
Roberto Robatto
Repo and Haircut
Consider a bond with face value $100
The bond trades at price $90
The haircut of the bond is 10%
The maximum amount that the investor (or ban
Wk 1/Discussion 1
Inc. magazine claims, Entrepreneurship is more mundane than it is sometimes portrayed . . .
you don't need to be a person of mythical proportions to be very, very successful in building a
company. Discuss whether you agree or disagree wi
1
Short questions
1. (Duration) You will be paying $10,000 in tuition expenses in 2 years.
(a) Whats the duration of your obligation?
(b) What maturity of zero-coupon bond would immunize your obligation?
ANSWER. The duration of the obligation is 2 years (
V ar (rS + rB ) = V ar (rS ) + V ar (rB ) + 2Cov (rS , rB )
Let c, d be constants
V ar (c rS ) = c2 V ar (rS )
V ar (rS + c) = V ar (rS )
Cov (r, c) = 0
Cov (r, r) =
2
r
(variance)
Cov (c rS , d rB ) = c d Cov (rS , rD )
Cov (c + rS , d + rB ) = Cov
FIN 320 - Problem Set 2
Professor Roberto Robatto
DUE February 24, in class.
Hand in one answer per group and write the name of all the group members on the first page. Clearly state
the formula/procedure/approach that you have used to obtain your results
FIN 320 - Problem Set 4
Professor Roberto Robatto
DUE Monday May 2, 2016, in class.
Hand in one answer per group and write the name of all the group members on the first page. Clearly state
the formula/procedure/approach that you have used to obtain your
FIN 320 - Problem Set 3
Professor Roberto Robatto
DUE Wednesday April 13, 2016, in class.
Hand in one answer per group and write the name of all the group members on the first page. Clearly state
the formula/procedure/approach that you have used to obtain
Limit order boook
1. Here is the NYSE limit order-book for Stock A:
Bid
Price
53.13
53.11
53.05
Ask
Quantity
2500
1250
3200
Price
53.15
53.19
53.25
53.27
Quantity
1500
2200
5250
3500
(a) What is the Bid-Ask Spread (measured using the best bid and ask pric
betting strategies. For now, we will allow only deterministic betting
strategies that do not look into the future and denote such a strategy
by cfw_g(t), t 0. This notation might look a little strange, but it is meant
to be suggestive for when we allow ce
Representation of Coherent Risk Measures 123 31 Further Remarks on
Value at Risk 125 Bibliography 129 Preface This set of lecture notes was
used for Statistics 441: Stochastic Calculus with Applications to Finance
at the University of Regina in the winter
to understand modern finance. Before we proceed any further, we
should be clear about what exactly a derivative is. Definition 1.2. A
derivative is a financial instrument whose value is determined by the
value of something else. That is, a derivative is a
both a random walk and a Brownian motion can be negative. Hence,
neither serves as an adequate model for a stock. Nonetheless,
Brownian motion is the key ingredient for building a reasonable model
of a stock 30 Brownian Motion as a Model of a Fair Game an
independent and identically distributed random variables with Pcfw_Y1 = 1
= p, Pcfw_Y1 = 1 = 1p for some 0 < p < 1/2. Let Sn = Y1+ +Yn denote
their partial sums so that cfw_Sn, n = 0, 1, 2, . . . is a biased random walk.
(Note that cfw_Sn, n = 0, 1, 2, .
g is any antiderivative of g 0 . Conclude that Z 1 0 g 0 (s)Bs ds N 0, Z 1
0 [g(1) g(s)]2 ds . In general, this exercise shows that for fixed t > 0,
we have Z t 0 g 0 (s)Bs ds N 0, Z t 0 [g(t) g(s)]2 ds . Exercise 10.6.
Use the result of Exercise 10.5 to
g : [0,) R be a bounded, continuous function in L 2 ([0,). If g is
differentiable with g 0 also bounded and continuous, then the
integration-by-parts formula Z t 0 g(s) dBs = g(t)Bt Z t 0 g 0 (s)Bs ds
holds. Remark. Since all three objects in the above ex
t 0 g 2 (s) ds . Definition 9.1. Suppose that g : [0,) R is a bounded,
piecewise continuous function in L 2 ([0,). The Wiener integral of g
with respect to Brownian motion cfw_Bt , t 0, written Z t 0 g(s) dBs, is a
random variable which has a N 0, Z t 0 g
know that Xn j=1 g(tj1)(Btj Btj1 ) It = Z t 0 g(s) dBs N 0, Z t 0 g
2 (s) ds from our construction of the Wiener integral in Lecture #9.
Thus, we conclude that g(t)Bt Xn j=1 g 0 (t j )Btj (tj tj1) It N
0, Z t 0 g 2 (s) ds in distribution as well. We now o
B()(t) = Bt() is the value of this function at time t. This is analogous to
our notation in calculus in which g is the function and g(t) is the value of
this function at time t.) Question. What can be said about I? On the one
hand, we know from elementary
g(s) dBs and It(h) = Z t 0 h(s) dBs. As the previous example suggests,
these two random variables are not, in general, independent. Using
linearity of the Wiener integral, we can now calculate their covariance.
Since It(g) = Z t 0 g(s) dBs N 0, Z t 0 g 2
answer this question in Example 17.2. 2 Financial Option Valuation
Preliminaries Recall that a portfolio describes a combination of (i) assets
(i.e., stocks), (ii) options, and (iii) cash invested in a bank, i.e., bonds. We
will write St to denote the val
constructed earlier. Since the discrete stochastic integral resembles a
Riemann sum, that seems like a good place to start. 7 Riemann
Integration Suppose that g : [a, b] R is a real-valued function on [a, b].
Fix a positive integer n, and let n = cfw_a =
cfw_Sn, n = 0, 1, 2, . . . is a simple random walk starting at 0. Now suppose
that I0 = 0 and for j = 1, 2, . . . define Ij to be Ij = X j n=1 Sn1(Sn Sn1).
Prove that cfw_Ij , j = 0, 1, 2, . . . is a martingale. Solution. If Ij = X j n=1
Sn1(Sn Sn1). then
world of finance. In fact, trillions of dollars worth of options trades are
executed each year using this model and its variants. In 1997, Myron S.
Scholes (originally from Timmins, ON) and Robert C. Merton were
awarded the Nobel Prize in Economics for th
Cov(Sn, Sn+1) = n. Discrete-Time Martingales 17 Exercise 4.7. As a
generalization of this covariance calculation, show that Cov(Sn, Sm) =
mincfw_n, m. Example 4.6 (continued). We now show that the simple
random walk cfw_Sn, n = 0, 1, 2, . . . is a marting
pricing options. Example 1.4. An example that is particularly relevant to
residents of Saskatchewan is the Guaranteed Delivery Contract of the
Canadian Wheat Board (CWB). See
http:/www.cwb.ca/public/en/farmers/contracts/guaranteed/ for more
information. T
exists by Theorem 7.2, which means that in order to determine its
distribution, it is sufficient to determine the distribution of the limit of
34 The Riemann Integral of Brownian Motion 35 the right-hand sums I =
limn 1 n Xn i=1 Bi/n. (See the final remar
increments of Brownian motion are independent, we have Var(I (n) t ) =
Xn j=1 g 2 (tj1)E(Btj Btj1 ) 2 = Xn j=1 g 2 (tj1)(tj tj1). We now
make a crucial observation. The variance of I (n) t , namely Xn j=1 g 2
(tj1)(tj tj1), should look familiar. Since 0 =