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Em Mldterm 2
Winter 20 l 6
Times: Wednesday 2016-03-09 at 10:00 to 11:20 MATBUS 470
Duration: 1 hour 20 minutes (80 minutes)
Exam ID: 3149670 . .
Sections: MAT
MATBUS 470 Midterm/.
MATBUS 470 midterm 2
Marking Scheme:
_Qllestion Scorj;
L[)_s] , _4
_2. [8 pts] ;
'3. L5 pts_] Lt
.4' [6 pts]
6. [6 pts]
Total [30 pts] _ Hi l. [5 pts] A trader has a put option contract to sell 100 shares of a stock for a strike
%1 WATE R Loo
m Examination
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m Test 3
Fall 2016
Times: Friday 2016-11-18 at 13:30 to 14:25 (1:30 to 2:25PM) MATBUS 470
Duration: 55 minutes
Exam ID: 3311439 . .
Sections: MATBUS 470 LEC 001 SpC1al Materials
Instructors: Keith Fre
W UNIVERSITY OF
K\ WATERLOO
m Examination
WatIAM/Quest Login Userid:
m Test 1
Fall 2016
Times: Friday 2016-10-07 at 13:30 to 14:25 (1 :30 to 2:25PM) M ATBUS 470
Duration: 55 minutes
Exam ID: 3311237 . .
Sections: MATBUS 470 LEC 001 Sp601a1 Materlals
Instr
for selling a put, you earn the premium. but:
when the price below 1150, you have to pay others by 1150 <- K
which is required by the question
therefore, you sell a put to make sure that you will pay at least 1250
if the price is higher than 1250, you hav
So > Fo no arbitrage,
So < Fo has arbitrage,
after purchase the option, the up-state and
down-state have the same value which is 80
or still use the probability one.
you will receive the same answer
MATBUS 470 Test 1
Marking Scheme:
,
l
Question _ Marks 1. [2+2+2+2=8 pts] A box spread is a combination of a bull spread (long call with strike price K I plus short call with
strike price K2) and a bear spread (long put with strike price KL plus shortea-l
MATBUS 470 Winter 2014 Test 1 1. [IConsider a 9month forward contract on a stock with a current price of $50. Assume that the stock
will pay dividends of $X per share at times 3 montlyand 6 months. Assume the riskfree rate of interest
compounded continuou
since it is a short position, you
have to hold all shorting
required amt first.
therefore, for the cost, the
remaining cost is under a
decreasing trend
if you wanna trade 1 unit of whatever derivative, you
have to know the quality that you need right now.
APPENDIX
B
Zero Rates, Forward Rates, and
Zero-Coupon Yield Curves
T
he n-year zero-coupon interest rate is the rate of interest earned on an investment
that starts today and lasts for n years. All the interest and principal is realized at the
end of n ye
598
APPENDICES
8
Zero rate (%)
7
6
5
4
3
2
1
Maturity (years)
0
0
0.5
1
1.5
2
2.5
3
FIGURE B.1 Zero Curve for Data in Table B.3
which gives R = 7.05%. The zero curve is usually assumed to be linear between the
points that are determined by the bootstrap m
600
APPENDICES
EXAMPLE C.1
Consider a six-month futures contract on the S&P 500. The current value of the index
is 1,200, the six-month risk-free rate is 5% per annum, and the average dividend yield
on the S&P 500 over the next six months is expected to b
606
APPENDICES
At each node:
Upper value = Underlying asset price
Lower value = Option price
Shading indicates where option is exercised.
Strike price = 50
Discount factor per step = 0.9917
Time step, dt = 0.0833 years, 30.42 days
Growth factor per step,
APPENDIX
C
Valuing Forward and
Futures Contracts
T
he forward or futures price of an investment asset that provides no income is
given by
F0 = S0 erT
where S0 is the spot price of the asset today, T is the time to maturity of the forward
or futures contra
596
APPENDICES
TABLE B.1
Zero Rates
Maturity
(years)
Zero Rate (%)
(cont. comp.)
0.5
1.0
1.5
2.0
5.0
5.8
6.4
6.8
borrowings can be rolled over at the end of 6, 12, and 18 months. If interest rates do
stay about the same, this strategy will yield a profit
APPENDIX
F
Valuing American Options
T
o value American-style options, we divide the life of the option into n time steps
of length t. Suppose that the asset price at the beginning of a step is S. At the end
of the time step it moves up to Su with probabil
APPENDIX
D
Valuing Swaps
A
plain vanilla interest rate swap can be valued by assuming that the interest rates
that are realized in the future equal todays forward interest rates. As an example,
consider an interest rate swap that has 14 months remaining a
APPENDIX
E
Valuing European Options
T
he BlackScholesMerton formulas for valuing European call and put options on
an investment asset that provides no income are
c = S0 N(d1 ) KerT N(d2 )
and
p = KerT N(d2 ) S0 N(d1 )
where
d1 =
ln (S0 K) + (r + 2 2)T
T
APPENDIX
A
Compounding Frequencies
for Interest Rates
A
statement by a bank that the interest rate on one-year deposits is 10% per annum
sounds straightforward and unambiguous. In fact, its precise meaning depends on
the way the interest rate is measured.
Appendix B: Zero Rates, Forward Rates, and Zero-Coupon Yield Curves
597
Bond Yields
A bonds yield is the discount rate that, when applied to all cash flows, gives a bond
price equal to its market price. Suppose that the theoretical price of the bond we
ha
593
Appendix A: Compounding Frequencies for Interest Rates
Suppose that Rc is a rate of interest with continuous compounding and Rm is the
equivalent rate with compounding m times per annum. From the results in equations
(A.1) and (A.2), we have
Rc n
Ae
(
APPENDIX
G
Taylor Series Expansions
C
onsider a function z = F(x). When a small change x is made to x, there is a
corresponding small change z in z. A first approximation to the relationship
between z and x is
z =
dz
x
dx
(G.1)
This relationship is exact
611
Appendix G: Taylor Series Expansions
When x = 2 and y = 1
z
= 0.35355
x
2 z
= 0.08839
x2
z
= 0.70711
y
2 z
= 0.35355
y2
2 z
= 0.17678
xy
The first order approximation to z, given by equation (G.3), is
z = 0.35355 0.1 + 0.70711 0.1 = 0.10607
The second
607
Appendix F: Valuing American Options
We estimate theta from nodes D and C as
Theta =
3.77 4.49
2 0.08333
or 4.30 per year. This is 0.0118 per calendar day. Vega is estimated by increasing the volatility, constructing a new tree, and observing the effe