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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
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
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Ema Midterm 2
Winter 201 7
Times: Wednesday 2017-03-08 at 14:30 to 15:50 (2:30 to 3:50PM) MATBUS 470
Duration: 1 hour 20 minutes (80
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 be
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. As
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
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 th
604
TABLE E.1
APPENDICES
Greek Letters for Options on an Asset That Provides a Yield at Rate q
Greek Letter
Delta
Gamma
Call Option
Put Option
eqT N(d1 )
eqT [N(d1 ) 1]
qT
N (d1 )e
S0 T
S0 N (d1 )eqT
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 be
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
o
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,
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 i
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
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 t
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,
consid
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
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 me
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 pric
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
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
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)
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 vola