2/11/15
Problem Set 2
Advanced Futures Strategies
Problem 1
C(S,X,t) + B(X,t) = S + P(S,X,t)
C(S,X,t) – P(S,X,t) = S – B(X,t)
C(S,X,t) – P(S,X,t) = 80 – 80 = 0
So, you buy the stock and pay with borrowed money. If
futures price is accurate predictor, then expected return
from holding stock is risk-free rate
If stock pays dividends, then futures price is 80e(r-d)t
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Problem 2
C(S,X,t) – P(S,X,t) = S – B(X,t)
$5.59 – $5.59 = 80 – 80 = 0
•  If you try different volatilities, will still find
puts and calls have equal value.
Problem 3
C(S,X,t) + B(X,t) = S + P(S,X,t)
C(S,X,t) + B(X,t) – S = P(S,X,t)
•  C(S,X,t) = S * N(d1) – B(X,t) * N(d2)
•  Then, P(S,X,t) = S*N(d1)–B(X,t)*N(d2)–S+B(X,t)
P(S,X,t) = S*(N(d1)–1) + B(X,t)*(1–N(d2))
•  If S/X = 80/40, then N(d1) and N(d2) both approach 1. So,
the put is nearly without value and need not be synthesized
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Problem 4
•  Yes, just fix“moneyness ratio” at 2 and
proceed from there
•  Would be easier to maintain hedge
Problem 6
•  The implied repo rate for a given term is
(F–S)/S
•  If equilibrium not maintained, one could
arbitrage repo agreements against futures
contracts
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Problem 7
•  With IMM Index of 93
Ø F = 100 – (100–93)(90/360) = 98.25
Ø For $1,000,000 contract, futures price would
total $982,500
•  Let’s arbitrage!
Bonds as commodities (Problem 7)
$982,500
%
o
Rep
d
e
i
l
Imp
$973,370
$973,067.09
4
= 7.3
e
t
a
R
7.10%
48 days
NY
Jan 28
F=98.25
91 days
7.17%
139 days
Profit = $302.81
NY
Mar 17
NY
June 16
$1,000,000
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Problem 8
•  The desired beta is half the beta for the portfolio
•  If the beta for the futures contract is one, the
amount involved in the futures position would be
half the value of the portfolio
•  If beta of futures contract not exactly one, then
multiply 0.5 times the beta of the futures contract
•  The beta of the futures contract on the stock
market index is e(r-d)t
Problem 9a
•
•
•
•
•
Starting beta is 1.25
The desired beta is 1
Index future price is 1250
Let’s assume the beta for the futures contract is 1
Then the amount involved in the futures position
would be ((1/1.25) –1)*12,500,000 = –2,500,000
•  Number of contracts to sell is
(2,500,000/1250) / 250 = 8
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Problem 9b
•  Portfolio drops from $12,500,000 to $11,750,000
•  Index future drops from 1250 to 1181.25
•  The index has dropped 5.5%, and the portfolio has dropped 6% from
their original values
Ø  (6/5.5 is only 1.09, so the drop in the portfolio was not as great as its beta
would predict, and so there may have been some alpha capture).
•  Profit from the futures position would be the number of contracts
(-8)times the multiplier (250) times 1181.25 – 1250
Ø  So, the profit would be $137,500
•  Added to the new value of the portfolio, you would have $11,887,500,
reducing the loss to 4.9% of the original value of the portfolio.
Ø  This is a bit less than the 5.5% drop on the index, which is consistent with a beta of
1 combined with some alpha capture.
Problem 10 (assume storage costs same both places)
Sell oil
€600,000
Buy oil
$1,000,000
FRA
today
$1 = € 0.60
5%
Replace for
€614,977.39
FRA
later
€629,240.41
Profit = €14,263.02
NY
today
3%
$1 = € 0.62
NY
later
Sell for
$1,014,903.88
What is not balanced?
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