Krzyś Ostaszewski, http://www.math.ilstu.edu/krzysio/
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SOA Sample Questions for Derivatives Markets, Problem No. 16, and Dr.
Ostaszewski’s online exercise No. 230 posted October 10, 2009
The current price of a non-dividend paying stock is 40 and the continuously compounded
risk-free rate of return is 8%. In addition, you are given the following table of call and put
option premiums for various exercise prices:
Exercise Price
Call Premium
Put Premium
35
6.13
0.44
40
2.78
1.99
45
0.97
5.08
You are interested in speculating on volatility in the stock price, and are comparing two
investment strategies. The first is a 40-strike straddle. The second is a strangle consisting
of a 35-strike put and a 45-strike call. For what range of stock prices in 3 months does the
strangle outperform the straddle?
A. The strangle never outperforms the straddle
B. 33.56 < ST < 46.44
C. 35.13 < ST < 44.87
D. 36.57 < ST < 43.43
E. The strangle always outperforms the straddle
Solution.
The straddle consists of buying a 40-strike call and buying a 40-strike put. The cost of the
straddle is 2.78 + 1.99 = 4.77 at t = 0, and that cost accumulated with interest at time
t = 0.25 is 4.77e0.02 ≈ 4.87. The strangle consists of buying a 35-strike put and a 45strike call. This costs 0.44 + 0.97 = 1.41 at time t = 0, and the cost grows to
1.41e0.02 ≈ 1.44 at time t = 0.25. Now us first consider the profit of the straddle. For
ST < 40, the straddle has a profit of 40 − ST − 4.87 = 35.13 − ST , while for ST ≥ 40, the
straddle has a profit of ST − 40 − 4.87 = ST − 44.87. Now let us consider the profit of the
strangle. For ST < 35, the strangle has a profit of 35 − ST − 1.44 = 33.56 − ST , while for
35 ≤ ST < 45, its profit is −1.44, and for ST > 45, the strangle has a profit of
ST − 45 − 1.44 = ST − 46.44. Comparing the payoff structures between the straddle and
strangle, we see that if ST < 35 or if ST ≥ 45, the straddle would outperform the strangle,
since 35.13 > 33.56 and −44.87 > −46.44. However, if 35 ≤ ST < 45, we can solve for
the two boundary points for ST , where the strangle would outperform the straddle by
considering the inequalities: −1.44 > 35.13 − ST and −1.44 > ST − 44.87. The first
inequality gives ST > 36.57, while the second inequality gives ST < 43.43. We conclude
that the range we are seeking is 36.57 < ST < 43.43.
Answer D.
© Copyright 2009 by Krzysztof Ostaszewski. All rights reserved. Reproduction in
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Exercises from the past actuarial examinations are copyrighted by the Society of
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