Baylor University Hankamer School of Business

Baylor University Hankamer School of Business

Baylor University
Hankamer School of Business
Department of Finance, Insurance & Real Estate
Options, Futures and Other Derivatives
Dr. Garven
Problem Set 6
Name:
SOLUTIONS
Problem 1.
Consider a European put option on a non-dividend paying stock with an exercise price of $50
that expires in 3 months. The (annualized) riskless interest rate (r) is 5%, the underlying
asset is worth $50, and its volatility (σ ) is 20%. Assume discrete compounding; i.e., the
one step discount factor is 1/(1 + rδt). Also assume that:
• 1 timestep, δt = 1/12 (i.e., there will be 3 monthly timesteps until expiration);
√
√
• The “up” move, u = 1 + σ δt, and the “down” move d = 1 − σ δt;
This option can be dynamically replicated by short selling ∆ shares of the stock and lending
L dollars at each of the nodes in the binomial tree. Use the dynamic replication approach
to find the current price (p) of such a put option. (Hint: you will need to compute values
for ∆, B, and p at: (i) node uu in 2 months; (ii) node ud in 2 months; (iii) node dd in 2
months; (iv) node u in 1 month; (v) node d in 1 month; and (vi) today.)
We begin by√computing the
p tree of stock prices over the course
p 3 months; note
√ of the next
that u = 1 + σ δt = 1 + .2 (1/12) = 1.0577, and d = 1 − σ δt = 1 − .2 (1/12) = .9423.
Thus,
Suuu = $59.17
Suu = $55.94
Su = $52.89
S= $50
Suud = $52.71
Sud = $49.83
Sd = $47.11
Sudd = $46.96
Sdd = $44.39
Sddd = $41.83
Since the exercise price for this option is $50, this implies that puuu = puud = $0, whereas
pudd = $3.04 and pddd = $8.17.
Next, we find the values for ∆ and B :
• At note uu, since puuu = 0 and puud = 0, it follows that ∆uu = 0 and Buu = 0.
puud − pudd
0 − 3.04
=
= −52.89% and Bud =
Suud − Sudd
52.71 − 46.96
upudd − dpuud
1.0577(3.04) − .9423(0)
=
= $27.77. Therefore, we have a short
(1 + rδt)(u − d)
(1 + .05/12)(1.0577 − .9423)
position of .5289 shares of stock and we lend $27.77. This portfolio has a net value of
VHud = pud = −.5289 (49.83) + $27.77 = $1.41.
• At node ud, we have ∆ud =
3.04 − 8.17
upddd − dpudd
pudd − pddd
=
= -100% and Bdd =
=
Sudd − Sddd
46.96 − 41.3
(1 + rδt)(u − d)
1.0577(8.17) − .9423(3.04)
= $49.79. Therefore, we have a short position of one
(1 + .05/12)(1.0577 − .9423)
share of stock and we lend $49.79. This portfolio has a net value of VHdd = pdd =
−1 (44.39) + $49.79 = $5.40.
• At node dd we have ∆dd =
puu − pud
0 − 1.41
upud − dpuu
=
= -23.03% and Bu =
=
2
u S − udS
55.94 − 49.83
(1 + rδt)(u − d)
1.0577(1.41) − .9423(0)
= $12.83. Therefore, we have a short position of .2303
(1 + .05/12)(1.0577 − .9423)
shares of stock and we lend $12.83. This portfolio has a net value of VHu = pu =
−.2303 (52.89) + $12.83 = $0.65.
• At node u we have ∆u =
pud − pdd
1.41 − 5.40
updd − dpud
=
= -73.40% and Bd =
=
2
udS − d S
49.83 − 44.40
(1 + rδt)(u − d)
1.0577(5.40) − .9423(1.41)
= $37.83. Therefore, we have a short position of .734
(1 + .05/12)(1.0577 − .9423)
shares of stock and we lend $37.83. This portfolio has a net value of VHd = pd =
−.734 (47.12) + $37.83 = $3.25.
• At node d we have ∆d =
0.65 − 3.25
upd − dpu
pu − pd
=
= -44.96% and Bd =
=
uS − dS
52.89 − 47.12
(1 + rδt)(u − d)
1.0577(3.24) − .9423(0.65)
= $24.32. Therefore, we have a short position of .4496
(1 + .05/12)(1.0577 − .9423)
shares of stock and we lend $24.32. This portfolio has a net value of VH = p =
−.4496 (50) + $24.32 = $1.85.
• Today, we have ∆ =
Thus we have the following binomial tree of put option prices:
puuu = $0
puu = $0
pu = $0.65
puud = $0
p = $1.85
pud = $1.41
pd = $3.25
pudd = $3.04
pdd = $5.40
pddd = $8.17
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Problem 2.
A stock price is currently $50. It is known that at the end of six months it will be either
$60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum.
Calculate the value of a six-month European call option on the stock with an exercise price of
$48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same
answers.
At the end of six months the value of the option will be either $12 (if the stock price is $60)
or $0 (if the stock price is $42). Consider a portfolio consisting of:
+∆ : shares
−1 : option
The value of the portfolio is either 42∆ or 60∆ − 12 in six months. If
42∆ = 60∆ − 12
i.e.,
∆ = 0.6667
the value of the portfolio is certain to be 28. For this value of ∆ the portfolio is therefore
riskless. The current value of the portfolio is:
0.6667 × 50 − f
where f is the value of the option. Since the portfolio must earn the risk-free rate of
interest
(0.6667 × 50 − f )e0.12×0.5 = 28
i.e.,
f = 6.96
The value of the option is therefore $6.96.
This can also be calculated using risk-neutral valuation. Suppose that p is the probability
of an upward stock price movement in a risk-neutral world. We must have
60p + 42(1 − p) = 50 × e0.06
i.e.,
18p = 11.09
or:
3
p = 0.6161
The expected value of the option in a risk-neutral world is:
12 × 0.6161 + 0 × 0.3839 = 7.3932
This has a present value of
7.3932e−0.06 = 6.96
Hence the above answer is consistent with risk-neutral valuation.
Problem 3.
A stock price is currently $30. During each two-month period for the next four months it
is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a
two-step tree to calculate the value of a derivative that pays off [max(30 − ST , 0)]2 where ST
is the stock price in four months? If the derivative is American-style, should it be exercised
early?
This type of option is known as a power option. A tree describing the behavior of the stock
price is shown in the figure below. The risk-neutral probability of an up move, p, is given by
e0.05×2/12 − 0.9
p=
= 0.6020
1.08 − 0.9
Calculating the expected payoff and discounting, we obtain the value of the option as
[0.7056 × 2 × 0.6020 × 0.3980 + 32.49 × 0.39802 ]e−0.05×4/12 = 5.394
The value of the European option is 5.394. This can also be calculated by working back
through the tree as shown in the figure below. The second number at each node is the value
of the European option. Early exercise at node C would give 9.0 which is less than 13.2449.
The option should therefore not be exercised early if it is American.
Tree to evaluate European power option in Problem 3. At each node, upper
number is the stock price and the next number is the option price
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