Practice Problems (with solutions) in Option - Mechanical and ...

PRACTICE PROBLEMS FOR REAL OPTIONS
MIE566F: DECISION ANALYSIS, FALL, 2003
DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, UNIVERSITY OF TORONTO
1. Suppose that the price of a non-dividend-paying stock is $40, the continuously compounded
interest rat is 8%, and options have 3 months to expiration. A European
call with a strike price $40 sells for $2.78 and a European put with a strike price $40
sells for $1.99. Are these prices fair If not, find an arbitrage opportunity.
[Solution] Using put-call parity,
$2.78 + $40e −0.08×0.25 = $1.99 + $40
Therefore, the prices are fair and there is no arbitrage opportunity.
2. Make the same assumption as the previous problem except suppose that the stock
pays a $5 dividend just before expiration. The price of the European call is $0.74 and
the price of the European put is $4.85. Are these prices fair If not, find an arbitrage
opportunity.
[Solution] Using put-call parity with dividends,
$0.74 + $5e −0.008×0.25) + $40e −0.008×0.25 = $4.85 + $40
Therefore, the put-call parity with dividends is satisfied and the prices are fair.
1

3. A stock currently sells for $32.00. A 6-month call option with a strike of $35.00 has
a premium of $2.27. Assuming a 4% continuously compounded risk-free rate and a
6% continuous dividend yield, what is the price of the associated put option
[Solution] We will use put-call parity with dividends to compute the price of the put option.
c + Ke −rT = $2.27 + $35e −0.04×0.5 = $36.5770
p + S 0 e −δT = p + $32e −0.06×0.5 = p + $32.81
Therefore, by equating two quantities, p = $36.5770 − $32.81 = $3.767.
4. A stock currently sells for $32.00. A 6-month call option with a strike of $30.00 has a
premium of $4.29, and a 6-month put with the same strike has a premium of $2.64.
Assume a 4% continuously compounded risk-free rate. What is the present value of
dividends payable over the next 6 months
[Solution] We will use put-call parity with dividends to computer the present value of the
dividends that will be paid during the lifetime of the options. Since
c + D + Ke −rT = p + S 0
We have
D = p + S 0 − c − Ke −rT = $2.64 + $32 − $4.29 − $30e −0.04×0.5 = $0.944
5. Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%,
2

and the dividend yield is 0%. A 1-year European call with a strike of $815 cost $75
and a 1-year European put with a strike of $815 costs $45. Consider the strategy of
buying the stock, selling the call, and buying the put.
(a) What is the rate of return on this position held until the expiration of the options
(b) What is the arbitrage implied by your answer to (a)
[Solution]
By buying the stock, selling the 815-strike call, and buying the 815 strike put, your balance
will be
−$800 + $75 − $45 = −$770
This debt will become −$770e −0.05×1 = $809.4787 in 1 year, when the options are expired.
If S T > $815, we will discard the put and the call will be exercised by the holder. Therefore,
we will have to sell the stock for $815. Since our debt is only $809.4784, we will still have
$5.5213. If S T < $815, we will exercise the put in order to sell the stock for $815. On the
other hand, the call holder will discard the call option. As a result we will have $815, which
is enough to clear our debt. The remaining balance will be $5.5213. We invest $770 and the
net profit is $5.5213. The present value of the net profit is $5.5213e −0.05×1 = $5.2520.
6. Let S = $41, K = $40, σ = 0.3, r = 8%, T = 0.25 (3 months), and δ = 0. Compute the
Black-Scholes call price.
[Solution] See class notes
3

7. Using the same inputs as in the previous problem, compute the price of a put whose
strike and time to maturity are the same as the call.
[Solution] See class notes
8. Suppose S = $41, K = $40, σ = 0.3, r = 8%, and T = 0.25(3 months). The stock will
pay $3 dividend in 1 month, but make no other payouts over the life of the option.
(a) Find the present value of the dividend.
(b) Compute the Black-Scholes call price.
[Solution] See class notes
4