Econ 136: Financial Economics
Problem Set #2 – Solutions
Due Date: September 13, 2018
General Instructions:
• Please upload a PDF of your problem set to Gradescope by 11:00 pm.
• Late homework will not be accepted.
• Please put your name, student ID & your GSI’s name at the upper right corner of the front page.
1. A 6-month at-the-money European call option on a non-dividend-paying stock is cur-
ently selling for GBP 3.79 while a put with the same terms is selling for GBP 2.33.
The underlying stock price is GBP 55.00, and the risk-free rate is 5% per year. Is
there an opportunity for you as an a
itrageur? If so describe the trades you would
undertake and the profit you would make; if not explain why not.
Solution: We begin by considering the relative prices of the portfolios that are on
each side of our expression for put-call parity
C + Ke−rt = S + P (1)
The portfolio on the left-hand side of this expression is cu
ently priced at
C + Ke−rt = 3.79 GBP + 55e−0.05×6/12 GBP (2)
= 3.79 GBP XXXXXXXXXXGBP = 57.43 GBP (3)
and the portfolio on the right-hand side of this expression is cu
ently priced at
S + P = 55 GBP XXXXXXXXXXGBP = 57.33 GBP (4)
So we will sell (go short) the left-hand-side portfolio, buy (go long) the right-hand-
side portfolio, and make a profit as illustrated on the next page.
1
• Today:
– Sell the call for 3.79 GBP
– Buy the stock for 55.00 GBP
– Buy the put for 2.33 GBP
– Bo
ow the 53.54 GBP (= XXXXXXXXXX) needed for the purchases
at the rate of 5% per year.
• At expiration:
– Sell the stock for 55 GBP. If the price of the stock is:
∗ Equal to 55 GBP: Sell the stock. Both options expire worthless.
∗ Less than 55 GBP: Exercise your long put to sell the stock at 55.
The call expires worthless.
∗ More than 55 GBP: Your short call will be exercised against you by
your counterparty who is long the call, and you will be required to
sell the stock to them at 55.00 GBP.
– Pay back the loan you took out to make your purchases. The loan has
grown to 54.90 GBP (=53.54e+0.05×6/12: of which 53.54 is the principal
that you bo
owed and 1.36 is the interest accrued over the 6-months).
You make this payment using a portion of the 55.00 you made on the
stock sale.
– You keep the difference of 0.10 GBP (= 55.00 GBP XXXXXXXXXXGBP).
2. Develop your own Black-Scholes option pricer and reproduce the Black-Scholes Excel
example shown in class to demonstrate that your pricer is working (Note: reproducing
a known result is called “validation”). Then
(a) Generate a screen-shot of your reproduction.
Solution: The screen-shot of the example from class is shown in the figure
elow.
2
(b) Calculate the call and put prices if the stock price increases to 110 and all othe
variables remain unchanged. Briefly explain the change in the call and put price.
Solution: If the stock price increases to 110 the call price rises to 12.97 and
the put price drops to 4.87. The increase in the call price is due to it now
eing in the money. Similarly, the drop in the put price follows because it is
now out of the money.
(c) Calculate the call price if the stock volatility decreases to 20% and all othe
variables remain unchanged. Briefly explain the change in the call price.
Solution: The call price has decreased to 4.79. This is due to the na
owing
of the probability density represented by the decrease in the volatility (the
standard deviation of the return density). This na
owing has put less proba-
ility mass in the positive regions of the call payoff diagram which results in an
decrease in the expected payoff. Since the discount factor did not change and
the call price is the present value of (or discount factor times) the expected
payoff, the decrease in volatility decreases the price of the call.
3. Create an extension of the option calculator you built in part 2 of this problem set to
eproduce the call diagram shown in slide 2 of lecture 5. The parameters for the options
shown are a strike of 50, time to expiration of 1 year, risk-free rate of 10%/year, and
volatility of 39.115%1. The red dashed curve co
esponds to a time-to-expiration of one
year. The solid blue line co
esponds to a time-to-expiration of zero years. However,
the Black-Scholes formula does not react well to zero time to expiration (you’ll get
“divide by zero” e
ors if you try this), so use a very small time to expiration instead.
I find that 1/1,000,000 years works well as a proxy for zero years. Your answer to
this question is (i) a copy of your reproduction of the call diagram shown in slide 2
of lecture 5 and (ii) a screenshot of your calculation: part of your spreadsheet if you
calculated this in a spreadsheet or part of your code if you did this using something
like MatLab or Python.
Solution: See figure on next page.
1Strictly speaking volatility should be quoted as 39.115%/yea
1
2 The reason for this strange time unit is
that the product σ
√
t needs to be dimensionless (i.e., have no units). Since the units of
√
t are the square
oot of time the units of σ must include the inverse of the square root of time so that these square roots
cancel in the product. However, in actual practice nobody speaks of the time aspect and simply refers to
the volatility as a percent.
3
Documentation of my Excel/VBA implementation is shown below. The reproduc-
tion of the call diagram is shown in the graph. The option price with a year to
expiration at an underlying level of 50 should be 10, and we see that the red curve
has a value of 10 when the x-axis is 50 which confirms this aspect of the valuation.
The highlighted cell shows that the function “mybsm” was used to generate the
option price in the cell and the panel in lower right of the screen shows the code
for this function.
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4. Review the derivation of digital calls and puts in the study guide and use these results
to derive a general expression for an option position that at expiration pays (i) nothing
elow K1, (ii) one (1) unit of cu
ency (e.g., 1 USD) between K1 and K2, and (iii)
nothing above K2. Note: digital calls have come up in job interviews and financial
economics exams. It is a good idea to be familiar with these derivations.
Solution: A digital call is an option that at expiration pays nothing (0) below
the strike and one (1) above the strike. Similarly, a digital put is an option that
at expiration pays nothing (0) above the strike and one (1) below the strike. The
expressions for these option prices are:
C(K) = e−rtN (d2,K) (5)
and
P (K) = e−rtN (−d2,K) (6)
where we’ve slightly modified the notation to include the strikes. As long as you
strikes are indicated in some way, any notation to this end is fine.
There are two ways one can create the payoff described above. Either of the
following ways is an acceptable; you didn’t need to do both.
(a) Long one K1-strike digital call and short one K2-strike digital call. For this
portfolio the long call provides the “zero below K1 and 1 above K1” features
of the payoff and the short digital call sets the payoff function to zero above
K2 by adding -1 to the +1 payoff of the K1-strike call. The expression fo
this option position is
Position = e−rtN (d2,K1)− e−rtN (d2,K2) (7)
o
Position = e−rt
[
N (d2,K1)−N (d2,K2)
]
(8)
(b) Long one K2-strike digital put and short one K1-strike digital put. For this
portfolio the long put provides the “zero above K2 and 1 below K2” features
of the payoff and the short digital put sets the payoff function to zero below
K1 by adding -1 to the +1 payoff of the K2-strike put.
Position = e−rtN (−d2,K2)− e−rtN (−d2,K1) (9)
o
Position = e−rt
[
N (−d2,K2)−N (−d2,K1)
]
(10)
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