The current price of a non-dividend-paying stock is S0 D 50. The annual interest rate (simple) is r D 2%. We also observe the market option prices, c.K; T /, for three European calls on the stock (T in years):
Problem Set 5FIN 424, Winter 2017
This problem set is due before the start of class on Wednesday, February 22.
You should work as a study group, but every one should be involved in answering
each question (this is really practice for the exams, which you will have to
do solo, so please do not just divide the problems up). Each group should hand in
one solution. If (and only if) you must all miss class that day, then please email
your group’s solutions before the start of class the day they are due.
Please think of your product/deliverable as something you might give your employer.
If you email solutions, I will open one and only one electronic file, and
push the print button exactly once (i.e., I won’t print multiple Excel sheets). If you
do the problems in an Excel, please do your “scratch work” on back worksheets
and make sure that your answers are all organized onto the first sheet. Include
your names. No question is deliberately designed to trick or confuse you, or be
overly complex, so assume the obvious if you feel that sufficient detail is missing.
1. (Constructing a simple implied volatility trees.)
The current price of a non-dividend-paying stock is S0 D 50. The annual
interest rate (simple) is r D 2%. We also observe the market option prices,
c.K; T /, for three European calls on the stock (T in years):
c.50; 1/ D 12:67
c.30; 2/ D 25:92
c.80; 2/ D 6:91:
(a) Construct a two-year, two-step Derman-Kani (implied) tree consistent
with the observed stock price, interest rate, and option prices.
You can solve the equations analytically if you want, but feel free
to use the Excel Solver tool to solve equations if you prefer (you’ll
need to use the “subject to constraint” function within the Solver
routine to deal with more than one equations).
(b) What is the tree-implied volatility at each node (i.e., the “local volatilities”),
assuming the simple, annual expected return of the stock is
Remember, the tree-implied volatility and annualized (continuously
compounded) expected rate of return for the underlying security
are given by
p/ ln Su
S C .1
where S denotes the underlying price at the node, Su and Sd denote
next period’s up and down prices, respectively, and p denotes
the objective probability (i.e., not the risk-neutral probability) of
the up-move at the node. Solving the first of those for p yields
loc t D p
S C .1
) p D
1 C loc
where loc D e
loc t is the simple one period expected stock
(c) Use the constructed tree to price a two-year ATM (i.e., K D 50)
American put option.
2. (The volatility “smile” for Cell Therapeutics (CTIC:Nasdaq) options.) Sometimes
the lognormal assumption is worse than others, and sometimes it’s
At the close on Friday March 4, 2005, CTIC was at $10 (its realized volatility
over the previous year was 68%), the one, three and six month t-bill
yields were 2.56%, 2.76% and 3.01%, respectively, and puts and calls with
one, two, five and eight months to maturity were prices according to the
Strike Call Put Call Put Call Put Call Put
5 5.35 5.65 5.75 5.95
0.35 0.55 0.7 0.825
7.5 3.85 4.25 4.4 4.7
1.2 1.7 1.925 2.125
10 2.775 3.3 3.6 3.75
2.7 3.25 3.5 3.65
12.5 2.05 2.6 2.8 3
4.5 5 5.2 5.35
15 1.475 2.1 2.25 2.45
6.35 6.95 7.05 7.25
17.5 1.05 1.325 1.575 1.875
8.45 8.75 8.95 9.15
20 0.65 0.95 1.225 1.475
10.6 10.8 11.1 11.2
19-Mar 16-Apr 18-Jun 17-Sep-05
Figure 1: Cell Therapeutics Option Prices
i. The number of trading days to maturity on each of the option
ii. The approximate annualized continuously compounded yield-tomaturity
for each expiration date.
(b) Calculate the implied volatilities for each put series, and plot these
volatility “smiles” on one graph.
(c) How do the implied volatility vary across strike and time to maturity?
i. Does it look like a smile, or a smirk? Or something else?
ii. How can you reconcile these implied volatilities with the historic
iii. What sort of stock price dynamics might support the volatility
surface you’ve just produced?
(d) In the “smile” lecture we made an ad-hoc adjustment for earnings announcements:
we added a few days to the contract life (the exact number
depends on the size of the firm’s historical earnings announcement
surprises). This was intended to adjust for the impact of the earnings
announcement volatility, which added uncertainty to the returns to the
underlying over the life of the contract.
If the expected surprise it big, this adjustment (which could then be
months) will also have a material impact on the time discounting,
something we would like to avoid. The holding period adjustment
return variance D
That is, the “
T ” that goes in Black-Scholes used the standard deviation
of non-earnings announcement day returns, but added
(You must annualize these numbers, of course, before you plug them
We could, instead, have done the adjustment on the volatility: letting
be the annualized normal (everyday) variance,
return variance D
T D T
. The disadvantage of this procedure is that
the volatility adjustment depends on the time-to-maturity; the advantage
is that the adjustment doesn’t throw off the time discounting implicit
in Black-Scholes. The advantages outweigh the cost when the
expected surprises are large.
Please plot the “pricing errors” (model prices minus market prices)
for each put series, first assuming “normal” volatility is 90% (i.e.,
D 0:9), and second assuming the 90% volatility, but that the market
also expects a normally distributed log-price jump with a standard
deviation of 70% (i.e., EA D 0:7).
3. (Forward rates and swaps)
Suppose you observe the following term structure of interest rates (zero
coupon bond prices, per $100 dollars of face):
Maturity (years) Zero Coupon Bond Prices
(a) What are the six-, twelve-, and 18-month ahead forward prices of the
six-month rate? What about the six- and twelve-month ahead forward
prices of the one-year rate?
(b) What is the two year swap rate for swaps making semi-annual payments?
What about the two year swap rate for swaps making annual
(c) What is the price of a two year receive-fixed swap making semi-annual
payments on a notional of $1 million at a fixed rate of 3.5%? What
about the two year pay-fixed swap making semi-annual payments on
the same notional a fixed rate of 3.0%?
(d) If you owned both the swaps from the previous part, what would be
the net cash flows? What would the position be worth?