## 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

Robert Novy-Marx

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.

1

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).

S0

S11

S22

S0

S10

S20

q0

q11

q10

(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

8%?

Remember, the tree-implied volatility and annualized (continuously

compounded) expected rate of return for the underlying security

are given by

loc D

p

p.1

p/ ln Su

p

Sd

t

c:c:

loc D

ln.p Su

S C .1

p/Sd

S

/

t

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

e

c:c:

loc t D p

Su

S C .1

p/Sd

S

) p D

e

c:c:

loc t

Sd

S

Su

S

Sd

S

D

1 C loc

Sd

S

Su

S

Sd

S

where loc D e

c:c:

loc t is the simple one period expected stock

return.

2

(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

really bad.

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

following matrix:

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

(a) Estimate:

i. The number of trading days to maturity on each of the option

series.

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.

3

(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

vol.?

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

followed

p

return variance D

q

n 2

NEAD C

2

EA

D

q

.n

1/ 2

NEAD C

2

NEAD C

2

EA

D

q

.n

1/ 2

NEAD C

2

EAD

D

vuut.n

1/ 2

NEAD C

2

EAD

2

NEAD !

2

NEAD

D NEAD

vuut.n

1/ C

2

EAD

2

NEAD !

That is, the “

p

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

into Black-Scholes.)

4

We could, instead, have done the adjustment on the volatility: letting

be the annualized normal (everyday) variance,

p

return variance D

q

2T C

2

EA

D

0

@

s

2 C

2

EA

T

1

A

p

T D T

p

T

where T

q

2 C

2

EA

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).

5

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

0.5 99.01

1 97.547

1.5 95.75

2 93.795

(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

payments?

(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?