Part 2: Empirical Work
The main objective of this project is to practice two main approaches to options pricing: The Binomial model, and the Black-Scholes model.
A.The problem
Chose a stock that interests you as an underlying asset for your options. Assume that you have a
substantial long position in this stock and would like to stay invested and maintain your position
in this stock for a long term. The market conditions are currently quite uncertain and you are not
sure whether the price of your stock will move substantially up or down, but you wish to benefit
from either outcome. The two possible option strategies to achieve your goal are:
1. Long straddle
2. Long strang
B. Deliverables
1.In your report, briefly explain why you like your chosen stock.
2.From Yahoo Finance (or any other data source), download daily prices and dividends on your chosen stock for the last 500 days (approximately two years).3.
Calculate daily returns as:
r
t
,
t
+
1
=
ln
(
S
t
+
1
+
¿
t
+
1
S
t
)
where
r
t
,
t
+
1
is a daily, continuously compounded return between day
t
and the next day,
t
+
1
.
¿
t
+
1
represents the dividend (if any), paid during this period, and
S
t
is the price of
your stock at time
t
.
4.
Estimate μ (annualized mean return) and
2
(annualized variance)
1
for your stock returns.
Present these estimates in your final report. Also report the current price of your stock.
5.
Choose one of the two option strategies mentioned above. In your report, describe which
options are involved (i.e. a call or a put, or both, European or American).
6.
Choose strike (exercise) prices for the options involved in your chosen strategy. Briefly
explain reasons that you used to choose these particular strike prices.
7.
Construct a
payoff table
and draw
the payoff diagram
for your option strategy, using
your chosen strike prices. Present this table and this graph in your report.
8.
Choose the expiration time
T
(less than a year); it should be the same for all options. Do
not forget to express this time in annual terms.
9.
Assume that the risk free rate is 0.5% per year,
10.Use a 4-period binomial model to value your options:
a.
Explain how you calculated parameters of the binomial model (
u
,
d
,
p
RN
,
∧
the
gross
risk
free
rate
per
period
,
R
f
¿
and show details of your
calculations. Use regular formulas in symbols and then show how you plug
numbers into them (Excel formulas are not substitutes for that).
b.
“Draw” binomial trees (you can use Excel for this and then just copy your tree for
the Excel file to your Word file).
c.
Using formulas, show how you find payoffs at expiration for each option.
d.
Show all calculations, which you do when you “work the tree backwards” to find
option prices. It is a lot of similar calculations, but it is a good practice.
11.
Use the Black Scholes option pricing formula to price your options
a.
Show all calculations in details using formulas in symbols first. For this you
clearly need to use the Word equation edito
b.
Then show how you plug your numbers into these formulas.
12.
Now that you have prices for your options (use those obtained applying the Black
Scholes formula), calculate and report the cost of your strategy.
13.
Construct a profit/loss diagram for your strategy using your payoff diagram from
Deliverable 5 and subtracting the cost from Deliverable 10. Present this graph in your
report
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