Task
In this assignment, you will solve problems on Binomial Option Pricing.
Instructions
- Use your textbook to answer the following questions from Chapter 12:
- Exercise 16, 17, 18, 19.
- Please, upload xls, xlsx file.
- Please, use the full computing power of Excel.
16. When there are no dividends, the early exercise of an American put depends on a tradeoff
between insurance value (which comes from volatility) and time value (a function
of interest rates). Thus, for example, for a given level of volatility, early exercise of the
put becomes more likely if interest rates are higher. This question provides a numerical
illustration
Consider a two-period binomial model with u = 1.10 and d = 0.90. Suppose the initial
stock price is 100, and we are looking to price a two-period American put option with
a strike of K = 95.
(a) First, consider a “low” interest rate of R = 1.02. Show that early exercise of the
American put is never optimal in this case.
(b) Nowconsider a “high” interest rate of R = 1.05. Showthat it nowbecomes optimal
to exercise the put early in some circumstances. What is the early exercise premium
in this case?
17. Consider a two-period example with S = 100, u = 1.10, d = 0.90, R = 1.02,
and a dividend of $5 after one period. Is early exercise of a call optimal given these
parameters?
18. We repeat the previous question with higher volatility and interest rates and with lower
dividends. Consider a two-period binomial tree with the following parameters: S = 100,
u = 1.20, d = 0.80, and R = 1.10. Suppose also that a dividend of $2 is expected
after one period.
(a) Compute the risk-neutral probability in this world.
(b) Find the tree of prices of an American call option with a strike of 100 expiring in
two periods.
(c) What is the early-exercise premium?
19. The payment of a dividend on the underlying stock increases the value of a put option
since it “lowers” the stock price distribution at maturity. This question provides a
numerical illustration.
Let a two-period binomial tree be given with the following parameters: S = 100,
u = 1.10, d = 0.90, and R = 1.05. Consider a two-period American put option with
a strike of 90. Note that this put is quite deep out-of-the-money at inception.
(a) What is the value of the American put given these parameters?
(b) Now suppose a dividend of $4 is paid at the end of the first period. What is the new
price of the put?