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Posted: January 19th, 2023
Chapter 12. Binomial Option Pricing
7. Suppose you have two states of the world and two assets. The
prices of both assets in each of the two states is known. What conditions are
needed for a derivative security that is a function of the two assets to be
replicable?
10.
(Di cult) In a two-period
binomial tree, let the volatility at any node be given by
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= ln(u=d)
where u and d are as usual, respectively, the up-shift and
down-shift values for stock price moves on the tree. Given a starting stock
price of $50, suggest one way to draw a two-period recombining stock tree when
the volatility of the rst period is = 0:20 and in the second period is 0:25.
11.
You are given the following tree of stock prices. In addition, the
rate of interest per period is constant at 2%. Find the risk-neutral
probabilities of the stock movements from each node on the tree. Are these
probabilities the same? If not, explain whether the tree is a valid one.
12. On the tree given in the previous problem, price an American call
and an American put. Both options are assumed to be at strike $45.
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13. Again, using the same tree as in the previous two questions, what
is the delta of the call and the put at times 0 and 1?
.
14. The initial stock price is $100. The stock moves up each period by
a factor of 1.3 and down by a factor of 0.8. If the simple interest rate per
period is 1%, what is the risk-neutral probability of an up move in the stock
price?
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Draw the stock price tree for three periods
and price an Europan call option for three periods at strike $105.
15. The initial stock price is $100. The stock moves up each period by
a factor of 1.3 and down by a factor of 0.8. If the simple interest rate per
period is 1%, nd the prices of three period European and American puts, and
state the early exercise premium amount.
:
Sundaram
& Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 163
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16. When there are no dividends, the early exercise of an American put
depends on a trade-o 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.
Sundaram
& Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 164
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.
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(b) Now consider a high" interest rate of R = 1:05. Show that it
now becomes optimal to exercise the put early in some circumstances. What is
the early exercise premium in this case?
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