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Posted: April 9th, 2024
Chapter 18. Exotic Options I: Path-Independent Options
1. What is the sign of the delta of a call option on a put option?
Why? What about a put on a call?
2.
Why does a call on a put cost less than the put?
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3. Find the price of a binary cash-or-nothing put option in a
binomial tree with the following
parameters: S = 100, u = 1:10, d = 0:90, R =
1:02, and K = 100. Assume that the binary pays a at amount of $10 if ST 100, and nothing otherwise.
4. Consider a digital call option, i.e., one that pays a dollar if at
maturity, the stock price ST is greater than the strike K.
(a)
What is the sign of the delta of this option?
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(b)
When will the delta of this option be the highest?
Sundaram
& Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . .
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5. You are given a three-period binomial tree with the following
parameters: S = 100,
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R = 1:02, u = 1:10, and d = 0:90. Consider a
claim whose payo at maturity is given by Smax Smin where Smax and Smin are, respectively, the highest and lowest stock prices observed
during the option's life (including the initial price of S = 100). What is the
initial price of this claim?
.
6.
Are ordinary American-style options path-independent?
7. Consider a stock with current price S = 50 whose price process can
be represented by a binomial tree with parameters u = 1:221 and d = 0:819.
Suppose the per-period gross interest rate is R = 1:005.
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(a)
Find the value of a two-period European put option with a strike
of K = 50.
(b) Using backwards induction on the tree, nd the value of a forward
start put option that comes to life in one period, is European, has a further
life of two periods, and will be at-the-money when it comes to life.
(c)
Verify that your answers to parts (a) and (b) coincide.
(d) Suppose the puts had been American. What are the answers to parts
(a) and (b)? Do they still coincide?
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8 Consider a stock currently trading at S = 80 whose price evolution
can be represented by a binomial tree with parameters u = 1:226 and d = 0:815.
Suppose the per-period gross rate of interest is R = 1:005.
Price a one-period call option on the tree with a strike of K =
76.
Using backwards induction, nd the price of a forward start call
option that comes to life in one period, has a further life of one period, and
has a strike equal to 95% of the stock price when it comes to life. Verify that
it is the same as your answer to (a).
Find the initial delta of the forward start.
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Now assume that the initial stock price is S = 1. Assuming the
same parameters for the binomial tree, nd the price of a one-period call with
strike K = 0:95. How does this price compare to the delta you identi ed in part
(c)? Why?
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