Math 3589 Spring 2016
Introduction to Financial Mathematics
Homework Assignment #1 Due January 19 Solutions
1. Suppose S0 = 4, S1(H) = 8, S1(T) = 2 and r = 0. The price of a European
Call option with strike price k =
...
Math 3589 Spring 2016
Introduction to Financial Mathematics
Homework Assignment #1 Due January 19 Solutions
1. Suppose S0 = 4, S1(H) = 8, S1(T) = 2 and r = 0. The price of a European
Call option with strike price k = 10 is V0 = 2. Using a replicating portfolio,
explain why there exists an arbitrage opportunity.
Solution There are many different ways we can construct a portfolio in
which we will earn something at t = 1 when we start with nothing at t = 0.
This is because at time t = 1 option is worth V1(H) = 0 = V1(T). In other
words, if we use the replicating portfolio, we find that the risk-neutral price
of the option is V0 = 0. Here is one example:
We sell ∆0 units of the option, and invest ∆0V0 = 2∆0 dollars in the stock;
i.e. we purchase ∆0/2 units of stock. At time t = 1, the option is worth
zero, and therefore it is not exercised, and we have 4∆0 if S1(H) occurs and
∆0 if S1(T) occurs. This is arbitrage because we started with $0 and at
t = 1, we had the possibility of making something without the possibility
of losing anything.
2. We have a stock with S0 = 2, S1(H) = 3 and S1(T) = 1. The interest rate
is r = .05. We wish to replicate a security that pays nothing if the coin
is heads and 5 if it’s tails. What is the initial investment, and how much
stock should we buy?
Solution
For the security, V , we have V1(H) = 0 and V1(T) = 5. The formula says
V0 =
1
1 + r
[˜pV1(H) + ˜qV1(T)] = 1
1.05
[.55 · 0 + .45 · 5] = 2.1429 .
Thus, the price of the security is $2.14. This is the “initial investment”.
Again, according to the formula,
∆0 =
V1(H) − V1(T)
S1(H) − S1(T)
=
−5
3 − 1
= −2.5 .
Since ∆0 < 0 we should short-sell 2.5 shares of stock if we wish to replicate
the security V . This m
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