108156.pdf

80 Mathematics for Finance

Exercise 4.3

Show that the No-Arbitrage Principle would be violated if there was a

self-financing predictable strategy with initial valueV(0) = 0 and final

value 0=V(2)≥0, such thatV(1)<0 with positive probability.

The strategy in Exercise 4.3 clearly violates the solvency assumption (As-
sumption 4.4), sinceV(1) may be negative. In fact, this assumption is not essen-
tial for the formulation of the No-Arbitrage Principle. An admissible strategy
realising an arbitrage opportunity can be found whenever there is a predictable
self-financing strategy (possibly violating Assumption 4.4) such thatV(0) = 0
and 0=V(n)≥0forsomen>0.

Exercise 4.4

Consider a market with one risk-free asset and one risky asset that follows

the binomial tree model. Suppose that whenever stock goes up, you

can predict that it will go down at the next step. Find a self-financing

(but not necessarily predictable) strategy withV(0) = 0,V(1)≥0and

0 =V(2)≥0.

This exercise indicates that predictability is an essential assumption in the
No-Arbitrage Principle. An investor who could foresee the future behaviour of
stock prices (here, if stock goes down at one step, you can predict what it will
do at the next step) would always be able to find a suitable investment strategy
to ensure a risk-free profit.

Exercise 4.5

Consider a market with a risk-free asset such thatA(0) = 100,A(1) =

110,A(2) = 121 dollars and a risky asset, the price of which can follow

three possible scenarios,

Scenario S(0) S(1) S(2)

ω 1 100 120 144

ω 2 100 120 96

ω 3 100 90 96

Is there an arbitrage opportunity if a) there are no restrictions on short

selling, and b) no short selling of the risky asset is allowed?