4 Mathematics for Finance
Exercise 1.
LetA(0) = 90,A(1) = 100,S(0) = 25 dollars and letS(1) =
{
30 with probabilityp,
20 with probability 1−p,where 0<p<1. For a portfolio withx= 10 shares andy= 15 bonds
calculateV(0),V(1) andKV.Exercise 1.
Given the same bond and stock prices as in Exercise 1.1, find a portfolio
whose value at time 1 isV(1) ={
1 ,160 if stock goes up,
1 ,040 if stock goes down.
What is the value of this portfolio at time 0?It is mathematically convenient and not too far from reality to allow arbi-
trary real numbers, including negative ones and fractions, to represent the risky
and risk-free positionsxandyin a portfolio. This is reflected in the following
assumption, which imposes no restrictions as far as the trading positions are
concerned.
Assumption 1.3 (Divisibility, Liquidity and Short Selling)
An investor may hold any numberxandyof stock shares and bonds, whether
integer or fractional, negative, positive or zero. In general,
x, y∈R.The fact that one can hold a fraction of a share or bond is referred to
asdivisibility. Almost perfect divisibility is achieved in real world dealings
whenever the volume of transactions is large as compared to the unit prices.
The fact that no bounds are imposed onxoryis related to another market
attribute known asliquidity. It means that any asset can be bought or sold on
demand at the market price in arbitrary quantities. This is clearly a mathe-
matical idealisation because in practice there exist restrictions on the volume
of trading.
If the number of securities of a particular kind held in a portfolio is pos-
itive, we say that the investor has along position. Otherwise, we say that a
short positionis taken or that the asset isshorted.A short position in risk-free