In the previous lesson we presented in Example
2 a binomial market where it is possible to find an
investment strategy that yields a positive profit with positive
probability but without any downside risk. Such a strategy is
commonly known as an **arbitrage opportunity**.

Before giving a formal definition of an arbitrage
opportunity
it is important to introduce some notation to clarify the
concepts of **trading strategy** and
most importantly
**self-financing trading strategy**.
A trading strategy
is a process that for any time *t* specifies the
quantity of
shares in the money market account *S*^{0}
(in our
examples this corresponds to the amount of money in dollar
currency) and the number of shares of the risky asset *S*
held by the investor between times *t-1* and *t*.
We
use the following notation for a trading strategy:

It is important to notice that *x*_{t}
and
*y*_{t} could take
negative values which corresponds
to borrowing money and short selling the risky
asset,
respectively. For instance the strategy presented in Example
2 could be written as *x*_{1}=1.5
and
*y*_{1}=-1.

With this notation *y*_{t}S_{t-1}
is the amount invested into
the risky asset at time *t-1*, while *y*_{t}S_{t}
is the resulting value at time *t*. We define the **value
of the portfolio** *(c,d)* at time *t*,
by

The term arbitrage is
commonly referred to as the practice of taking advantage of the price
differential between two markets by buying and selling assets.
This section is mainly dedicated to making this statement precise. A
market with asset prices that rule out these practices is called an **arbitrage-free**
market. An investor that is engaged in an arbitrage opportunity is
called an **arbitrageur**.

We will have a

**self-financing trading strategy**
if for any

*t* greater than or equal to 1 and less
than or equal to

*T-1*, the value of the portfolios

*(x*_{t},
y_{t}) and

*(x*_{t+1},
y_{t+1}) at time

*t*
are the same. By the observation made in the beginning of the
paragraph, this is equivalent to say that the fluctuations in the value
of the portfolio are equal to the gains and losses resulting from asset
pricing fluctuations only, i.e. there are no cash flows coming in or
out.

We define formally an **arbitrage
opportunity** (see Tangent) as a self-financing
trading strategy *(x,y)* such that the value of the
initial portfolio *(x*_{1},y_{1})
at time *0* is less than or equal to 0, but the value
of the final portfolio *(x*_{T},y_{T})
at time *T* is nonnegative with probability 1 and
positive with positive probability.

In order to clarify the concepts and notation introduced above we
present some examples.

## Example 2 (continued)

The strategy presented in the

previous
lesson corresponds to

*x*_{1}=1.5
and

*y*_{1}=-1. Since we
are facing a 1 period model the self-financing condition trivially
holds. Observe that the value of the portfolio

*(x*_{1},y_{1})
at time 0 is

However, the
value of the
same portfolio at time 1 is either 1.5*1.1+(-1)*1.65=0 with probability
0.5 (i.e. if the Euro goes up) or
1.5*1.1+(-1)*1.2=0.45 with the same probability (i.e. if the Euro goes
down). This is an example of an arbitrage opportunity in a one
step binomial market. We say in this case that the market is not
arbitrage-free.

## Example 3

Suppose that today the price per share of Stock for General Motors
Corp. (GM) is 10. For the next two days the rate of return is either 1%
with
probability 0.3 or -3% with probability 0.7. Suppose further that the
rate of interest in the money market is 2%. We claim that this binomial
market is not arbitrage-free. Consider the following trading strategy,

*(x,y)*
*x*_{1}=x_{2}^{u}=x_{2}^{d}=10,

*y*_{1}=y_{2}^{u}=y_{2}^{d}=-1,
where

*(x*_{2}^{u},y_{2}^{u})
and

*(x*_{2}^{d},x_{2}^{d})
are the portfolios when the price of the stock after the first day goes
up and down, respectively. Since the portfolio does not change at any
time this trading strategy is trivially self-financing. At time 0 the
value of the portfolio

*(x*_{1}, y_{1})
is 10*1+(-1)*10=0. If the price of the stock goes up on day 1 and day 2
then the value of the final portfolio

*(x*_{2}^{u},y_{2}^{u})
is equal to 10*(1.02)

^{2}+(-1)*10*(1.01)

^{2}=0.203.
This event occurs with probability 0.3*0.3=0.09 (see

Probability
Review). If the price of the stock goes down one day but up
the other day the value of the final portfolio

*(x*_{2}^{u},y_{2}^{u})
or

*(x*_{2}^{d},y_{2}^{d})
is equal to 10*(1.02)

^{2}+(-1)*10*(1.01)*(0.97)=0.607.
Each of these events occurs with probability 0.3*0.7=0.21. Finally, if
the price of the stock goes down on day 1 and day 2 the value of the
final portfolio

*(x*_{2}^{d},y_{2}^{d})
is equal to 10*(1.02)

^{2}+(-1)*10*(0.97)

^{2}=0.995
This event occurs with probability 0.7*0.7=0.49. Hence, regardless what
occurs during these two days the value of the final portfolio is
positive and this trading strategy is an arbitrage opportunity. This is
an example of a 2 step binomial market that is not arbitrage-free.

## Note on self-financing strategies

So far in our examples the self-financing condition holds trivially. In
order to better understand the concept, consider the market described
in the previous Example and suppose that at time 0 the investor short
sells one share of stock (i.e. borrows one share of stock from the
broker). At time 1, if the price goes down he pays the stock back to
the broker and buys one share in the market, and if the price goes up
he does nothing. With our notation the strategy on the stock can be
written as

*x*_{1}=10, y_{1}=-1,
y_{2}^{u}=-1, y_{2}^{d}=1.
Intuitively in order to have a self-financing strategy, the strategy on
the money account should balance off the fact that between times 1 and
2 the investor paid back stock to the broker and bought new stock in
the market. Formally with the notation developed we have to find

*x*_{2}^{u}
and

*x*_{2}^{d}
such that the following holds

We notice that

*x*_{2}^{u}=x_{1}=10
and

*x*_{2}^{d}=x_{1}+(-2)*(10)*(0.97)/(1.02)
is the unique solution to the system above.

The reasoning above can be generalized and
summarized by the following observation. Given any initial portfolio *(x*_{1},
y_{1}) and
any strategy on the risky asset *S*, *y*_{2},
..., y_{T}, there exists a unique
strategy on the money market, *x*_{2}, ...,
x_{T}, such that the trading strategy *(x,y)*
is self-financing. This strategy can be found by successive use of the
following formula

**Activities**
- Consider the
market described
in Example 3 but assume that the interest
rate
*r=0*. Given an arbitrary initial portfolio *(x*_{1},
y_{1}) and an arbitrary strategy on the
risky asset *S*, *y*_{2},
find a strategy on the money market, *x*_{2},
such that the trading strategy *(x,y)* is
self-financing.

*Hint:*
Recall that you have to consider two market states, one when the price
of the stock goes up the first day and the other one when the price
goes down. Hence, your strategy on the money market corresponds to two
variables *x*_{2}^{d}
and *x*_{2}^{u}
which should be given in terms of *x*_{1}, y_{1},
y_{2}^{d}, y_{2}^{u}.
- For the
strategy found in part
a), write down explicitly
the value of the initial portfolio and the possible values of the final
portfolios in terms of
*x*_{1}, y_{1},
y_{2}^{d}, y_{2}^{u}.
- Explain
why
it is impossible to
find
*x*_{1},
y_{1}, y_{2}^{d},
y_{2}^{u} such that
the value of the initial portfolio is nonpositive but the value of the
final portfolio is always nonegative and positive at least once. In
other words, explain why this market is arbitrage-free.

*Hint:*
In order to prove this you will have to consider a system of
inequalities and explain why it is not consitent, i.e. why there are no
values of *x*_{1}, y_{1},
y_{2}^{d}, y_{2}^{u}
that satisfy all the inequalities at the same time. At least one of the
inequalities should be strict; this corresponds to the fact that in an
arbitrage the value of the final portfolio is positive for at least one
market state.