Chapter 3: DiscreteTime Securities Market
1
Chapter 3: DiscreteTime Securities Market
There are many uncertainties in the securities market. For some securities such as callable
bond, there is time uncertainty on when call provision is exercised. In many other
situations, future cash flows are not certain as well. For example, a call option written on
a common stock has uncertain future cash flow. In the previous chapter, we explored an
example of security market consisting of two basic assets: one is riskless asset, another
one is risky and its price movement follows a binomial tree. In this simple example we
have shown that, by “noarbitrage” principle, how to price other security.
This chapter’s purpose is to extend previous discussion to a general security market
consisting of many basic assets. To make the notations simple while not losing the
fundamental finance ideas, we first focus on the single period securities market. Then we
extend the analysis to multiperiod security market. We shall define a concept
“no
arbitrage
” and then to put forward a
fundamental theorem
about whether a market is free
of arbitrage or not. In a noarbitrage securities market, valuation of other complicated
security depends on it can be replicated (dynamically) by those basic assets. This leads to
another important concept:
“complete
”. If the security can be replicated by basic assets, it
can be valued by a “
riskneutral probability measure
”, which is the last important concept
of this chapter.
Section 1. SinglePeriod Securities Market
1. Securities Market
There are two times:
one is time zero which is the current time; another one is time one
which denotes a
future time
. There is only one time period from time zero and time one.
Market participants observe the same information at time zero. However, they don’t
know the information in the future.
There are N securities in this market
, namely
N
S
S
S
,...,
,
2
1
. These securities are also
called basic assets.
How to model the uncertainties of the prices of S’ at time one? We make use of a simple
probability space
)
},
,...,
,
{
(
2
1
P
M
ω
ω
ω
=
Ω
. There are M possible scenarios for each basic
asset. P is a probability, but it is agentdependent. It other words, every investor of this
market agrees that there are M scenarios but they might have different view (probability)
of price movement of those basic assets.
Here is one example of the price movement of security.
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Chapter 3: DiscreteTime Securities Market
2
Time Zero Time One
30
20
10
The possible outcome of the first security is written as
)]
(
,
),
(
),
(
[
1
2
1
1
1
M
S
S
S
ω
ω
ω
°
at
time one, corresponding to state
i
ω
. The possible outcomes of all basic security are
written as a matrix:
The current price of these basic assets, denoted by
)]
0
(
,
),
0
(
),
0
(
[
)
0
(
2
1
N
S
S
S
S
°
=
, is
known at time zero for all investors in this market.
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 Winter '09
 Adam
 Probability theory, securities market, DiscreteTime Securities Market

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