- Discrete Time Market Models 85
By Theorem 4.4 the existence of a risk-neutral probability implies that there
is no arbitrage.
4.2 Extended Models.........................................
Securities such as stock, which are traded independently of other assets, are
calledprimary securities. By contrast,derivative securitiessuch as, for exam-
ple, options or forwards (in Chapter 1 we have seen some simple examples of
these) are legal contracts conferring certain financial rights or obligations upon
the holder, contingent on the prices of other securities, referred to as theun-
derlying securities. An underlying security may be a primary security, as for a
forward contract on stock, but it may also be a derivative security, as in the
case of an option on futures. A derivative security cannot exist in its own right,
unless the underlying security or securities are traded. Derivative securities are
also referred to ascontingent claimsbecause their value is contingent on the
underlying securities.
For example, the holder of a long forward contract on a stock is committed
to buying the stock for the forward price at a specified time of delivery, no
matter how much the actual stock price turns out to be at that time. The
value of the forward position is contingent on the stock. It will become positive
if the market price of stock turns out to be higher than the forward price on
delivery. If the stock price turns out to be lower than the forward price, then
the value of the forward position will be negative.
Remark 4.1
The assumptions in Section 4.1, including the No-Arbitrage Principle, are
stated for strategies consisting of primary securities only, such as stocks and
bonds (or the money market account). Nevertheless, in many texts they are
invoked in arbitrage proofs involving strategies constructed out of derivative
securities in addition to stocks and bonds. To avoid this inaccuracy the as-
sumptions need to be extended to strategies consisting of both primary and
derivative securities.
The setting of Section 4.1, involving portfolios of risky stocks and the money
market account, will be extended to include risky securities of various other
kinds in addition to (and sometimes in place of) stock. In particular, to cover
real-life situations we need to include derivative securities such as forwards or
options, but also primary securities such as bonds of various maturities, the