23-RiskManagement Page 738 Wednesday, February 4, 2004 1:13 PM
738 The Mathematics of Financial Modeling and Investment Managementselected portions of risk; and (3) by trading these contracts. From a sta-
tistical point of view, a key innovation is the attention paid to the ratio
between the bulk of the risk and the risk of the tails. The latter has
become a key statistical determinant of risk management policies.
Within the realm of finance, one has to make a broad distinction
between the management of risk in investment management and in
banking and finance at large. As we have seen in the previous chapters,
investment management is essentially a question of determining a prob-
ability distribution of returns and engineering the optimal trade-off
between risk and return as a function of individual preferences. There-
fore, risk management is intrinsic to investment management.
The risk management function, which is often associated with the
investment management process, has the objective of (1) controlling risk
when the investment process is not fully automated; (2) taking into con-
sideration special risks such as the business or operational risk; and (3)
controlling the global risk, especially the tails of the risk.
Banks and financial firms, however, engage in financial operations
other than pure investing. Many of these operations are profitable but
risky and their risk must be managed or eliminated. For instance, a
financial firm offering a customized derivative instrument to a client
assumes a risk that, in itself, might be suboptimal or excessive. Hence,
the need to transfer all or part of this risk to the market at large. The
risk management function controls this process.
The possibility of effectively controlling and managing risk depends
on the availability of instruments that allow for the transfer of risk. A
market is called complete if there are instruments able to cover any trad-
able risk.
In this chapter we discuss market completeness, risk measures, and
the notion of coherence of risk measures, and then present risk models
and their use in investment management. We begin the chapter with the
concept of market completeness because it is a necessary condition for
effective risk management. We first introduced this concept in Chapter
14, where we covered arbitrage pricing.MARKET COMPLETENESS
In finance, the effectiveness of risk management is essentially related to
the degree of market completeness. In a complete market any individual
risky position can be completely hedged, that is, its risk can be com-
pletely eliminated by purchasing appropriate contracts. In intuitive
terms, this means that any payoff, intended as a random variable, can