23-RiskManagement Page 743 Wednesday, February 4, 2004 1:13 PM
Risk Management 743and where stock prices evolve as geometrical Brownian motions. This is
a complete market. Therefore any European option can be perfectly rep-
licated by a portfolio of the underlying stock plus the risk-free asset.
In this market, investors can protect themselves from excessive
losses by purchasing options. However, in case of large losses someone
has to foot the bill. The risk transfer process is the following. Suppose
that an investor who owns a stock wants to buy protection against large
price movements of the stock by purchasing an option. In this way the
owner of the stock transfers the risk of eventual large movements to the
underwriter of the option. The underwriter might decide to bear the risk
or to transfer the risk by purchasing an appropriate self-financing strat-
egy. In the latter case, the risk of large movements has been transferred
in two steps from the initial investor to the option underwriter and then
back to the market.
In case of large negative movements, there will be a transfer of
money from owners of long stock positions to the original investor who
sought protection. The transfer will occur through the mechanism of
short positions. It would be a mistake to think that by replication every-
one comes out of large negative market movement unscathed. In this
case, in particular, if options are properly hedged, the final losers are
those who hold stock positions without hedging them.
Suppose, now, that price processes follow stochastic volatility dynam-
ics. In this case, markets are incomplete and options cannot be perfectly
hedged. The key difference with respect to the previous case is that the
underwriter of the options has to foot the bill of eventual large losses. In
this case, underwriting options is a risky business, while in the previous
case, ultimately the risk is borne by stock owners or stock “lenders.”
In the case of stock markets, risk does not disappear in aggregate.
Total market capitalization fluctuates and there is no way that this glo-
bal risk can be eliminated. In fact, on a global scale, no one profits if
markets move down or loses if markets move up. Profits and losses of
short and long positions are only local relative losses. In aggregate,
investors lose if markets go down and gain if markets go up.
However, the market as seen by each individual investor might be
complete or not as a function of the dynamics of price processes. Com-
pleteness dictates that risk can be arbitrarily apportioned but does not
change the fact that massive losses might occur in aggregate. In other
markets, however, there is a level of aggregation at which risk does not
exist or is very small. In this case, hedging has a different rationale as
for each movement there are winners and losers. Hedging is a stabiliza-
tion device as risk can be mutually exchanged. In this case, market com-
pleteness acquires a different meaning. In fact, in a complete market,