Mathematics Achieves Command of Financial Risks
The Mathematics
The analysis of financial derivatives involves mathematical
developments in probability, statistics, partial differential
equations, optimization, and computational methods. The
impact of finance problems on these fields, in terms of teaching
as well as research, has grown rapidly in recent years. Financial
mathematics has become a significant component of many university
curricula in the past ten years. Typically, it is taught
in mathematics, statistics or operations research departments,
at all levels, from undergraduate studies, professional Masters
programs to PhD research topics.
Starting with wealth 0, we
want to finish with wealth +100 if the NY Yankees win the series,-100
if the LA Dodgers win the series.
Let's work backward!
Suppose that after 6 games, we have a 3 - 3
tie. The winner of game 7 wins the series. The only way to
guarantee the desired post-game-7 wealth (+100/-100) is
to have pre-game-7 wealth 0,
and to bet 100 on
NY to win game 7.
Now suppose that after 5 games, NY leads 3 - 2.
If NY wins game 6, they will win the series; in thatcase we want
wealth+100.
If LA wins game 6, then we will have a 3 -3 tie;
in that case we want wealth 0 as discussed above. The only way
to guarantee the desired post-game-6 wealth (+100/0) is
to have pre-game-6 wealth 50,
and to bet 50 on
NY to win game 6.
Likewise, if after 5 games, NY trails 2 - 3,
then the only way to guarantee the desired post-game-6 wealth (0/-100)
is to have pre-game-6 wealth -50, and to bet on 50 NY
to win game 6.
Continuing inductively, we find that if after 1
game, NY leads 1 - 0, then we will want to have wealth
31.25,
but if after 1 game,
NY trails 0 - 1,
then we will want to have wealth -31.25 .
So we must bet 31.25 on
NY to win game 1.
This wagering strategy is an example of replication or hedging
of a derivative asset (the World Series bet) using basic assets
(the bets on individual games). The computational technique that
we used to calculate the replication strategy is known as backward
induction in a binomial tree.
where $\mu$ and the volatility $\sigma$ are constants.
Loosely speaking, it states that during a small time interval of
length dt ,
the proportional change $\dd S/S$ in the stock price St is normally
distributed with mean 
and
variance .
Under these assumptions, Black, Scholes, and Merton used techniques
from \emph{stochastic calculus} and \emph{partial differential
equations} to show that the value of the call option at any time t<T must
be C(St,
t) where the function C is
defined by 
.
where K is the strike ($120$ in our example), and $N$ denotes the
standard normal cumulative distribution function, and for simplicity
we have assumed zero interest rates.
The number of shares to hold in the replicating portfolio at each
time t $t$ is
Intuitively, if the call option value moves $40$ cents for each $1$ dollar move in the underlying shares, then we should replicate the option by holding $0.4$ shares.
This trading strategy -- known as \emph{delta hedging} -- is another example of replication or or hedging of a derivative asset (the call option on Amazon) using basic assets (shares of Amazon stock).
