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).