Solutions

World Series

Starting with wealth 0, we want to finish with wealth if the NY Yankees win the series, if the LA Dodgers win the series.

Let's work backward!

Suppose that after 6 games, we have a tie. The winner of game 7 wins the series. The only way to guarantee the desired post-game-7 wealth is to have pre-game-7 wealth , and to bet on NY to win game 7.

Now suppose that after 5 games, NY leads . If NY wins game 6, they will win the series; in thatcase we want wealth. If LA wins game 6, then we will have a tie; in that case we want wealth 0 as discussed above. The only way to guarantee the desired post-game-6 wealth is to have pre-game-6 wealth , and to bet on NY to win game 6.

Likewise, if after 5 games, NY trails , then the only way to guarantee the desired post-game-6 wealth is to have pre-game-6 wealth , and to bet on NY to win game 6.

Continuing inductively, we find that if after game, NY leads , then we will want to have wealth , but if after game, NY trails , then we will want to have wealth . So we must bet on NY to win game .


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.

 

Call Option

Unlike the case of the World Series bet, the solution depends on assumptions about the probability distributions of the underlying random variables. (In the World Series bet, the actual probability that NY wins each game was irrelevant!) The assumption of the Black-Scholes model can be expressed using the stochastic differential equation

where $\mu$ and the volatility $\sigma$ are constants. Loosely speaking, it states that during a small time interval of length , the proportional change $\dd S/S$ in the stock price 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 must be where the function is defined by .


where 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 $t$ 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 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).