1.18. Alice buys a put option from Bob (see Definition 1.13), strike price X = $39 , for some fair price P which you are to determine by arguments similar to those employed for call options. Develop one step of a binomial model estimating the value of the put option.

1. First, show that at the expiry of the put option, when the asset has price S, the put option has value P = max{0,X−S} by considering the cases X < S and X> S.

2. Second, at the start of a year Alice constructs a portfolio of one bought put option, value P, and hedges by buying φ units of the asset (selling if φ is negative), each of price S = $60 . Explain the value of the portfolio at the end of the year if the asset may either double or halve in value. Deduce the risk-free hedge ratio φ.

3. Lastly, use the principle of arbitrage to show the fair price P for the option;

assume inflation is rampant and that bonds increase in value by 20% over the year.