1.17. Alice buys a put option from Bob (see Definition 1.13), strike price X = $57 , 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 to estimate the value of the put option.

Definition 1.13. A European put option gives the buyer (of the option) the right but not the obligation to sell an asset at a previously agreed strike price on a particular date.

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 goes up in price by 25%, and if the asset goes down in price by 20%. Deduce the risk-free ???hedge ratio φ = 13.

3. Lastly, use the principle of arbitrage to determine the fair price P = $4 for the option; assume bonds increase in value by 416

% over the year.