Let us consider a protective put strategy. We hold an asset, with value (S_{0}=Sleft(t_{0} ight)), but we

Let us consider a protective put strategy. We hold an asset, with value \(S_{0}=S\left(t_{0}\right)\), but we are concerned with a possible loss over the holding period \(\left[t_{0} T\right]\). One way to hedge risk is buying a put option with strike \(K\). Then, the overall portfolio value at maturity is the sum of the asset value and the option value,

\[S_{T}+\max K \quad S_{T} 0=\max K S_{T}\]

If we look at the total payoff, it seems that the larger the strike, the better. Clearly, this is too good to be true. Indeed, we should not forget that the protection from the put option does not come for free, and it is a safe guess that a put option with a larger strike price will be more expensive, too. On the contrary, hedging with forward or futures contracts can be achieved at no initial cost. However, we give up the whole upside potential (if \(S_{T}\) grows), whereas this is partially retained by hedging with options.