# Are European call and put option useful ? [Cox-Ross-Rubinstein model]

I'm new to the world of option market, but after having studied CRR model I'm wondering if European call and put option are very useful since a talk with my professor that piqued ma curiosity. In the following I focus on call option. I mean, from what I have understood there are 2 uses of call option : 1) limit the amount one will have to pay for a given underlying asset 2) try to make money based on belief

Now, in order to price the option in an arbitrage free and complete market, the seller will take as price the initial value of a strategy but nothing says the seller has more information than the buyer of the call, so why the buyer cannot just buy this initial strategy in order to replicate the payoff of the call option ? This applies also for 1) since even if the buyer want to limit the amount he will pay at a given time $$T$$, by selling his strategy at time $$T$$ he will be able to limit this amount by clearing his position (and so he will be able to use the difference between $$S_T$$ and the srike $$K$$ if the price of the underlying has increased at time $$T$$)

I hope my explanations are clear, I would like to know if I'm right or not so don't hesitate to correct me

Thank you a lot

• This is the famous question whether or not "options are redundant securities". Undoubtedly in the Black Scholes theory options can be exactly replicated and so are redundant. However there are numerous papers arguing that the real world does not exactly follow this theory and therefore options may not be redundant after all. Keep in mind that the Black Scholes theory requires specific assumptions. The existence of large option makets suggests someone finds them useful. IMO there is no simple yes/no answer to your question, it is an open research area with papers to read... Jun 3, 2022 at 13:23
• If markets are complete, we do not need options (they are meaningless and add no economic value). If markets are inefficient, we may need options. However, if we need the assumption of complete markets to value an option, it is in fact impossible to price them. So either, we do not need them, or cannot price them. That's called the Hakansson's paradox. Jun 3, 2022 at 14:15
• In real world settings, replication is complex and tedious, while option pricing theories give you a convenient way to model / price / trade options. A somewhat sloppy analogy that may still help; you can always build cars yourself (replicate the steps needed) but it's way more convenient to rely on readily available products (transaction costs /overhead costs will also be lower). Jun 3, 2022 at 14:17
• Oh good, thank you a lot for your comments ! It helps me to set the record straight ! I will look the references you gave me ! Jun 3, 2022 at 14:48
• Hakannson's Paradox quant.stackexchange.com/a/223/16148 Jun 4, 2022 at 9:59