Why are call options necessary?

Question

My question is actually less ambitious and more specific then the title may have lead you to believe.

Suppose the interest rate is $25\%$ you have a stock at time zero price of $S_0=50$ and at time 1 its price will either be $S_1=25$ or $S_1=100$. I offer you a call option for a price of $60$ to buy three units of this stock at time 1 at the price of $50$ each, so $150$ total.

My understanding is that the reason you consider buying this option is to reduce your exposure to upward volatility in the stock price (I guess that's what a hedge is, sorry I'm new to this).

I take this $60$ dollars and borrow another $40$ from the money market, buy two units of this stock for a total price of $100$ dollars, and then if the price of the stock increases at time 1 I have to buy one more unit of stock at $100$ dollars, which puts me in the hole $1.25(40) + 100 = 150$ which I then sell to you and get $150$ back and thus make or lose nothing. If the price drops I sell my two units of stock for $50$ to pay off my $1.25(40)$ debt and again I make or lose nothing.

So my first question is why do I even bother selling you this call option. It involves no initial investment on my part, but there is no chance of risk or reward either, so why waste my time?

Second why do you bother buying the call option from me? Instead why don't you take your $60$ dollars, borrow another $40$ from the money market yourself, and then do the same thing I would have done with that $100$ and construct your own hedge?

Answer 1

This is a good question.

As @Macavity states, when jumping to the continuous model things are more complicated. Another important point, is that in practice, it may not always be possible to replicate due to the following reasons:

• Depending on the markets you are in, it may not always be possible to buy or sell the replicated assets when needed (liquidity issues)
• For legal reasons, it may not be possible to always trade in the stocks that you need to replicate the claim with.
• When you hedge say against some underlying asset, the assets in your portfolio may for practical reasons be only loosely correlated with the underlying -- its easier to buy an option than work out the exact relationship between the prices in your hedging portfolio and the asset you are trying to hedge against.

Hope this helps.