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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?

Set
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  • I originally posted this on quant stack exchange where it was deemed inappropriate, but now I realize I probably should have posted it on finance stack exchange and not on here. – Set May 23 '13 at 06:51
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    In this case - single period, binary stock model - there is no need, as you point out replicating its behaviour with stock and debt is simple. This is often used to explain option valuation basics. In more complicated cases, it is not so trivial to create a portfolio of stocks and bonds that replicates an option. – Macavity May 23 '13 at 06:52
  • @Macavity, is it always theoretically possible though? – Set May 23 '13 at 06:54
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    In most cases, yes. However practically, imagine creating such a portfolio for the continuous time case, where every instant you need to rebalance the portfolio, and instead consider the ease of providing a call option! – Macavity May 23 '13 at 06:56
  • Interesting I'll have to think on that some, thanks. – Set May 23 '13 at 06:58
  • You may also want to think about a tri-state one period model, where there are three possible states for the stock in one period. Here just debt and the stock are not enough to provide a complete basis, so there will be derivatives you can construct which cannot be mimicked by just those. Perhaps you should check at the fin site for more e.g. – Macavity May 23 '13 at 07:03

1 Answers1

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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.

Don Shanil
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