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Assume I write a call option on one share of the stock that I have. After selling the option I have an obligation to sell one share of the stock at some future time. I already have the stock, why would I want to hedge anything here? The idea of hedging is meaningless in this example. Is there any other set up where the party who sells the option has to hedge the risk it gets exposed to (that is, obligation to sell the underlying)?

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    $\begingroup$ How do you define hedging here? If you are short one option but long one unit if stock, then you're only well hedged when the option is deep in the money. The rest of the time you're long delta and short gamma and vega. $\endgroup$
    – will
    Sep 7, 2020 at 8:46
  • $\begingroup$ You would want to hedge because your instantaneous profit & loss is everything but null and your unrealized profit & loss is everything but secured (frozen). $\endgroup$
    – Lisa Ann
    Sep 7, 2020 at 9:52

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I think you have a small misunderstanding. The hedge for a call (or put) is rarely 1:1 with stock. When you are selling this option you are actually selling the future movements of the stock and specifically the future price jumps that will happen.

To hedge an option you would enter into a position that will offset the movements in price of the option you sold. In your case you are not (most likely) hedged.

Here's a simple example:

  • You buy 1 sh of stock for \$100.
  • You then sell a \$110 strike November call for \$1.
  • The price of the stock goes steadily down to \$90 by November expiry.

You made \$1 on the call you sold but then lost \$10 on the stock (less divs and + any borrow costs). So you weren't hedged at all.

The right (or better) way to look at this is that when you sell an option you need to look at the exposure that your option's price has to moves in its underlying asset. In the case above:

  • When you sell that November call, determine what is your exposure to a move up or down if the stock is to move the average of it's one day volatility.
  • Say that the average up/down daily move is 2%. So then you need to figure out what you expect the price move of the option to be if the stock goes up or down two percent.
  • Hedge an amount of stock that will give you a matching gain or loss to the option's gain or loss.

So in your case you have way to much stock. You are not hedged (as you can see by the simple example above).

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First of all, option contracts normally specify 100 (or in some cases, 1000 contracts of the underlying instrument) to the one option contract, so you are unlikely to encounter the 1:1 scenario you mention.

Let's ignore the ratios for now, and look at the risk profile portfolio you described (1 long underlying and 1 short call).

The portfolio you have is a synthetic SHORT PUT

This is the profit profile of a short put:

  1. Price of underlying > call strike : keep call premium collected + (potential) profit from selling underlying
  2. Underling price purchase price > Price of underlying < call price : slightly positive
  3. Price of underlying < Underling price purchase price : Potentially "unlimited" losses - well till price goes to zero

So your portfolio needs to be hedged, otherwise you have vulnerable to downward price movements of the underlying.

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