delta hedging strategy for OTM option

Question

Wondering how you would think about the following thought experiment - suppose you sell an OTM call option and plan to implement a delta hedging strategy whereby if the price of the stock were to increase and reach the strike, you hedge 100% by buying the shares at the strike price, and then subsequently if it were to come back down you take off the hedge - i.e. the hedge is always 100% or 0%. Assuming you could do this without any transaction costs, then you could just collect the premium from selling the option and guarantee you will be hedged at expiry.

There must be a flaw in this logic as it would imply a free collection of premium, so I am wondering where that flaw is and how it might relate to discrete vs. continuous time hedging.

Answer 1

suppose you sell a K = 105 call. When the stock reaches exacty 105 you buy 1 stock at 105. Now suppose the stock moves to 104.99, using your logic you sell 1 share at 104. You lost $0.01.

Again, after a while stock reaches 105 you buy 1 stock. After some time it goes up, but eventually it goes down again below 105. Thus you sell 1 share below 105. Again realizing a loss.

rinse repeat. You bought high and sold low.

Hope this helps.

Answer 2

While implementing this strategy , you're taking risk of downside movement of your long equity position, as others have pointed out.

If your market is quite volatile , and you can use the stop losses effectively , you can have minimal profit.

Answer 3

The problem is that when price is "back down" and "you take off the hedge", at this moment you already lost the premium but you still have that risky short position in call.

Answer 4

To add to the above answers , there is also an overnight risk which can expose you to unlimited loss . If the market opens way beyond beyond your strike/breakeven you will not be able to implement the hedge the way you earlier thought you could.

Answer 5

Delta hedging implies, loosely speaking, buying a proportion (delta) such that small movements in underlying have no net impact. What you have done with 100% and 0% is, in effect, bought the shares to COVER your position, if the deal goes south.

Let's work this out with an example. Say there is a stock trading at \$1 and you WRITE a call with strike \$10. You dont own the stock so this is a naked call. Let's be generous and get 50 cents premium on this call.

Now, if the price of the stock hits \$10 (or more), you are on point and are able to buy the stock at \$10. At this point, assuming the stock is above \$10 and this is American, the BUYER of the call will exercise and you will sell him the stock at $10 per share. NET: +\$0.50 - \$10 + \$10 = +\$0.50

if the stock never hits \$10 you keep the $0.50 per share premium because the calls expire. If the stock goes over $10 at expiration, then you better hope you were able to cover at $10, otherwise you incur losses at any point above $10. Theoretically, you could never lose money if you are able to cover exactly at the strike price of short option (assuming nominal transaction costs). But you would have given up all the gains from $1 to $10.