How to adjust delta hedging if stock price decreases?

Question

Question: You are long a call option no MITCO stock. You have delta hedged your position. You hear on the radio that the CEO of MITCO has just been arrested for running a massive Ponzi scheme. The stock price plunges $10. How do you adjust your hedge (qualitatively)? That is, do you borrow and buy stock or sell stock and lend? Explain carefully.

I know that delta is the rate of change of option's value with respect to stock price.

Since stock price decreases, so is delta.

But I do not know how to adjust the delta hedging.

Answer 1

You would be over hedged in your call position if it was delta neutral before the stock cratered. Since you are long delta on the call, you would have shorted stock to make the original position delta neutral.

When the stock fell, your long call delta would have fallen, and you would buy to cover some of your short stock hedge. However, being long the call, you would have been long gamma. As such, this trade should have been a great trade for you.

Your position would have been overhedged. But as you were short the stock, your short delta on your hedge would not have been changing as the stock tanked. Meanwhile, your delta on your long call would have been decreasing. As a result, your short delta would have been increasingly short as the stock was crashing--just the way you want to be positioned as the stock is tanking.

You would also have been long Vega in your long call position. As the stock tanks, the implied volatility should have gone up which would have benefited your long call position slightly. Also, skew would have worked in your favor.

Answer 2

You are long a vanilla option, so long gamma (positive gamma). If the stock price decreases, so does the delta of your option.

Since you short-sold the stock to hedge, you now have short-sold too much since delta has decreased. As a consequence, you must buy back some stock.