# Gamma and delta neutral hedging

1- Gamma is largest when option is in the money. 2When stock price decreases , should we use more or less put option for the delta-neutral hedge and why? (this is from EOC )

Gamma tells you how delta changes when the stock price changes. Assume you have a call option on a stock with strike \$20 and the stock is trading now at \$5. Clearly delta is very small because even if the stock rises by \$1, the strike price is still far away…in fact the call option may not go up by even 5 cents if teh stock price rises \$1. If you calculate Gamma = Change in Delta / Change in Stock, it will be very small (because Change in Delta is very small as we have just seen).

Now suppose the stock is trading near \$20. Clearly delta is getting large because if the stock rises to let us say \$20, the option starts becoming attractive…it’s now at the money and may soon be in the money. Here gamma starts to get larger, near one. Assume the stock is now trading at around \$50 (way deep in the money). here delta is probably = 1.0…for every \$1 rise in stock price the option will gain \$1. But notice that if the stock rises again by \$1 to \$51, delta is still 1.0, so Change in Delta = 0, and Gamma is very small, zero in fact. So as you can see, Gamma is large when the stock is at the money (around \$20).

Now do this for the put and you’ll easily answer part 2.

Thanks a lot Dreary. I must say it was very very helpful…

I have a question from that reading. This is about problem 10 in reading 56 in CFAI. I understand that when underlying decreases, put option gains value and its slope is more negative. But why should one sell options to maintain the hedge constructed when underlying was high?

Thank you,

P

It helps to think about this as a real life situation…if you own 1000 shares of the a stock and it drops by \$1, you’re down \$1000. That’s bad and you should do something about it. One thing you could do is sell some options (calls or puts) against your stock holding. If you had done that you would’ve offset some of the \$1000 loss. So, how many puts should I sell? Divide 1000 shares by the delta of the puts (or the delta of the calls if you intend to sell calls). The theory is that the proceeds from the options sale will offset any dcrease in the stock price (but only up to a point).

So, to answer prodigal question: " …But why should one sell options to maintain the hedge constructed when underlying was high?", The reason is if the underlying continues to slide down, the proceeds you got from your earlier sale of the puts is not enough to offset further losses, so you have to sell more options, calls or puts, all the time…that’s why it’s called dynamic hedging. It is not perfect but it does work to a great extent.

Quiz: As the stock continues to drop, do you need to sell more or fewer put options than you did before?

Dreary, Just a note, when you hold stock you do not sell puts in order to hedge position, you buy puts. Put delta is negative.

jpsi1, that’s the idea behind delta hedging, that you don’t buy the puts but sell them. You are talking about something else, not delta hedging.

dreary if you hold a stock and sell puts, both of your positions will lose if the stock price goes down. This isn’t a hedge.

Come on folks! This isn’t about classic hedging. This is called delta hedging, and it works in the way explained above. If transaction costs are negligible and the volatility of the stock is not too big, you’d have a good hedge.

This topic is making my head spinning Any way to understand it conceptually Dreary?

dreary, you’re wrong on this. selling puts will NOT hedge a long stock position.

the idea of delta hedging is to hold a combination of stocks and options in the right proportion so that if the stock price changes, your portfolio value doesn’t change.

ie you need a combination where one wins when stocks rise/fall, and the other loses.

Folks, stop thinking about perefct hedging and you’ll get this quickly. In a perfect hedge, you pay money to get the protection you need. For example, you own stock and then you buy (i.e., spend more money) to buy the insurance via a put. With delta hedging, you do not want to pay for insurance and you are willing to take some risk in case dynamic hedging doesn’t work very well.

For example, buy 1000 shares of stock for \$20 a share, that’ll cost you \$20,000. Now you can immediately offset your cost by selling some calls or puts. When you sell an option you get paid. Let us say your put delta =0.50, with a strike of \$20, so if the stock drops by \$1, the put will rise by \$0.50, and asume the put is selling for \$1 right now. Then you know that 1000/.0.50 = 2000 puts, i.e., you need to sell put 20 contracts, that’ll get you \$2000. So, you investment is costing you only \$18,000. That’s the first sign of a hedge right there.

The dynamic part comes when the stock drops again or rises back to \$20 or more, etc. When these things happen, delta changes (something you can measure the rate of using gamma by the way), and now your hedge may be in danger, meaning if the stock drops by \$2 to \$18, you are at risk. Why? Because your investment will be worth \$18000, but then the puts you sold are in the money, so if it’s near expiration, you may have to pay \$2 to the put buyer. That means your investment is worth only \$16000 (versus \$18000 net investment at the beginning). So, here because of the big drop in stock price your delta hedge didn’t work very well, and so that’s why you have to dynamically adjust that by issuing more puts to continue to hedge.

If the stock drops to \$19 and it’s expiration day, how has delta hedging helped? You try it.

Dreary I’m sorry but you have this totally wrong. Maybe read the book again.

Hedging is about reducing/eliminating your exposure to the underlying. (stocks, currency, etc.) Surely you can see that in your example above, the investor holds two positions (long stock, short put) that move in the same direction when prices change? That is adding exposure, not reducing it.

http://financial-dictionary.thefreedictionary.com/_/dict.aspx?word=Delta+Hedging

God knows I tried

I give up.

A JPM guy just lost \$2 billion in a bad hedge…who are we to do better? huh. http://dealbook.nytimes.com/2012/05/10/jpmorgan-discloses-significant-losses-in-trading-group/?nl=todaysheadlines&emc=edit_th_20120511

Hey Prodigal,

this is a tough question… hopefully this clears it up for you

from question 10 from the CFA options book, it says we are going to buy put options to hedge our stock price drop.

we want to completely neutralize our stock price change, so we care about delta hedging it rather than just buying 100,000 put options to create a floor… a floor position would would not completely neutralize our position, but protect against a drop in price below the strike on our shares.

what the question asks us to do is use option z, with a strike price of 36. shares are currently trading at 38, so this option is currently out of the money. now we know that an option that is out of the money has a lower delta than an option that is in the money. how do we know this? Intuitively, as an option becomes more and more in the money its delta approaches 1 for a call and -1 for a put to the point where every single \$1 move in the stock is going to have the exact same price move on the option. now lets make up some numbers… (they give you numbers for delta, but lets make some up anyways). lets say as we stand at 38 per share, with a strike of 36, we can assume the put option delta is -0.4. In order to hedge our shares, we will need to buy 100,000 / -0.4 put options, for a position of -250,000 put options (negative means we are long, so if price drops, we make money). if now the stock drops to 36, our put delta will approach negative 1 (because it is becoming more valuation) so lets assume the option delta is now -0.5. now with our 250,000 put option position, a move further down in the share price will cause more of a gain on our option position than a loss on our stock. so what we need to do is “rebalance” our option position. Our new option position with delta -0.5 is 100,000/-0.5 = -200,000, so basically our option position changed from 250,000 to hedge 40 delta, to 200,000 to hedge 50 delta. so in essence, we need to sell 50,000 put options in order to rebalance our delta hedge.

My bad. Sale of puts was related to rebalancing of the hedge not to initial hedged position (long stock and long puts).

People!

Thank you for all your comments. I got it. Carthurj, thanks for pointing out that it is indeed a hard question. Felt I was alone in thinking this way. Thanks for the explanation. I like the way you think.

P

long stock and long puts? That’s not delta hedging. Oh I forgot, I said I would just give up.