Delta hedging

CFA Level III Derivatives
#1

Can someone help me understand the concept of delta hedging? CFA vol 5 SS 15 ready 37.

buy delta*shares for each short call option, so u will earn riskfree rate. e.g. delta =0.50 and short 200 calls. I need to buy 200*0.5= 100 shares. If shareprice goes down by 1, I will lose 100 on portfolio and option will decline by 0.50 each so also 100 for the position, this loss in value will be a gain for the shortposition, so eventuelly breakeven.

the delta is pricechange of option due to price change of underlying. But if I go short the calls, I collect my premiums. So if the price of underlying change, how to I gain from the the short call position? Can someone explain?

#2

hedging covers the risk of loss . If you are short call options you will lose when the market rises above the strike ( you will have to reimburse the call holders for the moneyness in the option ). The way you hedge this loss is to go long the stock , so when the stock appreciates and you have to pay on the option you can select the gains on the stock and be neutral .

if the stock appreciates $1 , your position in the call option loses $0.50 or a loss of $100 on the call options.

If the stock appreciates $1 , you are richer on the stock by $100 .

So your gain and loss are covered and you are neutral ( hedged)

#3

A long call position has unlimited profit and conversely a short call position has unlimited loss.

You take short call positions (i.e. you sell ) because you believe the underlying will not exceed the exercise price. But to protect yourself from theoreticaly unlimited loss or practically a huge loss in case you’re wrong, you hedge your short call positions by covering it with the actual underlying by the quantity of delta shares. Your call and equity positions will then offset in value. Your gain is the premium less transaction cost (for the stock position and without saying the original option position)

#4

You can buy back the calls at a cheaper price. The difference between the price at which you sold them and the price at which you can buy them back is your gain, which should exactly offset the loss on the underlying.

#5

S2000magician, when you delta hedge, do you have to match the hedge period with the maturity of the options being used to hedge? i.e. if I have 2,000 stocks that I wish to hedge for 3 months, do I have to short call options that have maturity of 3 months as well?

Thanks!

#6

Not necessarily. I suspect that you’d prefer having the expiration on the options to be longer than your expected holding period, though that may not always be possible.

You’re welcome.

#7

Using the same example, if I want to delta hedge a long stock position for 3 months, I can short 1 month calls 3 times as the options expire after each month, correct? This is why you don’t have to short 3-month calls?

I presume re-establishing positions like these 3 times would incur significant transaction cost compared to going short 3-month call once.

#8

There will be transactions costs, of course.

Additionally, when the options expire, there’s the chance that they will be exercised. That’d be bad.