Forums  > Risk Management  > How many shares to delta hedge a long at-the-money option?  
     
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unscientific


Total Posts: 7
Joined: Sep 2014
 
Posted: 2015-01-06 19:08
An option that is at-the-money would have a delta of +50 if it's a call option, -50 for a put option.

Suppose you long a call option and it is currently at-the-money, so delta = +50. To hedge your delta, you short 0.5 stock to make overall delta = +50 - (0.5*100) = 0.

For a long put option, you buy 0.5 stock to make overall delta = -50 +(0.5*100) = 0.

However, why question 2.40 from the book "Heard on Wall Street" suggests 1.5 stock?

athletico


Total Posts: 951
Joined: Jun 2004
 
Posted: 2015-01-06 19:57
Read that answer again very carefully -- it does not suggest "1.5 stock".

silverside


Total Posts: 1408
Joined: Jun 2004
 
Posted: 2015-01-06 20:01
Also, I'm not sure if the delta of an ATM option is exactly 0.5

It is often useful to fall back on put call parity: C-P==Fwd-PVStrike.

silverside


Total Posts: 1408
Joined: Jun 2004
 
Posted: 2015-01-07 00:41
It can be a surprisingly deep question ...

I suppose the "delta ATM=0.5" is completely correct for a pure Normal diffusion, is roughly correct for Black-Scholes for short expiries, and in the presence of skew, is model / skew dependent ( if you define delta as "how many units of the stock should I buy for a locally minimum variance hedge"). For example, a local-vol model and a stoch-vol model are likely to give different deltas.

But then no real-life option remains ATM for long (pin risk)

I hadn't actually thought about this in a while but I'm sure it's discussed in various books (Rebonato, etc)

Baltazar


Total Posts: 1760
Joined: Jul 2004
 
Posted: 2015-01-07 08:50
And if asked in interview, maybe bring up the fact that ATM can mean different thing (ATM = delta 50? ATM means K=S ?, K=F?).

Indeed vols dynamics will impact your delta so maybe it is good to nuance BS deltas versus complete deltas.

Qui fait le malin tombe dans le ravin

unscientific


Total Posts: 7
Joined: Sep 2014
 
Posted: 2015-01-07 11:27
Of course, I misread one-half as 1.5. Silly me!
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