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aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 10:27
Hi!

I got a question regarding the P&L of a delta hedged long Straddle position.
Is it equal to the P&L of a delta hedged option, i.e.

(daily),

(at maturity)
respectively?

Is it the same if i assume that \Pi is my portfolio consisting of the long straddle and a position that delta hedges this portfolio?

In practice, this works, but how about in theory?

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-02 11:06
Call put parity says that you can do a straddle by buying a call and a put or 2 calls and sell a stock or 2 puts and buy the stock

so the pnl of the three is the same yes. (different cashflows though as by selling the stock you have a positive cashflow when you enter the trade)

Qui fait le malin tombe dans le ravin

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 12:21
Thanks for your response.
I'm glad you agree to my statement, but i must admit that i can't follow your argumentation.

In practice, this works, but how about in theory?

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-02 12:29
my argument is a delta hedge straddle is the same as 2 delta hedged options (calls or puts doesn't matter).

so the pnl has to be the same

Qui fait le malin tombe dans le ravin

AIC


Total Posts: 167
Joined: Apr 2008
 
Posted: 2008-07-02 13:18

just watch out for deep in the money americans.


Train yourself to let go of everything you fear to lose- Master Yoda

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 13:28
This makes sense. I understand.

My next question is now. Why should i buy a straddle if i want to trade volatility? A delta hedged option leads to the same P&L and moreover is much cheaper.

In practice, this works, but how about in theory?

tbone


Total Posts: 6
Joined: May 2007
 
Posted: 2008-07-02 13:31

"my argument is a delta hedge straddle is the same as 2 delta hedged options (calls or puts doesn't matter)."

Is this true in the real world?

If the market tanks, doesn't a hedged ATM straddle underperform two delta hedged ATM puts due to skewness of returns (limiting argument to equity index futures)?


"Giving money and power to government is like giving whiskey and car keys to teenage boys." - P.J. O'Rourke

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-02 14:35
Tbone, if the options are european (or american but very unlikely to be exercise as cokehead mentions), it is strickly the same pnl.

and, as far as i know, it is true for real world, it is a static hedging argument. It just the call-put parity after all.

an hedged option is cheaper?
I don't think that buying puts and hedging them with stock will be cheaper then buying a straddle.

Qui fait le malin tombe dans le ravin

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 14:52
Why do you think that buying a put or call will not be cheaper then buying a straddle?

First of all you have to buy two options instead of one.
And second you have a higher Delta, i.e. the delta of the straddle is 2 times the call delta. So you have to buy more stocks.

In practice, this works, but how about in theory?

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-02 14:59
first comment this belong to basics and not pricing.

aschon, pay attention:
* when you buy a straddle, you also buy two options, a call+ a put.

*"the delta of a straddle is twice the delta of a call".
This is just wrong

*you say cheaper but I dunno what that mean in that context:
You mention buying a delta hedged option, right? so you should also take into account the cost of trading the stock.
If you do with puts you need to buy the stock so bigger negative cashflow, if you do with calls you need to sell stocks so positive cash flows.

Qui fait le malin tombe dans le ravin

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 15:20
Of course. You're right!

1. The delta of a straddle is twice the delta of a call minus 1.
2. Put => buy the stock (-), Call => sell the stock (+)

I must have been a bit confused.

Thanks for helping me out!

In practice, this works, but how about in theory?

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 15:27
One question is left.

Why trading straddles instead of delta hedged options when taking a long/short position in volatility? Since P&L is the same.

In practice, this works, but how about in theory?

Dimatrix


Total Posts: 539
Joined: May 2006
 
Posted: 2008-07-02 15:42


In a straddle you have two options with the same vega, in a delta neutral you have one option with a vega. The stock doesn't have vega.

Ctrl - L.

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-02 15:44
why? there is no absolute reasons why,
Maybe the absolute value of the cashflow paid upfront is smaller so people prefer it.
Maybe to save some transaction costs wrt to the stock, even though if you plan on delta hedge it...
Maybe they have tighter quotes on the straddle.

din: the vega of 2 options+ stock or 1 straddle will be the same so i don't think it will matter here

Qui fait le malin tombe dans le ravin

Johnny
Founding Member

Total Posts: 4333
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Posted: 2008-07-02 15:45

aschon, the answer is that it usually doesn't matter. As Balt said (and so did you actually) right at the start of this thread.

edit: crossed with Balt


From the subprime to the ridiculous

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 17:34
Another thing just came into my mind.

What looks the P&L of a delta-gamma-hedged portfolio like?

In practice, this works, but how about in theory?

Dimatrix


Total Posts: 539
Joined: May 2006
 
Posted: 2008-07-02 17:42


the vega of 2 options+ stock or 1 straddle will be the same so i don't think it will matter here

Oh, I didn't see that he had 2 calls in his neutral position.



Ctrl - L.

DocAdam7


Total Posts: 153
Joined: Nov 2007
 
Posted: 2008-07-02 17:46
I assume you mean statically delta and gamma hedged, meaning that your portfolio sensitivities will change presumably as the underlying changes. In that case the PnL depends on whats in your portfolio, not what your greek sensitivities are at a static underlying level.

That which counts cannot always be counted. That which can be counted does not always count.

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-02 18:34
Sorry, but i dont understand your point.

In practice, this works, but how about in theory?

DocAdam7


Total Posts: 153
Joined: Nov 2007
 
Posted: 2008-07-02 18:37
Just because your portfolio shows zero gamma and zero delta doesn't mean it's flat. Say you're long one atm call and short some number of 20 delta calls against it assuming this portfolio has zero gamma and zero delta. As the underlying moves higher you're going to get short gamma and short deltas. Assuming you hold no position in the underlying and aren't hedging dynamically (just looking at a PnL diagram) you're going to approach a position thats just naked short X number of calls.

That which counts cannot always be counted. That which can be counted does not always count.

AIC


Total Posts: 167
Joined: Apr 2008
 
Posted: 2008-07-03 08:56

for stock options i suggest buying delta hedged ATM puts for long vol positions and short calls (delta hedged) to get short vol. reason for the strategy being you dont have to short stock which can get tricky sometimes.

Baltazar...i know we talked about not being able to short stock before. What happens to european (for simplicity) puts & calls on the same strike for such a stock?

 


Train yourself to let go of everything you fear to lose- Master Yoda

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-03 09:18
well i dunno, I guess you need to find an instrument correlated with the stock and short that instrument instead.

otherwise you cannot use the parity directly. But this also mean you cannot delta hedge a call with stock, only with synthetics. I wonder if you see a lot of option traded on that type of stocks.



Qui fait le malin tombe dans le ravin

DocAdam7


Total Posts: 153
Joined: Nov 2007
 
Posted: 2008-07-03 15:03

When shorting becomes a problem the conversion/reversal values can get WAY out of whack. In names that are virtually impossible to short (like some biotechs get, I think overstock.com was like that, some of the now defunct mortgage lenders, etc..) the marketplace can wind up paying to convert and collecting to reverse, even with no dividend stream. At that point the actual short stock rebate rate doesn't even matter since getting long stock becomes so important. It becomes outside the normal model universe. These are things you'll never hear about, learn about, or see in academia - only in practice.


That which counts cannot always be counted. That which can be counted does not always count.

aschon


Total Posts: 164
Joined: Jun 2008
 
Posted: 2008-07-03 17:53
Sorry, but I have to return to my question concerning the P&L of a delta-gamma-hedged portfolio once again.

Is it the same as that of a delta hedged portfolio?

Gamma hedging requires the use of a non linear instrument, ie an option. So the portfolio will be accumulate with options and P&L will be the same. Right?

In practice, this works, but how about in theory?

Baltazar


Total Posts: 1764
Joined: Jul 2004
 
Posted: 2008-07-04 06:43
This is a puzzling question.
in a bs world you don't need to gamma hedge since continuous delta hedge is enough to remove all the risk relative to stock.

If you take into account discreetness of the delta hedging scheme, it makes sense to hedge your gamma but i don't think you can come up with a simple pnl formula. I guess it depends if you plan on gamma hedging once or rebalancing this hedge.

but to answer your question, no the pnl will not be the same:
your profit is a direct function of gamma times the difference between implied vol and realiwed ones. Reducing gamma reduces this pnl.

Qui fait le malin tombe dans le ravin
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