Delta of Futures and Forward

[Liming]

Dear David,

I don't understand why delta of forward for non-dividend stock could be 1 when the forward price equals to (So)exp(rT), meaning every unit change in So should be multiplied by exp(rT) to really affect the forward price.
As to futures contract, I think that due to the daily margining practice, gains or losses arising out of daily stock movement can be immediately realized, (ie. $1000 underlying stock loss translates directly to $1000 loss to investor's margin account), shouldn't the delta of futures equal to 1?

Do these relationships also hold for dividend-paying stock?

Thank you very much for your enlightenment!


Cheers!
Liming

10/09/2009

 

[David Harper CFA FRM]

Hi Liming,

Ha! This was a long discussion last year (I cannot find the thread??) and, frankly, I was neither unable to defend Hull nor convey a persuasive explain! I fear I may not understand...my own library fails me here ... can nobody rescue us?

For the forward, Hull uses the *value* of a forward

value of f = S - delivery*EXP(-rT)
on the math, the derivative of this is simple:
df/dS = 1.0

then for the futures, he switches (sly master!) to the *price* instead of value:
F = S * EXP(rT)
df/dS = EXP(rt)

(in my opinion, as long as the formula for the foward/futures price is exactly the same, the delta with mathematically be the same...and using Hull's they would both = 1.0)

Liming, the best i can do with my intuition is lean on the daily settlement. Just as you say, in the case of the future, the $1 is "received" immediately. So the key difference is, given a +$1 spot:
forward: +$1 at end of contract
future: +$1 immediately

in present value terms,
forward = EXP(-rt)
future = 1

in future value terms:
forward = +1
future is EXP(rt); i.e., invest dollar at r

(so, to be honest, my intuition leads to delta future = 1 and delta forward = exp(-rt) ... but that is not helping matters!)
...sorry it is not the enlightment you are looking for!

regarding dividends, says Hull (for div yield = q)
forward delta = EXP(-qT)
futures delta = EXP((r-q)*T)

David

 

[Sidharth]

Hi David

I have a doubt here..

if Stock price increase by $1 then forward increase in value ,($1),will not be given to us ...but will stay in contract...and should not that money would grow by risk free rate ...as we always assume.....

then also payment at the end of time will be equivalent to $1*exp(rT)....which is payment in case of future also....which make both delta equal..


what do you think ??