What's new

# What will be the Delta Normal VaR for a Long Call option & a short Call option

#### rahul.goyl

##### Member
VaR of a long call option (delta normal)

Delta = .75
Position= 50
price= 8
Vol= .25

VaR of a Short call option (delta normal)

Delta = .75
Position= 50
price= 8
Vol=.25

Please revert with calculation, I wish to know if the VaR for both is excatly same.
Don't we have higher VaR for short call compare to long call.

#### ShaktiRathore

##### Well-Known Member
Subscriber
long call P/O= max(S-X,0)
from A, vol(short call)=-vol(long call)...1
also from A above, delta(short call P/O)=-delta(long call P/O)...2
1 => Value*vol(short call)=- Value*vol(long call)....3
Multiply 2 and 3 =>
delta(short call)*Value*vol(short call)= Value*vol(long call)*delta(long call)
or z*delta(short call)*Value*vol(short call)= z*Value*vol(long call)*delta(long call)
or VaR of long call= VaR of short call

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I added two follow-up questions, to this problem, to create an "FRM fun question:" http://www.bionicturtle.com/forum/threads/frm-fun-17-stock-option-var-p1-only.6128/

ShaktiRathore shows that the VaR of long call = VaR of short call. Here are two follow-up questions:
• We are not given the option price (or the strike price, for that matter). If this were a computation of bond dollar VaR, for example, we'd need the bond price. Yet this seems to compute an option (dollar) VaR without an option price. Why is this possible, or is the problem incorrect?
• Assume we added an additional assumption: option Gamma = 0.05. If we extended the VaR to include gamma, will the VaR of a long call still equal the VaR of a short call?

#### caramel

##### Member
For Shakti
long call P/O= max(S-X,0)

Payoffs do not include premiums. Only maximum gains and losses
I guess it would be better to say that
long call P/O= max(S-X,0)
short call P/O= -max(S-X,0)
hence short call P/O=- long call P/O

#### ShaktiRathore

##### Well-Known Member
Subscriber
Caramel its just for the bringing out the understanding sake of it that Var of long call= Var of short call however i have not tried to put emphasis on whether the formula i used for deriving it is essential. But in the end as you said there may be this error in this formula which i think in the end shall not affect the result. You can ignore the premium in the P/Off.
Caramel wish you understand me thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I gave caramel a star because IMO she makes an excellent point, IMO. The last two three of Shakti's "proof" appear to nicely summarize the symmetry given by the long and short call only with respect to linear delta. (I do not mean to disagree with Shakti's conclusion) However,
• option profit = payoff +/- premium; by convention, without respect to time value of money
e.g., OTM short-call profit [at expire] = 0 payoff + premium received, and
• delta(short call P/O) could be confusing because the VaR of an option refers to the price risk of an option, where current option price = (current) intrinsic value + time value. It might be a little better to use delta(option price) = delta (intrinsic value + time-value) if only to explicitly make the point this is current price risk, not just payoff. thanks,