FRM Fun 17 (stock option Var), P1 only

[David Harper CFA FRM]

P1 only. Rahul posted an interesting question, if we assume:

• Option delta = 0.75
• Position = 50
• (asset) Price= 8
• Volatility = 0.25

Then asked:

1. What is the (delta normal) VaR of a long call option?
2. What is the (delta normal) VaR of a short call option?

ShaktiRathore shows that the VaR of long call = VaR of short call. Here are two follow-up questions:

1. 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?
2. 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?

 

[ShaktiRathore]

1) VaR of stock= z*sigma*Price*position
delta=d(optionprice)/d(stockprice)
or delta*d( stockprice )=d( optionprice )
taking VaR on both sides;
delta*d(VaR of stock)=d(VaR of option)
delta*(VaR of stock)=(VaR of option)
from derivation above its clear that we don't require option price to calculate option VaR.

2) dC= delta*dS+.5*gamma*dS^2
taking VaR on both sides,
VaR dC=delta*VaR of stock+.5*gamma*dS*VaR of stock
VaR of call option=(delta+.5*gamma*dS)*VaR of stock
Now since delta of short call=-delta of long call and also gamma of short call=-gamma of long call
VaR 0f long call=(delta+.5*gamma*dS)*VaR of stock
VaR 0f short call=-(delta+.5*gamma*dS)*VaR of stock
implies that VaR of short call=-Var of long call

thanks

 

[David Harper CFA FRM]

Thanks Shakti!

1. Agree: I think this is an instructive difference between duration and option delta. Duration is "infected" with 1/P, however option delta is a pure first derivative; i.e, delta=d(optionprice)/d(stockprice) ... it is a common mistake (even made in the Linda Allen text by the author!) to use delta as a %, but as you show, delta gives the change in option price with respect to change in stock price: as it's not a percentage, we don't require the stock price!
2. Disagree: only because you lost the dS^2, from your first step to the next . If you restore the dS^2, then we can see why the gamma is not symmetrical. In the case of delta, as the impact is a linear delta*dS, the delta-only short call has the same downside as the delta-only long call.
   
   However, if we add the gamma (i.e., extend the Taylor), the gamma correction is (as your first line shows) 0.5*gamma*dS^2. Due to the dS^2, an increase in stock and a decrease in stock price (e.g., +$0.10 or -$0.10) are both squares such that the gamma term is always a positive. (Visually, this is the concave up, or convexity, in the option vs stock price plot). Consequently, the delta+ gamma VaR of a long call will be LESS THAN the delta+gamma VaR of the equivalent short call. I hope that is interesting...

 

[rahul.goyl]

Thanks a lot Shakti & David.

 

[bhar]

1) VaR of stock= z*sigma*Price*position
delta=d(optionprice)/d(stockprice)
or delta*d( stockprice )=d( optionprice )
taking VaR on both sides;
delta*d(VaR of stock)=d(VaR of option)
delta*(VaR of stock)=(VaR of option)
from derivation above its clear that we don't require option price to calculate option VaR.

2) dC= delta*dS+.5*gamma*dS^2
taking VaR on both sides,
VaR dC=delta*VaR of stock+.5*gamma*dS*VaR of stock
VaR of call option=(delta+.5*gamma*dS)*VaR of stock
Now since delta of short call=-delta of long call and also gamma of short call=-gamma of long call
VaR 0f long call=(delta+.5*gamma*dS)*VaR of stock
VaR 0f short call=-(delta+.5*gamma*dS)*VaR of stock
implies that VaR of short call=-Var of long call

thanks

The other post

long call P/O= max(S-X,0)
short call P/O= premium-max(S-X,0)=premium-long call P/O...A
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


In the other post- you mention that VaR of short call = VaR of Long Call. The 'minus sign is missing. Could you please clear the doubt. Is there any difference ?

 

[ShaktiRathore]

Bhar,
please dont get confused by signs.It is possible that I considered signs in one derivation and not in other but the real significance of the result is that the magnitude of VaRs of a long call option and a short call option are equal. I mean vol(short call)=-vol(long call) i mistakenly taken the negative sign which should be positive so make it vol(short call)=vol(long call) so as to make the result VaR of short call=-Var of long call.thanks for identifying the mistake but in the end this will not affect the final result in which we want to show that VaRs of long call and short call are symmetric that is they are equal.

thanks

 

[bhar]

Ok great That clarifies. I was just going through the derivation. I understood the end result. But since there was some difference wanted to clarify.
Thanks.

 

[David Harper CFA FRM]

As per the point of the 2nd question above, I think it is incorrect to say that universally the VaR of short call = -VaR of long call.

Indeed, this is true with respect to an option VaR when only the delta (linear approximation) is employed to estimate the VaR.

But, while this equality holds for the linear approximation (which is indifferent to the +/- direction), this is not true of the non-linear approximation that includes the gamma adjustment: the delta-gamma VaR of short option is greater than (>) the delta-gamma VaR of long option, with equivalent features.

Potential loss in long call option position = +delta*(-ΔS) + 0.5*gamma*(-ΔS)^2 = neg delta adjustment + positive gamma (due to ^2)
Potential loss in short call option position = -[gain in long position] = -[+delta*(+ΔS) + 0.5*gamma*(+ΔS)^2] = -delta*(-ΔS) + 0.5*gamma*(+ΔS)^2 = neg delta adjust - gamma adjust
i.e., the VaR is symmetrical with respect to delta but not when including the gamma (a visualization of option price versus asset price will confirm). Where VaR = abs(loss), the VaR of the short position must be greater

Thanks,