Mapping options to risk factors

[afterworkguinness]

Hi,
The study notes indicate a long option is broken down (mapped to) a long position in the underlying asset of asset price * delta and a short position in the underlying financed by the amount of long position - option cost.

I'm a bit confused here, as I don't see how being both synthetically long and short at the same time replicates the long option. Jorion says the long option is broken down into a long position in the stock of asset price * delta and a short position in a bill of long position - option cost.

Thanks in advance

 

[afterworkguinness]

@David Harper CFA FRM CIPM , thanks for the star. I take it that means it is a typo in the notes and my understanding is correct?

 

[David Harper CFA FRM]

@afterworkguinness Yes (sorry, I was writing a reply but had to receive a phone call that interrupted me ...). It's a mistake, I apologize. It should parrot what Jorion says:
"... the call option is equivalent to a position of Δ in the underlying asset plus a short position of (ΔS - c) in a dollar bill, that is: Long option = long Δ asset + short (ΔS - c) bill"
Thanks!

 

[afterworkguinness]

Much appreciated, thanks for clearing that up. (Sorry to be picky, but there is a minor typo in your reply above long option should be followed by = not minus )

 

[NNath]

Is this statement correct?

the delta-normal VaR cannot be expected to provide an accurate estimate of true VaR over ranges where deltas are unstable. Deep out-of-the-money and deep in-the-money options have relatively stable deltas. Over these ranges, the relationship between the value of the underlying instrument and the value of the option is very much like a forward currency contract

 

[David Harper CFA FRM]

Hi @NNath

Yes, I think that statement is mostly correct (is the source our notes on Jorion, is this ultimately Jorian). "Stable deltas" should be associated with low gamma: indeed, deeply in-the-money options have deltas approaching 1.0 (and gamma approaching zero); deeply out-of-money (OTM) options have deltas and gamma approaching zero. Regarding the third statement, I think this is true of deeply in-the-money options. BSM says, for a non dividend paying stock, that call price, c = S(0)*N(d1) - K*e^(-rT)*N(d2). A deeply in the money option has N(d1) and N(d2) tending toward --> 1.0, so the c ≈ S(0) - K*e^(-rT), which is the value of a forward contract! (with respect to OTM options, the difference is that forward contracts very much can have negative value so I wouldn't extend the statement to deeply OTM options, unless i am missing the statement's intention). Thanks!

 

[NNath]

This i got from schweser notes.