Definition of Delta?

[kenndanielso]

Hi David,

I'm getting confused about the different (or not) definitions of delta among the different readings. In the Greek Letters chapter it says that delta is within 0 - 1 for call options but in the reading on "Putting the VaR to work" delta for call options in the example are way above 1.

What's the difference and how will I identify the need to differentiate during the exam?

 

[David Harper CFA FRM]

Hi kenndanielso,

It's time for GARP to retire L. Allen readings. If you will trust me with the short version (there is a long dissection of Allen in this forum, but it's so old i can't find it ...), you can ignore L. Allen on delta, in favor of J. Hull. Because L. Allen defines delta as an elasticity and option elasticities will be much greater than 1.0; e.g., her true option delta of 0.6X * $100 stock price/$6.89 call price = 8.7. It's not "wrong" to use elasticity, but it's just less (or not at all) relevant to the FRM exam.

Briefly within the consistent body of the FRM, which is basically John Hull, I think we care about:

• broadly, delta is change in (position) value with respect to change in asset price, dV/dS. It is unitless, like beta, correlation, and marginal VaR.
• Positions include futures, so can be slightly greater than 1.0 per delta(futures) = exp[(r-q)T], but I think this should be an upper bound on almost anything that isn't levered
• In practice, we tend to apply it to options, where call option delta, N(d1), must be between 0 and 1.0 (and put option delta between -1 and 0).
  Please do know the intuition as to why deeply OTM calls --> delta 0 and deeply ITM calls --> delta 1.0.
• This N(d1) delta, between 0 and 1, is what Carol Alexander calls "percentage delta" and i like to think of as "per option" delta. To hedge and work with position, we want POSITION DELTA (position Greeks) where position delta = percentage delta * quantity. This is the confusion that can ensue when we see "300 delta" or "-500 gamma" (i thought gamma was always positive??). For example, an ATM call option will tend to have necessarily positive (percentage) delta ~= 0.6 and necessarily positive (percentage) gamma ~ 0.01 , such that:
  
  long 1,000 of these has position delta = 0.6 * 1000 = 600 position delta, or
  short 1,000 has position delta = 0.6 * -1000 = -600, and is how "(percentage) gamma always positive" can lead to:
  short 1,000 with position gamma = +0.01 * -1000 = -10 position gamma.

I hope that helps, thanks,

 

[Aleksander Hansen]

Agree, I skipped Allen altogether. Too much "fluff".