Saturday, July 21, 2012

a greek's value converges during expiration (my take

Let's be clear what I mean -- for a given greek (say, delta) of European options, regardless of which underlier, what strike or expiration month, the value of that greek tends to move towards predictable levels as we approach expiration.

Well before expiration, a greek is like a real-value variable which could take on any numerical value within a range. However, during the last days, BS formula predicts that a greek's value (say delta) always gravitates towards some well-defined convergence points.

delta -- either 1.0 or 0 (negative for puts). However, if underlier spot price fluctuates around my strike level (IF you remain very close to ATM), then I'm switching sides between ITM and OTM, so my delta would swing wildly between the 2 convergence 1.0 and 0.

gamma -- either infinity or zero. ATM (IF you remain very close to ATM) would go to positive infinity gamma (for long positions); deep ITM/OTM go to zero gamma. However in the last moments there's no point talking about infinity or zero gamma. If in the last minute spot is still changing around your strike level, then you basically close your eyes and wait to see if you finish ITM or OTM. Even if you finish ITM, the payoff would be a small [1] percentage of strike, and probably a small percentage of premium.

[1] unless you have a Binary option. If you finish ITM with a binary option, you effective win a million dollar bet.

theta -- either infinity or zero. Similar to gamma, ATM would go to negative infinity theta; deep ITM/OTM go to zero theta.

option valuation -- either $0 or stock price.

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