I'm trying to wrap my head around Greeks, and I'm getting a little bit confused. For example, let's say my portfolio holds a long 5 month ATM call with strike \$20, and short 2 month OTM call with strike \$60. Now, if my underlying rises to \$40, what happens to my $\Delta$, $\Gamma$ and $\nu$ega exposures? I'm not exactly sure how much information I can say looking at the Black-Scholes formula since I don't have information about $\sigma$, so what can I say about the response of the Greeks to this underlying change?
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1$\begingroup$ You don't mention what your starting value is (also of Greeks like delta, seems you are very far away from ATM so potentially neither delta nor gamma changes much). Without knowing sigma, you cannot talk about Greeks really. $\endgroup$– AKdemyCommented Oct 14, 2021 at 12:43
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$\begingroup$ @AKdemy thanks for the comment, which starting value do you mean? Is it still not possible to get an intuition for how the Greeks (exposures, I rephrased it a little bit) change without plugging into the B-S equation? $\endgroup$– Ice TeaCommented Oct 14, 2021 at 12:50
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$\begingroup$ The $\Gamma$ gives you insight into how $\Delta$ changes if $S$ changes. This is true on a portfolio level and not just for a single option. What is the sign of your overall Gamma? $\endgroup$– nbbo2Commented Oct 14, 2021 at 13:15
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1$\begingroup$ What is your current underlying price? Could be any real number. without an option pricing model (or any idea what your values are), how would you want to compute something that purely depends on a model? What is the purposes of the question even? If you are asking how Greeks of two options (portfolio) work, that's simply the sum of the individual options. $\endgroup$– AKdemyCommented Oct 14, 2021 at 13:16
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1$\begingroup$ Ok, missed the ATM part. Delta will be somewhere close to 0.5 for ATM. The other is so far otm it will likely be zero.if your underlying doubles, Greeks will be useless as anything that jumps like this will not qualify for infinitesimally small changes (what Greeks show). In any case, you need a model to talk about Greeks. What is the context/ background of this question? $\endgroup$– AKdemyCommented Oct 14, 2021 at 13:38
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1 Answer
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The greeks are non-linear and only give you the instantaneous rate of change. The larger the change in underlying is, the less accurate the change based on the greeks will be. A doubling of the underlying will certainly not be predictable by the greeks alone. Delta and Gamma can be used to estimate the new price using a second-order taylor series approximation:
$P \approx P_0 + \Delta * dS + \frac12 * \Gamma * (dS)^2 $
But it's still an approximation, and with large changes in S it may be significantly different than the actual change.
But you asked about the effects on $\Delta$, $\Gamma$, and vega. Gamma will give you an approximate change in delta ($d\Delta \approx \Gamma*dS$). There are greeks that will tell you the change in Gamma and Vega when the underlying changes (Speed and Vanna, respectively), but they are not as widely used.
Of course, the practical way to calculate the difference based on a change in the underlying is to reprice the option with the new spot price, but that assumes that the volatility is the same, which is probably not true in reality (large changes in the underlying often coincide with large changes in implied volatility)
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$\begingroup$ @Ice Tea, this answer has graphical examples of Greeks and how far they are from actual values in case of large moves on the underlying (still ignoring changes in ivol which will be substantial if your underlying doubles). $\endgroup$– AKdemyCommented Oct 15, 2021 at 0:48