Can gamma of an option be greater than its delta?

Question

I have a currency pair usdinr put option with strike price at 73.5 INR, risk free rate 0, underlying price of 75.4025, days to expiry is 15 and iv is 5.9%. Delta of this option is -0.019 and gamma is 0.051. So I am interpreting gamma as change in delta with 1% change in underlying. So by this logic if I increase the underlying by 1% and option being put the delta should decrease by 0.051. So that means new delta would be 0.031. This would be wrong because delta of put option can't be positive. Now I have increased the underlying by 1% in my excel working. And delta is decreased by 0.017 whereas my gamma was 0.051. So is this because of the low volatility or is there any mistake in my interpretation of gamma?

Answer 1

Your greeks are derivatives, not absolute price differences, when your underlying changes by 1%. Also, the put price and delta are not linear in the underlying (in fact in your case they are highly non linear) and you cannot expect for a large change (which the 1% change is) that the delta increases by the linear amount that you expected. You need to look at smaller underlying changes for this to hold approximately.

Answer 2

In addition to the good answer given below: think of options with almost no maturity left (i.e. 1-day before expiry): for these options, the time value of the option is almost zero, so the option value as a function of the underlying has almost fully converged to the "step-function" of the pay-off at maturity (i.e. the classical "hockey-stick" pay off).

In other words, as you cross from OTM to ITM, your delta changes "violently" as a step function from zero to one. Gamma is the change in this "Delta", and the Gamma around this step as you cross from OTM to ITM can be very large: indeed larger than 1!

Suppose that Gamma around the ATM point is 2: does that mean that if the underlying increases by "1 unit", the Delta will become 2? Of course not, because Delta cannot be larger than 1.

Gamma itself is non-linear, and for these short-maturity options, it spikes up rapidly around the ATM mark, but then decreases rapidly to zero.

So in other words, suppose the value of Gamma is equal to (say) 2: this is only over an "infinitesimal" (or indeed very small) domain of the underlying, say 0.01 units around the ATM mark.

That's how you should view Gamma intuitively in general: as a non-linear function in the underlying. Try to plot your gamma and you will get it.