Exam1:Option Greeks

[Anonymous]

Hi,

Can someone tell me how gamma of the ATM varies with time to maturity of option.

It is known that ATM option with less time to maturity have more gamma and ITM or OTM option with less time to maturity will have less gamma.

I am not clear on what is the reason behind this.

Can someone explain.

Thanks,
Arun

[Arun]

Exam1:Option Greeks

[Anonymous]

It’s not that easy to see this without a diagram. Unfortunately I can’t insert one here. But think of a call option and its payoff diagram at expiry (see Figure 8.1 from Finance Theory handbook). Say the strike is $10. The payoff graph for the option is a flat horizontal line for values of Spot below $10 and a straight sloping line at 45 degree for values of Spot above $10. Of course we are not interested in the graph at expiry, we are interested in what it looks like prior to expiry. But a day before expiry the graph will probably look pretty close to the shape of the expiry payoff graph. And this has a “sharp corner” at $10 which implies infinite gamma. This is because the delta change from a value of 0 for any value below $10 to a value of 1 for any value greater than $10. Of course, 1 day before expiry we don’t quite have the sharp corner, but it’s pretty close to a sharp corner. So it’s sharp at the strike, i.e., high gamma, but delta is almost unchanging above or below the strike (it’s either 0 or 1). Almost unchanging delta means negligible gamma (i.e., for OTM and ITM close to expiry). You could use Excel and put together a Black Scholes formula for a call option with strike $10, vol 50% and get the value for the option when the spot is 8.00, 8.20, 8.40,……10.00,10.20…..12.00. Do this with time to expiry = 6 months (0.5) and do it again with time to expiry = 1 day (1/365 =0.00274). The graph of the values for “1-day to expiry” willl be very straight below $10 and above $10 and sharp at $10. But for the “6 months to expiry” option the graph should show some curvature below and above $10, and the curvature at $10 should not be as pronounced as with the “1 day to expiry” option. If you can’t see this visually (I haven’t actually tested this myself) you might try changing the vol or the range, say choose $5 to $15 rather than $8 to $12. Like I said, if I could insert a rough graph here I could explain it much better. I had a look at the Handbook quickly but I couldn’t see any good graphs that explain how the option value graph (option $value as a function of the spot price) changes as expiry approaches.
Regards

Jim

[Anonymous]

Jim,

Thank you for the explanation.

Just to add to what Jim has already said – an ITM or OTM option close to maturity will have a delta close to 1 and will effectively behave like the stock itself, and will go up and down with the price of the underlying. Since the delta remains the same, the gamma will be quite low. Looking at it another way, when we are close to maturity the outcome of the ITM or OTM option is more or less known, as there is by definition a big difference between the strike price and spot price, and we are close to expiry. So gamma has to be low.

For an ATM option, the gamma would be much higher because as the price of the underlying fluctuates either side of the exercise price, the option will go in and out of the money and the delta will change rapidly, meaning that the gamma will change too and will be high.

Mukul

[Anonymous]

Thanks a lot Mukul and JIM.

[Author]

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