The Gamma Riddle; The Solution

A week and a half ago I published the Gamma Riddle asking for two solutions on the following situation:

I am long the 50 call once at 25% volatility, with expiry in 112 days

I am short the 50 call twice at 25% volatility, with expiry in 235 days

The underlying is a non dividend paying asset and interest rate is set at 0%, the underlying trades at 50 and stays at 50

We know that at inception of the trade we will be short gamma and short to expiry for the shortdated option we will be long gamma in the combination.

My questions were:

1: How many days before expiry for the short dated option are we having a flat(tish) gamma position in this combination?

The underlying has stayed at 50 for quite some time, so most probably volatility would have come off to 15%.

2: How many days before expiry for the short dated option are we having a flat(tish) gamma position when volatility would have dropped towards 15%?

Many people have read the post, thank you for that!

Besides the right answer I asked for the right argumentation as well. It was interesting to see that most answers came from the “quants”, having solved the riddle mathematically rather than philosophically.

Below I will explain the philosophical solution:

As shown in the chart above, gamma, for an at the money option, is inversely proportional to the square root of time. This means that if maturity will quadriple, the gamma for an at the money option will halve. In this example one can see the gamma for the 50 strike being at 0.08 when maturity is a quarter year. When maturity will be one full year, being four times as long, the gamma will be halve the value (as shown in the chart the gamma is now 0.04 for the 50 strike).

So if we are long once a short term option and short twice a long term option, both at the money, we must have a flat gamma book when the time to maturity for the long term option is four times as large as the time to maturity for the short term option. The longer term has half the gamma, but we're short two of them.

At inception the maturity for the short term option is 112 days

The maturity for the long term options is 235 days, hence 123 days after the short term expiry.

When using T1 for time to expiry for the short term option and T2 for the two longer term options we can conclude the following:

T2 = T1 +123 and

T2 = 4 T1

Thus:

4 T1 = T1 +123 >

3 T1 = 123 >

T1 = 41 and T2 = T1 + 123 > T2 = 164

The one to four ratio is now intact 41:164 = 1:4

The answer on the first question is thus 41 days

For question 2 the volatility has been lowered. Gamma (for an at the money) is inversely proportional to the change of volatility, meaning that if volatility would halve that gamma would double, as shown in the chart below:

At 20% volatliity the gamma (0.04) will be half the gamma as compared to at 10% volatility (0.08), so yes; gamma will change when vol would drop from 25% to 15%. In the mean time the fact that gamma is inversely proportional to the square root of time will not change and therefore the gamma flat position will occur at 41 days before expiry of the short term option.

Hence the same answer as in question 1.

The winners of a signed copy of my book are:

Aleksandar Petrakiev, Senior Quantitative Analyst at Triple Point Technology

And

Christopher Merrill, Quant Developer at FT Options (he came with both, the mathematical as well as the philosophical solution)

Thank you all and looking forward to see you back at my next riddle!

Best regards

Pierino