The Vega Riddle: The Solution!
Last Week I published a riddle on vega:
If a call with a theoretical value of 1.82 has a vega of 0.06 and the Implied Volatility rises one percentage point from, say, 17 percent to 18 percent, the new theoretical value of the call will be 1.88 – it would rise by 0.06, the amount of the vega. If, conversely, the IV declines 1 percentage point, from 17 to 16 percent, the call value will drop to 1.76 – that is, it would decline by the vega.
My question to you was:
When interest would be set at 0% and the text is referring to a non-dividend paying asset, would this call option be:
In the money?
At the money?
Out of the money?
Besides giving the right answer, I also asked the right argumentation why the other two answers will not be possible.
Thank you all for taking the time to think about the riddle!
The answer is 1, an in the money option
It cannot be an at the money option; an at the money option doesn’t exhibit vega convexity, meaning that at 17% volatility this option is built up with 17 incremental steps, all of the same value, the vega. The chart below shows that for the at the money call (and put) each step of 1 volatility point has a value of 0.06, at 10% volatility the option has a value of 0.60, at 50% volatility the option has a value of (approx) 3.00.
Hence, at 17% volatility, the at the money call would have a value of 1.02
An out of the money call does exhibit vega convexity, meaning that it will have a 0 vega when the strike of this option (X) falls out of the probability distribution (the so called boundaries), but the vega will start to grow as soon as the option will be within the probability distribution. The more it will be inside the distribution (or closer to the at the money) the higher its vega will get. This will be the case when volatility will increase. This is also called Vomma.
As shown in the chart above, strike X will have no vega below 10%, around 10% volatility it has a vega of 0.01 and around 17% it will have reached a vega of 0.06.
In this particular case (depending on underlying level and time to maturity) the value of strike X will be determined by the total value of all 17 percentage steps times its respective vega, or in other words, 17 times the average vega. On average the vega will be, in this particular example, around 1.5 cents, resulting in a value of approximately 0.25 for the call. The average vega of strike X will always be lower than 0.06 and thus the value of the out of the money call with a vega of 0.06 and a volatility of 17% must always be smaller than 1.02 (no skew applied).
The only possibility left is that the solution of the riddle is that it must be an in the money call.
The two winners of a signed copy of my book, How to Calculate Options Prices and Their Greeks, are:
Rob Lees, Energy Options Trader at Mercuria
Tze Shao, Director Market Risk at UBS
Rob and Tze, I will contact you for your details.
Thank you all, have a nice weekend!