Why do Options have Convexity (gamma)?
December 6, 2006
Why do options have convexity (which also goes by the name of "gamma")? This was one of the first questions posed to us by one of the trainees (working for a hedge fund) in one of our recent training sessions. Convexity is the chief characteristic of all financial instruments that have non-linear payoff, options being one of them. And come to think of it, with convexity, the discipline of financial derivatives and financial engineering would not exist.
Thus this question needed to be answered correctly and completely. And we were required to answer this without using any math. The person wanted an intuitive understanding of convexity.
The two best explanations that we have so far come across of why options have convexity are given in Nassim Taleb's Dynamic hedging, where he talks about contamination principle and Paul Wilmott's Quantitative Finance where he explains the notion of randomness and convexity as interlinked concepts. Let's take Wilmott's approach and explain why options have convexity.
Intuitive Explanation of Convexity:
If you knew nothing about the theory of option pricing but knew what a call option is by definition then you might say, well since the average of the stock price in six month's time would $10 (average of $12 and $8) the value of this option in a year's time should be zero because call = max (Average (stock price) - strike, 0) . Even with discounting this value will be zero. Is this correct? No, it is not. But why not, where is the flaw in the reasoning? One way to explain the flaw could be that it is not guaranteed that there is exactly a 50% chance for either of the above to happen, and the probabilities of either outcome happening could be very different thereby making the probability weighted average different from $10. Moreover, even if the outcomes have 50% probabilities for half the time the call option will have some payoff, and therefore, it is quite likely that the final value will be greater than zero.
Then again, you could have reasoned that a better way of calculating the value of the call option is by taking the payoff of each outcome separately and then averaging the payoffs. You could calculate the payoff of the option by using call = max (stock - strike, 0) and thus when the stock finishes at $12 the payoff is $2 and when the stock finishes at $8 the payoff is zero and hence the average of the payoff is $1 which when discounted back should give you $0.97 (ninety seven cents). Therefore, using this method we find that the option value today is greater than zero, even if the stock has a fifty-fifty chance of ending up at $12 or $8.
What is the difference in the above two methodologies? The difference is the way we did the averaging. In the first method we averaged the stock price and then calculated the call option payoff using the average stock price and in the second method we calculated the call option payoffs first and then averaged those payoffs. The first method gave us a value of zero, which is incorrect because there is always some probability - chance - that the stock price will finish in such a way that the call option will have value greater than zero. Other way of putting it is to say that there will always be some payoffs which will dominate other payoffs and hence it is essential to calculate all the payoffs separately.
And so we can say that convexity is that measure of value that gives the call option a non-zero value today. If the value of an at the money (ATM) call option is zero today then no one will buy the option and yet ATM calls are the most liquid and traded of all options in any asset class. In fact, almost all of this value for an ATM option comes from this notion of "convexity". So when you are buying an ATM option you are actually buying convexity or the gamma.
Convexity enters the option pricing model via the way the payoffs are averaged. Call option payoff of an expected value of stock is zero, as we saw above, but the expected value of the call payoff is non-zero and this non-zero value is due to convexity.
We strongly recommend the reader to read Taleb's explanation of convexity as well which is linked to the notion of contamination principle. That is also a very intuitive way of understanding the notion of non-linear payoffs.
Reference: Paul Wilmott's Quantitative Finance .
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