Looking For Gamma
Sep 06, 2005
Option traders have sleepless nights when there is negative gamma on their books and are generally more relaxed when they run positive gammas. Gamma is a feature of any derivative product. In fact, gamma is an important feature of any profit and loss profile that is not a straight line. All curves have gamma. There was once a senior option trader in a bank who would tell his associates that “more the gamma you can buy the more will be your bonus”.
Gamma is the second mathematical derivative of the price of a derivative instrument with respect to the underlying. In other words, gamma tells us how much curvature is there in a curve. Or, how the slope of a straight line changes in order to bend it. When we buy derivatives – options, structured products, etc. – we essentially do that. We take the straight line P & L payoff of equities and cash and add curvature to it, all via adding gamma to it.
We can understand gamma by looking at the Taylor series expansion. If profit and loss of an option (or any derivatives security) is a function of the price, x of the underlying stock (or asset) then by Taylor Series expansion of the price function around a point h where h = x-x0 and x0 being the original price of the stock, we can write:
This means if we perturb the original price x0of the stock by an amount equal to h then the price function of the option will be given by the above equation. As we can see that in the above equation is the delta of the option and represents the first order risks. The second term is called the gamma of the option (or convexity or curvature) and represents the second order risks of the option position.
This implied that if , i.e. if Gamma is negative, the trader will lose whichever way the underlying moves. For an option seller, the gamma is always negative and the Taylor Series expansion for such a delta hedged position will look like:
This is the dilemma that the option sellers face. When they sell options their gamma becomes negative and their best case scenario is that the market goes nowhere, i.e. the stock price does not move. Any movement in the stock price from the initial level will add to the magnitude of the second term in the above equation. In other words, any movement in the stock price will mean bigger values of h and smaller values of the function f(x).
On the other hand, positive Gamma adds to the price of the instrument. This means that the more Gamma an option has, the better it is to hold and higher will be the value of the function f(x), i.e. the trader’s P & L.
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