Gamma as change in the rate of change of value

Psychosocial Implication in Gamma Animation (Part #10)

In the context of this exploration it is extraordinary to discover the importance of gamma in financial trading. There it offers a measure of the rate of change in the delta with respect to changes in the underlying asset's price. In a world focused on change, gamma is an indicator of the change in the rate of change. With respect to finance, this must necessarily be understood as the change in one of the most tangible forms of value -- if notional and symbolic. Given the argument for the value of playing (as noted above), it is strange that "playing the markets" is a well-recognized phrase and that gamma should be so fundamental to the skills involved.

As the second derivative of the value function with respect to that underlying price, gamma is an important measure of the convexity of a derivative's value, in relation to the underlying price. It is important because it corrects for the convexity of value. Convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.

Ironically, in a period of the severest financial confidence in the country of its origin, the use of the term gamma in mathematical finance derives from "the Greeks". The name is used because the most common of these sensitivities are often denoted by Greek letters. These are the quantities representing the sensitivities of the price of derivatives to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The range, precision and subtlety of such little known measures is incredible, given the fuzzy inadequacies and oversimplifications in the case of debate regarding change in psychosocial systems. Wikipedia offers entries on:

First-order Greeks
1. Delta: measures the rate of change of option value with respect to changes in the underlying asset's price.
2. Vega: measures sensitivity to volatility. It is the derivative of the option value with respect to the volatility of the underlying asset.
3. Theta: measures the sensitivity of the value of the derivative to the passage of time: the "time decay."
4. Rho: measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term)
5. Lambda (Omega): is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.
Second-order Greeks
1. Gamma: measures the rate of change in the delta with respect to changes in the underlying price.
2. Vanna: (or DvegaDspot and DdeltaDvol): is a second order derivative of the option value, once to the underlying spot price and once to volatility.
3. Vomma (Volga, Vega Convexity, Vega gamma or dTau/dVol) measures second order sensitivity to volatility. It is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.
4. Charm (or delta decay, or DdeltaDtime): easures the instantaneous rate of change of delta over the passage of time.
5. DvegaDtime: measures the rate of change in the vega with respect to the passage of time. It is the second derivative of the value function; once to volatility and once to time.
6. Vera (or Rhova): measures the rate of change in rho with respect to volatility. It is the second derivative of the value function; once to volatility and once to interest rate.
Third-order Greeks
- Color (gamma decay or DgammaDtime): measures the rate of change of gamma over the passage of time
- Speed (or the gamma of the gamma or DgammaDspot): measures the rate of change in Gamma with respect to changes in the underlying price.
- Ultima (or DvommaDvol): measures the sensitivity of the option vomma with respect to change in volatility.
- Zomma (or DgammaDvol): measures the rate of change of gamma with respect to changes in volatility.

Collectively these have also been called the risk sensitivities, risk measures, or hedge parameters. Together "the Greeks" are vital tools in risk management. Ironically again, "the Greeks" (including gamma), have been fundamental to the development of the continuing financial crisis -- especially given the manner in which these were based on the dubious packaging and marketing of financial derivatives.

So-called "cross gamma" is important for the financial trader because in many hybrids products, the value in terms of higher coupon comes from huge cross gamma. So when pricing these products the trader needs to analyse how much of this cross gamma can be captured in the market. Correlation assumptions are a way traders can decide how much of the value from cross gamma can be passed on in terms of enhanced coupons while trading the product. In a delta-hedge strategy, gamma is sought to be reduced in order to maintain a hedge over a wider price range. A consequence of reducing gamma, however, is that alpha too will be reduced.

A highly controversial study by a former risk manager, Nassim Nicholas Taleb (Antifragile: how to live in a world we don't understand, 2012), has addressed the conditions which are a source of strategic surprise -- developing an argument made in earlier work (The Black Swan: the impact of the highly improbable, 2007). Considerable attention is given to the management of convexity (Black Swans and Antifragility: a vivid reconceptualization of risk and resilience, NPQ: nonprofit quarterly, 20 March 2013). Taleb argues for recognition of antifragility, in contrast to fragility, where high-impact events or shocks can be beneficial. He coined the term because he considered that existing words used to describe the opposite of "fragility," such as "robustness," were inaccurate. Antifragility goes beyond robustness; it means that something does not merely withstand a shock but actually improves because of it.

However, as argued in a critical review by Eric Falkenstein:

The book is really a big spread argument that it's good to be long gamma, bad to be short it. Gamma is the essence of an option, why there's 'time decay' or theta, a predictable expense that anticipates the payoff times the probability. Whether or not this theta is adequate for the gamma is whether an option is priced fairly or not, and generally people pay too much for gamma... Being long options (positive gamma), especially out-of-the-money options, has been a losing strategy. (Taleb Mishandles Fragility, 27 November 2012)

The recognition of the importance of gamma with respect to one fundamental form of change raises valuable questions as to how it -- together with the other "Greeks" -- could usefully be applied to other forms of change in value, especially those involving a degree of risk. The critics of Taleb's thesis -- as one of the few who warned of the financial crisis -- should be compared with the perspective of others who did not, as an indication of that dynamic with respect to more intangible values (Jared Woodard, Why Taleb is wrong about markets and uncertainty, Condor Options, 26 Novemeber 2012).

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Gamma as change in the rate of change of value