Realized Gamma as Gamma-Dollar Exposure Integrated Over Time
Summary
The document develops a definition of realized gamma from the second-order term in an option’s price change. It defines gamma dollars as one half of spot squared times option gamma, then uses the diffusion component of spot’s change to show that the option’s gamma contribution accumulates in proportion to gamma dollars times variance over time. Integrating gamma dollars over the life of the position gives the proposed realized-gamma measure.
The derivation distinguishes this exposure from realized volatility and says it does not depend on choosing a particular pricing model or probability measure. Its final claim that the portfolio value change equals realized gamma times a difference in variances needs additional assumptions: the integral by itself is an exposure measure, and the variance factor must be specified consistently across time. The document poses the definition as a question and does not provide a worked numerical example or establish that final relationship in general.
Key ideas
- The second-order option price change depends on gamma and the squared spot move.
- Gamma dollars are defined as one half of spot squared multiplied by gamma.
- Under a diffusion model, the gamma contribution accrues with variance over time.
- Integrating gamma dollars over time yields a proposed measure of realized gamma exposure.
- A variance change cannot generally be factored out of the integral without further assumptions.
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Full text
# Definition of Realized Gamma
# Definition of Realized Gamma
What is the formal definition of "realized gamma" and how would one go about calculating it?
$$ \Gamma_{R} := \int_{t}^{T} \frac{1}{2} S^2 \gamma dt $$
The question is different than delta-hedging altough from the same application area in that it goes deeper into the aspect of "realized gamma" and how it is defined. Also do not confuse realized vol with realized gamma - both multiplied give the portfolio variance.
Also note we don't make any assumptions about expectation under Q (ref implied volatility) or P-measures (ref market volatility), and as such this is valid under any assumption (not only Black-Scholes but any other model like Bachelier, etc..).
My starting point is based on this article that I found here. No relation whatsoever with the author - it's just one of the few google hits I got.
So we start with the definition of gamma dollars:
$$ \Gamma_{DV}:=\frac{1}{2} S^2 \gamma$$
The variational principle tells us that the price of an option changes by dV when the spot changes by dS:
$$ dV = \delta \cdot dS+\frac{1}{2}\gamma\cdot dS^2+...$$ with $$ \delta:= \frac{\partial V}{\partial S} $$ $$ \gamma := \frac{\partial^2 V}{\partial S^2} $$
We are interested in the second term only which picks up the sigma. $$ dS:=\mu S dt + \sigma S dW $$ $$ dS^2 = \sigma^2 S^2 dt$$
Replacing back, and discretizing, we have the change in value of the option due to the second term only: $$ dV = ... + \Gamma_{DV} \sigma^2 dt + ...$$
As far as I can tell then the definition of "realized gamma" is then: $$ \Gamma_{R} := \int_{t}^{T} \Gamma_{DV} dt = \int_{t}^{T} \frac{1}{2} S^2 \gamma dt$$ with the implied fact that the portfolio change due to that is: $$ \Delta V = \Gamma_{R} \cdot (\sigma^2_1 - \sigma^2_0) $$
Is this correct?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.