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Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 (long call option) and 0 from −1 (long put option). The inverse is true for short options. [11]
That is, the value of an option is due to the convexity of the ultimate payout: one has the option to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a hockey stick shape) is convex – "optionality" corresponds to convexity in the payout.
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
Black and Scholes' insight was that the portfolio represented by the right-hand side is riskless: thus the equation says that the riskless return over any infinitesimal time interval can be expressed as the sum of theta and a term incorporating gamma. For an option, theta is typically negative, reflecting the loss in value due to having less ...
This would yield a limited loss if the options expire with the underlying near or above 110, a large loss if the options expire with the underlying far below 95, and a limited profit if the underlying is near or between 95 and 105. [1] A short ladder is the opposite position of a long ladder. Thus, for the first example above, the corresponding ...
Gamma correction is a type of power law function whose exponent is the Greek letter gamma (γ). It should not be confused with the mathematical Gamma function. The lower case gamma, γ, is a parameter of the former; the upper case letter, Γ, is the name of (and symbol used for) the latter (as in Γ(x)). To use the word "function" in ...
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