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# Gamma Pricing Model

## What Is the Gamma Pricing Model?

The gamma pricing model is an equation for determining the fair market value of a European-style options contract when the price movement on the underlying asset does not follow a normal distribution. The Gamma model is intended instead to price options where the underlying asset has a distribution that is long-tailed ("skewed"). This, for instance, is the case for a log-normal distribution, where dramatic market moves to the downside occur with greater frequency than would be predicted by a normal distribution of returns relative to large upside swings.

The gamma model is one alternative for pricing options other than the original Black-Scholes model, which requires the assumption of a normal distribution. Others include the binomial tree, trinomial tree, and lattice models, among others.

### Key Takeaways:

• The gamma model for pricing options is used to more accurately represent the distribution of asset prices that are asymmetric and is thus a better reflection of an option's fair value.
• The model utilizes an option's gamma or curvature to changes in its price sensitivity as the underlying asset moves.
• The model is used to price options on assets that have a distribution that is either fat-tailed or skewed, such as the log-normal distribution.

## Understanding the Gamma Pricing Model

While the Black-Scholes option pricing model is the best known in the financial world, it does not actually provide accurate pricing results under all situations. In particular, the Black-Scholes model assumes that the underlying instrument has returns that are normally distributed in a symmetrical manner.

As a result, the Black-Scholes model will tend to misprice options on instruments that do not trade based on a normal distribution, in particular, under-valuing downside puts. In addition, these errors lead traders to either over- or under-hedge their positions if they seek to use options as insurance, or if they are trading options to capture the level of volatility in an asset.

Many alternative options pricing methods have been developed with the goal of providing more accurate pricing for real-world applications, such as the Gamma Pricing Model. Generally speaking, the Gamma Pricing Model employs the option's gamma, which is how much fast the delta changes with respect to small changes in the underlying asset's price (where the delta is the change in option price given a change in the price of the underlying asset).

## Gamma and Volatility Skew

By focusing on the gamma, which is essentially the curvature, or acceleration, of the options price as the underlying asset moves, investors can account for the downside volatility skew (also known as the volatility "smile") resulting from the lack of a normal distribution. Indeed, stocks' price returns tend to have a far greater frequency of large downside moves than upside swings. Furthermore, stock prices are bounded to the downside by zero, whereas they have unlimited upside potential.

Most investors in stocks (and other assets) tend to hold long positions and use options as a hedge for downside protection. This creates more demand to buy lower strike options than higher ones.

The gamma model modifications allow for a more accurate representation of the distribution of asset prices and, therefore, a better reflection of options' true fair values.

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