I am only familiar with plain vanilla Black Scholes, and not stochastic vol and garch models. Wondering if they can help reflect a materially non-normal view I have. Say I have a view for the price of an underlying that is heavily skewed to the right. Very high probability that it will not perform well, but a nonnegligible probability that it will shoot to the moon; its price distribution should look like an early-stage startup, such that a diversified portfolio of such underlyings should resemble a power distribution. What models should I look into?
chap gpt says The third moment in statistics, also known as skewness, measures the asymmetry of a probability distribution around its mean. Here are some simple models or distributions that can reflect the third moment, indicating whether the distribution is skewed to the left (negative skewness) or to the right (positive skewness): Log-Normal Distribution: This distribution is positively skewed, commonly used to model data that is bound by zero but can potentially inflate to large values, such as income or certain types of scientific data. Exponential Distribution: Also positively skewed, this distribution is typically used to model time until an event occurs, such as the lifespan of an electronic component or time until next call in a call center. Gamma Distribution: This can exhibit either positive skewness (most common) or be symmetrical, depending on its shape parameter. It’s used in various fields, such as modeling insurance claims and rainfall amounts. Weibull Distribution: This flexible model can exhibit both positive and negative skewness depending on its shape parameter. It's widely used in reliability engineering and survival analysis. Beta Distribution: This distribution can take on a variety of forms including left-skewed, right-skewed, and symmetric shapes, depending on its two shape parameters. It is useful in modeling variables that are bounded at both ends, such as proportions and percentages. Pareto Distribution: Characterized by significant positive skewness, it is often used to model the distribution of wealth, where most of the wealth is held by a small fraction of people. Each of these models can be a good choice depending on the specific characteristics of the data you're analyzing, particularly if the data shows signs of being skewed rather than symmetrical.
SDE transformation into a Volterra integral eq -> derive conditions for moment boundedness. Jane St uses it in SN.
Structure something into the second moment (volga) proportional to third and hedge for the flip in modality.. IOW, market approaches strike and second, third moment trades flat/flips sign as spot > strike. I have something for that.