# Is there a stochastic equation which can model returns according to its four moments?

The normal stochastic equation only models mean and standard deviation.

For now, I'm randomly picking returns from a historical CDF of the returns. I'd like to have some flexibility when it comes to the next two moments - Skewness and Kurtosis.

What I'm looking for:

$$dS_t = \mu S_t \, dt + \sigma_t S_t \, dW_t + \mbox{skew-term} + \mbox{kurtosis-term}$$

I'm open to other ways to generating the random returns.

Cheers!

I suggest you have a look at the paper:

• Schloegel, Erik (2010) "Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order", Journal of Economic Dynamics and Control, Vol. 37, No. 3, pp. 611-631

available on SSRN.

A random variable $Y$ that follows a Gram/Charlier Type A series distribution has the probability density function

$$f_Y(x) = \phi(x) \sum_{j = 0}^\infty c_j \mathrm{He}_j(x).$$

Here, $\phi(x)$ is the standard normal density function and the $\text{He}_j(x)$ are the Hermite polynomials. The coefficients $c_j$ are related to the cumulants of the distribution. When you truncate the infinite sum at $j = 4$, set $c_0 = 1$, $c_1 = c_2 = 0$, $c_3 = \mathcal{S} / 6$ and $c_4 = \mathcal{K} / 24$, then the resulting distribution has zero mean, unit variance and an approximate skewness of $\mathcal{S}$ and excess kurtosis of $\mathcal{K}$.

We can now define the logarithmic price process $X$ as

$$X_t = X_0 + \gamma t + \sigma \sqrt{t} Y,$$

where $\gamma$ is chosen such that $S_t = S_0 \exp \left\{ X_t \right\}$ is a martingale under the bank account numeraire. This approach can be extended to multi periods. Please see the above paper for details. I can also recommend the author's book

• Schloegel, Erik (2014) "Quantitative Finance - An Object-Oriented approach in C++", Chapman & Hall

Levy models do that to some degree. They have the iid look and feel of the standard Gaussian models, but allow for higher moments. You can check the papers of Dilip Madan on Variance Gamma as a starting point.

Your question can be read several ways:

1. what about introducing terms in the model to obtain skewness and kurtosis of returns?
2. how to have a feedback loop on pas skewness and kurtosis in returns dynamics?

The easiest answer is on the first way to understand your question: how to introduce skew and kurtosis in a model? In math finance, it is usually done thanks to two effects:

• introducing a dynamics on the volatility
• adding jumps to the dynamics

Typically, the model would be $$\left\{\begin{array}{lcl}% dS_t &=& \mu S_t \, dt + \sigma_t S_t \, dW^S_t + dJ_t\\% dV_t &=& \kappa\cdot(\theta - V_t)\,dt + \xi \sqrt{V_t}\,dW^{V}_t% \end{array}\right.$$ Modelling volatility dynamics allows to change the moments of the returns; its usual name is the Heston model. The jump added as a third term to the dynamics can be arbitrary as soon as it is "properly defined" (i.e. a consistent mathematical way). It is related to Lévy models (as suggested by @Kiwiakos ), the related mathematics needed in math finance can be found in the Shyriaev-Jacod book Limit Theorems for Stochastic Processes.