Higher Moments in Option Pricing

The Limitations of Black-Scholes and the Volatility Smile

The Black-Scholes model, a cornerstone of modern financial theory, has been the dominant model for pricing options for decades. However, the model is based on a number of simplifying assumptions, including the assumption that asset returns are normally distributed. In reality, asset returns are often skewed and exhibit fat tails, which can lead to significant pricing errors when using the Black-Scholes model.

One of the most well-known manifestations of the limitations of the Black-Scholes model is the volatility smile. The volatility smile is the empirical observation that options with the same expiration date but different strike prices have different implied volatilities. This is in direct contradiction to the Black-Scholes model, which assumes that implied volatility is constant across all strike prices.

Explaining the Volatility Smile with Higher Moments

Higher moments, such as skewness and kurtosis, can be used to explain the volatility smile. The negative skewness of equity returns, for example, can help to explain why out-of-the-money put options tend to have higher implied volatilities than out-of-the-money call options. The high kurtosis of equity returns can help to explain why both out-of-the-money put and call options tend to have higher implied volatilities than at-the-money options.

The Heston Model

The Heston model is an extension of the Black-Scholes model that incorporates stochastic volatility. The model can be further extended to incorporate skewness and kurtosis, as follows:

dS(t) = (μ - q)S(t)dt + sqrt(v(t))S(t)dW1(t)
dv(t) = κ(θ - v(t))dt + σv * sqrt(v(t))dW2(t)

dS(t) = (μ - q)S(t)dt + sqrt(v(t))S(t)dW1(t)
dv(t) = κ(θ - v(t))dt + σv * sqrt(v(t))dW2(t)

Where:

• S(t) is the asset price at time t
• μ is the expected return of the asset
• q is the dividend yield
• v(t) is the variance of the asset's returns
• κ is the rate of mean reversion of the variance
• θ is the long-term mean of the variance
• σv is the volatility of the variance
• dW1(t) and dW2(t) are Wiener processes with correlation ρ

Building More Accurate Option Pricing Models

By incorporating higher moments into option pricing models, it is possible to build models that are more accurate and robust. These models can be used to:

• Price options more accurately: By taking into account the skewness and kurtosis of asset returns, it is possible to obtain more accurate option prices.
• Hedge options more effectively: By understanding the sensitivity of option prices to changes in skewness and kurtosis, it is possible to construct more effective hedges.
• Identify mispriced options: By comparing the market price of an option to its theoretical price from a higher moment model, it is possible to identify mispriced options.

Conclusion

Higher moments are essential tools for understanding and modeling the volatility smile. By incorporating skewness and kurtosis into option pricing models, traders can build more accurate and robust models that can be used to price and hedge options more effectively. In the next article in this series, we will explore the use of higher moments in the analysis of cryptocurrency markets.