Ralph Elliott's Measuring the Variance Risk Premium: A Comparative Analysis of Model-Free and Model-Based Approaches

This is a draft of the second article. I will continue to refine and add more details as I write the other articles.

Measuring the Variance Risk Premium: A Comparative Analysis of Model-Free and Model-Based Approaches

Accurately measuring the variance risk premium (VRP) is a important first step for any investor seeking to systematically harvest this potential source of return. The choice of measurement methodology can have a significant impact on the perceived magnitude and stability of the premium, as well as the performance of any trading strategy based upon it. Two primary approaches have emerged for quantifying the VRP: model-free and model-based methods. This article provides a comparative analysis of these two approaches, highlighting their theoretical underpinnings, practical implementation, and relative merits.

Model-Free Implied Variance

The model-free approach to calculating implied variance has become the industry standard due to its theoretical elegance and robustness. This method, pioneered by Britten-Jones and Neuberger (2000), demonstrates that the risk-neutral expectation of the future variance of an asset can be synthesized from a portfolio of European options across a continuum of strike prices. The CBOE Volatility Index (VIX) is the most prominent example of a model-free implied volatility measure. The VIX is calculated using a weighted average of the prices of a wide range of S&P 500 options, and its squared value represents the 30-day risk-neutral expected variance of the S&P 500.

The formula for the VIX squared is:

σ² = (2/T) * Σ [ (ΔK_i / K_i²) * e^(RT) * Q(K_i) ] - (1/T) * (F/K₀ - 1)²

σ² = (2/T) * Σ [ (ΔK_i / K_i²) * e^(RT) * Q(K_i) ] - (1/T) * (F/K₀ - 1)²

Where:

• T is the time to expiration
• F is the forward index level derived from index options prices
• K₀ is the first strike below the forward index level, F
• K_i is the strike price of the i-th out-of-the-money option
• ΔK_i is the interval between strike prices
• R is the risk-free interest rate
• Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i

Model-Based Implied Variance

Before the advent of model-free methods, implied variance was typically calculated using a specific option pricing model, most commonly the Black-Scholes model. This approach involves inverting the Black-Scholes formula to solve for the implied volatility that equates the model price of an option to its market price. While this method is more straightforward to implement for a single option, it is fraught with theoretical and practical challenges. The Black-Scholes model assumes that volatility is constant, which is inconsistent with the empirical observation of the volatility smile. This means that different options on the same underlying asset will have different implied volatilities, making it difficult to arrive at a single, consistent measure of implied variance.

A Comparison of Methodologies

The following table provides a comparison of the model-free and model-based approaches to measuring implied variance:

Actionable Example: Calculating Realized Variance

To calculate the VRP, we need to compare the implied variance to the subsequently realized variance. Realized variance is typically calculated as the sum of squared daily log returns over a specific period. For example, to calculate the 30-day realized variance of the S&P 500, we would use the following formula:

Realized Variance = Σ (ln(P_t / P_{t-1}))²

Realized Variance = Σ (ln(P_t / P_{t-1}))²

Where:

• P_t is the closing price of the S&P 500 on day t

By comparing the VIX squared at the beginning of a 30-day period to the realized variance over that same period, an investor can quantify the VRP and assess the profitability of a systematic VRP selling strategy.