What is the Volatility Risk Premium? | by Roman Paolucci | Towards Data Science

After the derivation of the Black-Scholes model, the discussion is open to its place in pricing vanilla options. The market dictates the interest-rate and option price for a particular strike and expiration by the laws of supply and demand. The only parameter not readily available as an input in the Black-Scholes equation is volatility. Since every parameter except volatility is available and dictated by the market, the inverse of the Black-Scholes equation allows us to find the volatility expected by market participants: implied volatility. Empirically, the interesting thing about implied volatility is its tendency to overestimate realized volatility largely due to the expectations of market participants and its realized counterpart.
Consider the following quote for a European call on the SPY.
Using a generic Monte Carlo pricing simulation allows us to illustrate this phenomenon. We’ll assume the underlying asset follows a diffusion process (geometric Brownian motion). Consider the following Python code to compute the expected net present value of cash flows to be generated by the European call option. In other words, the fair price of the option.
The resulting call price given by the Monte Carlo simulation is:
Monte Carlo Euro Call price:  5.09259969279995
Now, consider the following Monte Carlo simulation price using the market’s implied volatility.
Monte Carlo Euro Call price:  6.450104207093876
In a perfect world, the value of the option using the implied volatility will converge to the Black-Scholes or market value of the option. The real question is why is there a discrepancy between the fair price of the option and the market price of the option? Looking at volatility, we can ask another question: why does the volatility implied by the market tend to be higher than the realized volatility?
In a study from Yale, market participants tend to overestimate the probability of a significant crash, leading to a higher demand for options. This heightened perception of risk may lead to a higher willingness to pay for these options to hedge a portfolio. In other words, investors are willing to pay a premium for options to have protection against significant market crashes even if statistically the probability of these crashes is lesser or even negligible.
The notion that the volatility risk premium exists implies the ability to extract and benefit from both profits and diversification. The purest way to extract the VRP lies in a short put option position on the S&P500. In the long run, this strategy yields positive returns while also maintaining diversification benefits as it exhibits an extremely low correlation with equities.
The volatility risk premium is the notion that implied volatility tends to be higher than realized volatility as market participants tend to overestimate the likelihood of a significant market crash. This overestimation may account for an increase in demand for options as protection against an equity portfolio. Consequently, this would raise the price of these options, and account for the tendency of implied volatility being higher than realized volatility. The existence of the volatility risk premium can be exploited as a profitable strategy while assisting in diversification as a short option position in the S&P500 tends to maintain a low beta.
Marroni, L., & Perdomo, I. (2014). Pricing and hedging financial derivatives: A guide for practitioners. Hoboken: Wiley.
Natenberg, S. (2015). Option volatility and pricing: Advanced trading strategies and techniques. New York: McGraw-Hill Education.
Sullivan, R., Israelov, R., Ang, I., & Tummala, H. Understanding the Volatility Risk Premium. Retrieved September 29, 2020, from https://www.aqr.com/Insights/Research/White-Papers/Understanding-the-Volatility-Risk-Premium