Nonparametric Option Pricing: Theory and Empirics

Principal Investigator:

Professor Xiao QIAO
Assistant Professor, School of Data Science

Co-Principal Investigator:

 Professor Qi WU
Associate Professor, School of Data Science

Project Period:  1 November 2022 - 31 October 2024

Option contracts give their holders the right but not the obligation to buy or sell the underlying asset on some predetermined date. These instruments are widely used for investment and risk management. Corporate managers often use options to control or eliminate any undesirable risk they are exposed to. For example, airlines may use option contracts on crude oil to limit their risk exposure to changing jet fuel prices, because the price of jet fuel closely depends on the price of oil. Since option prices govern the cost of risk mitigation, the ability to determine whether an option is cheap or expensive would be immensely valuable for corporations that use options to hedge their risks.

How can we make this determination? To answer this question, we must take a stance on the value of an option. Our task then becomes building an option pricing model and applying it as a benchmark to evaluate prevailing prices in the marketplace. A central idea in modern finance is the Law of One Price: If two investments provide the same cash flows in the future, then they must have the same price today. If this condition were violated, then an investor can take a long position in one investment, a short position in the other, and construct a portfolio free from any risk that still pays some money up front. As this type of “free lunch” should not exist in a well-functioning market, we can invoke the Law of One Price to determine asset prices. Applied to option valuation, if we can construct a portfolio that offers the same payoff as an option, the price of such a portfolio would be the price of the option (Merton, 1973).

Most existing option pricing models follow three steps: 1) specify a return process for the underlying asset, 2) derive the pricing relation between the underlying and options, 3) use data to calibrate model parameters. Celebrated models such as Black and Scholes (1973) fit this description. Different models often differ in their assumptions about the underlying return process. However, it is rarely the case that the researcher can be certain about the underlying asset’s return dynamics, so the above approach is prone to model misspecification. A growing list of empirical shortcomings of existing option pricing models reflects the manifestation of such misspecification. 

In this project, we seek to develop a nonparametric option pricing model that does not require extensive assumptions about the underlying. We directly implement Merton’s original insight to compute the fair value of an option, which we call breakeven volatility (BEV), as the value of implied volatility (IV) that sets the profit and loss of a delta-hedged option position to zero. To the extent implied volatility differs from BEV, the market price contains either a premium or a discount. If the IV were higher (lower) than the BEV, a delta-hedged short (long) position would result in a positive trading profit. In this sense, BEV provides the fair value of an option to both sides of the contract. 

We seek to construct a large historical database of breakeven volatilities to understand their behavior. The finance literature has established that there exists a volatility risk premium (VRP): Implied volatility is on average higher than realized volatility. However, we currently have a limited understanding of the drivers of VRP and an incomplete documentation of its behavior. A comparison of breakeven volatility and implied volatility at the granularity of individual options can shed light on the empirical behavior of VRP and hint at possible theoretical explanations. 

We also strive to build a predictive a model for BEV. Through establishing a relation between option characteristics, such as moneyness and time to expiration, and breakeven volatility, our predictive model provides a nonparametric approach to price options without the need to specify the underlying price process. BEV predictions from our model can then be compared to implied volatilities to identify whether options are overpriced or underpriced. 

Our goal of identifying cheap and expensive options closely relates to issues in risk management. Any corporate managers who use options to manage their risk exposure can benefit from our findings. Through the identification of cheap and expensive options, our model reveals which options can more optimally be used as hedges. Managers who are aware of the relative attractiveness of options as hedges are empowered to make more productive and informed decisions in their risk management process. As such, our project has the potential to make a real-world impact on business practices.

The primary objectives of this project are listed below:

1. Understand the stylized facts around the volatility risk premium arising from the difference between breakeven volatility and implied volatility.
2. Construct a nonparametric option pricing model based on breakeven volatility values, explore different predictive methods and their implications.
3. Provide a theoretical foundation for explaining the volatility risk premium through the relationship between realized volatility and breakeven volatility.
4. Apply the proposed option pricing model to identify the best options for risk management, make a comparison to alternative approaches.