Option Pricing via Breakeven Volatility

Abstract

The fair value of an option is given by breakeven volatility, the value of implied volatility that sets the profit and loss of a delta-hedged option to zero. We calculate breakeven volatility for 400,000 options on the S&P 500 and build a predictive model for these volatilities. A two-stage regression approach captures the majority of the observed variation. By providing a link between option characteristics and breakeven volatility, we establish a non-parametric approach to pricing options without the need to specify the underlying price process. We illustrate the economic value of our approach with a simulated trading strategy based on breakeven volatility predictions.

Keywords:

• options
• prediction
• trading strategy
• volatility

PL Credits: 2.0:

Acknowledgments

The authors thank William Goetzmann (Executive Editor), Luis Garcia-Feijoo (Managing Editor), Daniel Giamouridis (Co-Editor), two anonymous referees, Petra Bakosova, Yeguang Chi, Jake DeSantis, Ruslan Goyenko, Steve Heston, Ray Iwanowski, Craig Iseli, Kris Jacobs, Jason Wei, Lei Wu, Qi Wu, and Zonghao Yang for helpful comments. We are especially grateful to Ben Albert for extensive discussions and data processing.

Disclosure statement

The authors report no funding sources. The authors alone are responsible for the content and writing of the paper. Hull Tactical Asset Allocation, LLC is a registered investment adviser that deploys strategies, such as the one described in this paper in investment products including separately managed accounts and an exchange-traded fund. All errors are our own.

Notes

1 Traditional option pricing models typically try to match the market price of options, whereas our pricing model provides the fair value. In this sense, traditional models are positive models that capture how the world is, whereas our model is normative.

2 Parametric models, such as Black-Scholes can be constrained in matching certain empirical patterns. Stochastic volatility models may have difficulty eliminating pricing errors in short-term option prices because the distribution of the underlying does not have enough kurtosis. Jump models may have issues with longer-term options because they revert too quickly to the Black-Scholes prices as the time to expiration increases. Nonparametric models can be more flexible in capturing the features of the data, which must be pre-specified and built into parametric models.

3 For stock indices, deviation from put-call parity is typically small. For individual stocks that are hard-to-borrow, deviation from put-call parity could be much larger.

4 This expression is approximate. To be precise, the summation is missing a borrowing or lending term equal to (${\Delta }_{t-1}{S}_{t-1}-c_{t-1}).$ At the daily frequency, this term is very close to zero, especially in a low-interest environment that our data span.

5 The iterative approach we take was used for a different purpose by Dumas et al. (1998), who try to determine whether the delta-hedging component matched the changes in option prices. They show that the Black-Scholes model provides the best approximation for option price changes.

6 Our calculation of breakeven volatility requires knowing the future path of the option, so it cannot be implemented in real time. To compute BEV in a real-time implementation, we would require a predictive model that uses currently available information to forecast the eventual BEV values.

7 There are a number of alternatives ways of hedging. One could choose to hedge at market open rather than at market close, use delta calculated from the at-the-money volatility or forecasted volatility, hedge only when the net delta exceeds a certain threshold, or adjust the hedge ratio depending on whether the order flow is believed to be informed. Different hedging methods correspond to different objectives, and there is no single best method. On a practical note, the BEV calculation should match the specific trading desks’ preferred hedging policy.

8 More frequent delta hedging (i.e., intraday) can alleviate the issue of large changes in gamma in the volatility calculations near maturity, especially for ATM options.

9 Extending our calculation to the expiration date will not significantly change our results. We do find that numerical computation of breakeven volatilities in the last several days just before expiration to be more difficult, as more volatility calculations do not converge in the days immediately before expiration. More frequent delta hedging can improve this convergence issue.

10 $\mathit{Normalized} \mathit{Strike}= \frac{\mathit{log}\left(\frac{S{e}^{r(T-t)}}{K}\right)}{\mathit{ATM}.IV\sqrt{T-t}}$ for underlying price $S,$ strike price $K,$ the at-the-money implied volatility $\mathit{ATM}.IV,$ and time to expiration $T-t.$ The annualized interest rate $r$ is set to 0.5%.

11 $d_1$ in the Black-Scholes model is given by the following expression: $$d_1= \frac{1}{\sigma \sqrt{T-t}}\left[\mathit{log}\left(\frac{S_t}{K}\right)+\left(r+\frac{{\sigma }^2}{2}\right)(T-t)\right]$$ where $S_t$ is the price of the underlying, $K$ is the strike price, $r$ is the interest rate, $\sigma$ is the volatility of the underlying, and $T-t$ is the time to expiration.

12 The exponential function, $e^{(·)},$ is convex. Jensen’s inequality states that for a convex function $f,$ $f\left(E\left[X\right]\right)\le E\left[f\right(X\left)\right].$

13 We compare our supervised approach with the parametric methods of Duan (1983) and we find that our approach achieves the lowest mean squared error.

14 With 396,899 observations and 17 predictors, the adjusted R-squared is 1.00004 times larger compared to the unadjusted R-squared.

15 We thank an anonymous referee for this suggestion.

16 If Equation (9) were violated, a costless arbitrage profit may be earned by writing a put with strike price $K_2,$ buying $\frac{K_3-K_2}{K_3-K_1}$ units of the put with strike price $K_1,$ and buying $\frac{K_2-K_1}{K_3-K_1}$ units of the put with strike price $K_3.$

17 We also explore thresholds of $0.50, $2, and $3, and we find our results are qualitatively unchanged. A higher trading threshold has two offsetting effects: We have more conviction on each trade, but we may miss some profitable opportunities that do not exceed the threshold. In our sample, these two effects apparently have similar magnitudes such that changing the threshold has only a marginal effect on the strategy performance.

18 Our return calculation is on the conservative side. By taking the notional value as the denominator, we have made the implicit assumption that the option positions are fully collateralized with 100% margin requirement. In practice, margin requirements typically range between 5 and 30% for puts and calls on the S&P 500 Index.

19 Private correspondence with option traders indicates that unless one wants immediate execution, transacting one tick from the midpoint price is achievable with very high probability.

Additional information

Notes on contributors

Blair Hull

Blair Hull is a founder and chairman of Hull Tactical Asset Allocation, LLC, Chicago, IL.

Anlong Li

Anlong Li is a senior financial engineer at Hull Tactical Asset Allocation, LLC, Chicago, IL.

Xiao Qiao

Xiao Qiao is an assistant professor at the City University of Hong Kong and Hong Kong Institute for Data Science, Hong Kong, China.