Probability Theory

Implied volatility (IV) and historical volatility (HV) are essential measures in probability theory. While IV is widely used, HV offers a grounded approach for calculating the probability of future pr

Implied volatility (IV) and historical volatility (HV) stand as two primary measures employed in the domain of probability theory, each with its strengths and limitations. While IV is widely used and quoted, HV offers a more grounded and accurate choice for calculating the probability of future price.

In this article, we will explore the technical nuances and mathematical foundations behind this assertion, examine specific algorithms, methodologies, and concepts used in advanced probability theory and financial modeling.

How Option Alpha Calculates the Probability of Future Price

If you've been around OA before, you already know that math and probabilities encompass a large part of how we trade. If interested, we have entire articles explaining the gritty details of how probability is calculated and how we generate true Expected Value from defined-risk trades. But in short, we use the probability distribution derived from Black-Scholes that describes the chance of the underlying stock price being below a target price level at some point in the future.

The inputs to that function are: the current underlying price, target strike price, days to maturity, risk-free rate, and volatility.

The volatility parameter is what I'd like to focus on. We use annualized volatility of the underlying as the volatility input to the equation, specifically the annualized 30-day standard deviation of returns, and it's important to understand why. IV has its place, but it's not entirely relevant to the problem we're trying to solve here. Recall that IV is solved for as the missing variable when reversing the option pricing equation, BSM or similar. But that IV is representative of a single option contract.

In Trade Ideas, for example, the Probability of Profit (POP) for an Iron Condor trade is the probability that the underlying price falls within the breakevens at expiration. An Iron Condor has 4 legs. The cumulative probability function has 1 input for volatility. So what do we do?

We are trying to figure out where the underlying price will be in the future, which means the IV of a single option leg won't help us. We could find some average of the individual legs' IVs, roll our own IV calculation that represents the underlying, rely on a black-box proprietary calculation for underlying IV, or use the annualized historical volatility, which is what we choose to do.

Assume in this article that Probability of Profit (POP) algorithms refers to a mathematically robust representation of probability calculation that is not simply a function or restatement of a Reward/Risk ratio.

Historical Volatility in Probability Equations

In POP algorithms, historical volatility is used instead of implied volatility to calculate the likelihood of profit. HV, representing the standard deviation of an asset's historical returns, offers a more accurate and predictive measure of the cumulative density area due to its reflection of actual price movements.

Since HV captures the realized price fluctuations of the underlying asset, and by incorporating HV into the POP calculation, we consider the complete historical distribution of the asset's returns. This comprehensive view enables a more accurate assessment of the likelihood that the underlying price will remain within the desired range based on its observed behavior over time.

The use of HV is a practical choice. It ensures that the full range of price movements observed in the past is taken into account. Consequently, a more representative and precise estimation of the likelihood of the underlying price falling within the desired range is achieved.

In "Index Option Prices and Stock Market Momentum," Amin, Coval, Seyhun (2004) examined the performance of HV and IV in options pricing and aimed to "test the prediction of standard option pricing models." They posited "that there should be no relation between option prices and past stock market movements." However,

"... evidence suggests that taking past stock returns into account results in more-accurate estimates of implied volatilities from index option prices, which have been used by many authors to explore arbitrage opportunities, measure risk premia in the stock markets, or forecast future volatilities."

The findings of the study revealed that incorporating HV as a measure of volatility led to more accurate pricing models and improved estimations of probabilities compared to using IV. HV captured the historical distribution of price movements, which provided a more complete view of the underlying asset's behavior and better reflected market dynamics. The researchers concluded that HV-based probability calculations resulted in superior predictions and a more reliable assessment of option prices.

Advantages of Historical Volatility

Incorporation of actual price movement. Historical volatility directly incorporates observed price movements and variations over time. This comprehensive view captures the impact of market conditions, shocks, and other factors on volatility dynamics.

Analysis of skew and term structure. HV allows for the direct analysis of the skew and term structure of volatility. We can observe variations in volatility levels across different strike prices and expiration dates, gaining insights into market perceptions of risk.

Tangible measure of realized volatility. Historical volatility provides a tangible and reliable measure of realized volatility based on actual price data. It serves as a robust foundation for estimating future volatility.

Reproducibility. In a later section we will discuss how deriving a single IV number from a landscape of options chains across expirations can be a very black-box endeavor. Anyone can produce an obscure, proprietary IV with a flurry of marketing material about how, "Our IV is better than their IV." In fact, there are several popular financial platforms that refuse to reveal or provide information on how their IV is calculated. Go ahead and ask them. What's more difficult is providing something that's truly reproducible and verifiable, and that is something we at OA stand behind.

IV typically overstates HV. IV is derived from option prices and reflects the market's expectation of future volatility, while HV is based on historical price movements. The options pricing models assume constant volatility, which is unrealistic in practice as market conditions change over time. Additionally, IV includes a risk premium to account for uncertain market conditions or perceived risk of extreme volatility.

HV is based on observed price movements that include both predictable and unpredictable factors, excluding anticipation events that can significantly impact prices. The forward-looking nature and market expectations embedded in IV tend to result in a higher estimate of future volatility compared to the retrospective nature of HV.

Limitations of Implied Volatility

In contrast, implied volatility is derived from option prices and reflects market expectations, not reality, of future price volatility. Calculated using options pricing models like Black-Scholes or similar, implied volatility is not directly observable and varies across different options and strike prices based on market participants' expectations and sentiment regarding future price movements.

In complex options trades such as iron condors, using IV directly becomes impractical as the probability algorithm requires a single volatility parameter representing the entire underlying asset. However, implied volatility can differ across options and strike prices due to factors such as time to expiration, strike price, supply and demand dynamics, and market sentiment specific to each option.

Incorporating implied volatility into the POP calculation necessitates deriving a single value that represents the overall volatility of the underlying asset. Yet, there is currently no standardized method for combining multiple implied volatilities into such a value accurately.

This single value represents an average or aggregate implied volatility across the selected options, typically encompassing different strike prices and expiration dates. The specific options chosen are based on factors like liquidity and moneyness (in-the-money, at-the-money, out-of-the-money), but regardless of the selection, a heuristic approach is employed.

Here's an overview of the process:

Option selection. Choose a representative set of options for the underlying stock. These options should have different strike prices and expiration dates, covering a range of maturities and moneyness levels. Commonly, at-the-money options (where the strike price is close to the current stock price) are included.

Option price input. Gather the current market prices of all selected options.

Implied volatility calculation. Apply an option pricing model, such as Black-Scholes, to calculate the IV for each option in the selected set. This is done by iteratively adjusting the volatility until the calculated option prices match the market prices. There are several common numerical algorithms used to do this:

• Newton-Raphson method, an iterative root-finding algorithm for solving nonlinear equations.

• Bisection method, a bracketing method to find the root of a function within a specified interval.

• Brent's method, a combination of the above methods that offers robustness and efficiency.

Weighted average. Calculate a weighted average of the implied volatilities obtained from the previous step. The weights can be based on various factors, such as option liquidity or market capitalization of the underlying stock. The exact methodology for assigning weights may vary depending on the provider or source of the quoted IV. These methods are usually unadvertised (black box).

Final IV value. The resulting weighted average implied volatility is the single IV value quoted for the entire underlying stock.

It's important to note that this single IV value is an approximation and represents the market's overall expectation of future volatility for the underlying stock. It provides a concise measure of the aggregate market sentiment for the stock's potential price movements.

Providers of implied volatility data, however, may use different methodologies in selecting options and calculating the weighted average, so the specific details of how the IV is derived are few and there can be a wide variance between sources. Comparing probability numbers between different softwares is never apples-to-apples, and this is why.

Due to this practical limitation, historical volatility is often used as a reasonable substitute in the algorithm for calculating POP. Historical volatility provides a single measure of price fluctuations observed in the past, which can be utilized as a consistent volatility input for the underlying asset in the context of the probability calculation.

Peering Beyond the IV

The process of calculating a single implied volatility value for an underlying has potential sources of misrepresentation. There are manipulative market forces at play influencing the price of an option that traders, especially retail traders, may not be aware of. Or, if they are aware of it, have no way to measure.

In the context of the POP equation, relying on the weighted IV can have certain significant drawbacks:

Simplified representation. The single IV value of an underlying represents an average or aggregate measure of expected volatility across a range of options. It assumes that volatility is constant across all strike prices and expiration dates, which may not be the case. In reality, volatility can vary depending on the specific option contract and market conditions. By condensing the IV into a single value, certain nuances and variations in implied volatility across different options may be overlooked or abstracted away.

Limited option sample. The contracts selected for calculating IV might not capture the full range of available options on the underlying stock. If illiquid or less actively traded options are excluded from the sample, the resulting IV value may not fully reflect the true market sentiment or expectations. Again, this part is not up to or controlled by you, the data consumer.

Weighting methodology. The weights assigned to individual options when calculating the weighted average IV can vary depending on the methodology used by different providers or sources. Different weighting schemes can lead to different IV values, potentially biasing the overall representation of volatility.

Skew and term structure. Implied volatility can exhibit a skew or smile pattern, where options with different strike prices have different implied volatility levels. This skew reflects the market's perception of potential asymmetric risks. A single IV value may not adequately capture the shape of the skew or the term structure of implied volatility, which could lead to misrepresentation of the underlying volatility dynamics.

Assumption of constant volatility. Implied volatility assumes a homogeneous volatility distribution across all strike prices and expiration dates. Variations in volatility across different options can be overlooked, leading to a simplified representation of volatility dynamics.

Supply and demand dynamics. Market makers and institutional investors, who often trade large volumes of options, can influence the supply and demand for specific option contracts. By actively trading or overtrading options, they can affect the prices and implied volatility levels. This may even be done strategically to impact the IV and potentially benefit their trading positions.

Volatility positioning. Market markers may also have an incentive to influence the IV to benefit their trading strategies. For example, if a market maker or a large investor holds a significant position in options, they might take actions that can temporarily affect the supply or demand for those options, thereby impacting the IV.

False quotes or rumors. Market participants can spread false quotes or rumors to manipulate perceived market sentiment. This can create artificial demand for options, affecting their prices and subsequently their IV.

Market maker activities. Market makers, who facilitate options trading and provide liquidity, have the ability to influence option prices and implied volatility through their trading activities. They adjust their quotes and hedging strategies based on their assessment of market conditions, which can not only impact calculated IV, but it is an unwanted inherent bias baked into the IV number.

It's extremely important for retail traders to be aware of these factors and exercise caution when interpreting and relying on implied volatility data. Multiple sources, analysis of option chains, and consideration of broader market factors can provide a more comprehensive view of the underlying volatility dynamics.

Conclusion

In this article we've discussed the motivations behind the use of Historical Volatility (HV) as the input parameter to Black-Scholes probability instead of the less trustworthy weighted Implied Volatility (IV) of the underlying security.

HV provides a more robust measure of volatility because it considers a broader range of market conditions and price movements over a specified time period. Unlike IV, which can be influenced by short-term market sentiment, HV takes into account a longer time frame, providing a more comprehensive view of volatility dynamics.

This empirical foundation lends credibility to HV as a reliable measure of realized volatility, as it is derived from a large sample size rather than relying on theoretical assumptions or market sentiment.

Since HV is based on readily available historical price data, it remains consistent over time and can be recalculated for any given period. It provides valuable insights into the potential range of price movements and assists in making informed decisions about risk exposure and position sizing. There is practicality and real-world relevance of HV that is simply not present in momentary IV.