The Black-Scholes model, initiated in 1973 by Fisher Black, Myron Scholes, and Robert Merton, gave the world its first mathematical basis for modern options pricing.
The method takes on all of these key variables—including stock price, strike price, expiration, interest rates, dividends, and implied volatility—to find the theoretical value of an option.
However, while this model revolutionized the field, it does assume a rather overly simplistic world—one of continuous volatility and frictionless markets.
In time, the newer models, such as Cox-Ross-Rubinstein and others, have built on some of its limitations and, yet, none of them manages to capture the complexities in the real markets.
How Retail Traders Experience Differentials
When retail traders employ pricing models, the discrepancies between theoretical and actual options prices may be substantial enough.
Here’s why:
- Market Dynamics vs. Model Assumptions: Models assume a set of idealized conditions that may not reflect real market behavior. For instance:
- Volatility is assumed constant, but in reality, it changes frequently.
- Interest rates and dividends may vary slightly depending on the source.
- Bid-Ask Spreads: Market makers buy options at the bid price and sell at the ask price. This spread makes a difference between theoretical and executable prices, favoring market makers who profit from this margin.
- Proprietary Models: Different brokerage firms use proprietary tweaks to standard models, so the Greeks and calculated prices will differ minutely.
- Implied Volatility Variations: One of the most important inputs in option pricing is IV but also the most challenging to pin down. IV is actually derived from market prices rather than the other way round. This can cause mismatches when traders input an estimated IV that doesn’t agree with the current market conditions.
Key Elements in Option Pricing
To address these differences, it’s important to go back to the key elements of option pricing models:
- Stock Price: Quotable.
- Strike Price: The trader selects it.
- Expiration Date: Days until expiration, which has a bearing on time value.
- Interest Rates: Usually a short-term money market rate and easily accessible.
- Dividends: Available publicly if the stock pays dividends.
- Implied Volatility (IV): A calculated number representing what the market believes future volatility will be.
While most of these inputs are set or available, IV is a moving variable that often most confuses people.
Historical vs. Implied Volatility
The difference between historical volatility (HV) and implied volatility (IV) needs to be understood.
Historical Volatility
- Measures past price movements in a stock.
- Calculated using the standard deviation of price changes over a time period, say 30, 60, or 90 days.
- Always annualized unless otherwise stated.
Implied Volatility
- Reflects the market’s expectation of future volatility.
- Solved backward from the current price of an option in the market using pricing models.
- Annualized unless otherwise stated (e.g., 30-day implied volatility).
HV measures past movements, whereas IV supports the current price of an option based on market expectation.
The Role of Market Makers
A price disparity is very crucially brought about by market makers.
A retail trader will trade directionally to make a profit; however, market makers need to create their margin based on the bid-ask spread and arbitrage opportunity.
They may fine-tune IV and other parameters within the models to remain competitive in terms of pricing.
Retail traders must remember that market makers are not bound to theoretical prices; they interact with real-time orders, supply, and demand, often influencing prices dynamically.
Addressing the “Mismatch”
If your calculated price doesn’t match the market price, consider the following:
- Implied Volatility Adjustment: Reassess the IV you’re using. Since IV is derived from the market, it might differ from your assumptions.
- Execution Realities: Be aware that you go long the ask and sell short the bid. Your algorithm might use a mid-point price which isn’t a representation of that spread.
- Other Models: Examine leading pricing models which include adjustments in variance or other features associated with reality.
- Practical Take: Retail traders profit more when using direction and trading plan without paying attention to tiny mismatches in the prices of bids and asks.
Conclusion
Although good, Black-Scholes and similar models for pricing options are far from reality.
There exist other variables that make the spread between bid and ask to differ in value, including the variations in implied volatility and even the proprietary brokerage models, among others.
Instead of getting bogged down in all the details, traders will use these models as tools in making decisions but are aware of the limits.
Knowing the reasons why mismatches happen will enable you to be more confident in trading in the options market and make informed decisions.