Intro to Option Pricing, Greeks, and the Implied Volatility Surface

[WITH CODE] Option pricing accuracy and market insights in the IV surface from academic research

Reviewed and updated 20 July 2026

Hello!

This week, I’m bringing you a different kind of market analysis. Paid subscribers expressed interest in exploring option pricing and potential market inefficiencies identified in academic research.

Today, I’ll be covering basic option pricing models, the option Greeks, and the implied volatility (IV) surface.

Even if you’re already familiar with these concepts, I wanted to start with the fundamentals before diving into more advanced topics. I’d also love to hear your feedback. Let me know which aspects of this topic you’d like to see explored further.

Let’s get into it.

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Option Pricing Models and History

I am sure that most of you are familiar with binomial pricing and the Black Scholes Model (BSM). However, I’ll give a quick overview to those that are interested.

Prior to 1973, the pricing of options was slow and complicated. Often, there was little edge to be captured. Keep in mind that options markets before 1973 were not nearly as popular (or liquid) as they are today.

Historical growth in U.S. options average daily volume by expiration type. Source: OCC and Nasdaq Economic Research.

In 1973, Fischer Black and Myron Scholes introduced the Black–Scholes model for option pricing. Robert Merton separately extended the framework, now commonly called Black–Scholes–Merton. This revolutionized the industry, and the framework is still widely used today (even with its limitations). It allowed traders to systematically hedge option positions (for example, delta hedging) and understand the impacts of various inputs.

Binomial pricing serves as an intuitive model to foundationally understand the BSM. The binomial model uses backward induction to value an option today based on future movements (steps) of the underlying.

As each time step becomes smaller—and the number of steps increases—the binomial model converges to the continuous-time process used in the Black-Scholes framework.

The formula for the BSM is shown below to refresh your memory. The original version takes in the inputs of the underlying spot price, the strike price, the time to maturity (in years), the risk free rate, and the implied volatility of the underlying.

Black–Scholes formula for European options without dividends.

A common extension for valuing options on stocks that pay dividends adds a continuous dividend yield (St turns to Se-qt and subtract q from r in d1). The framework was also adapted for futures (Black Model) and currency options (Garman-Kohlhagen Model).

Options are primarily constructed in two ways: American and European options. The major difference between these two types of option structures are that American options allow for early exercise of the option by the buyer, and European options are only able to be exercised on expiration.

The other major type of option structure is the Bermudan option, in which the option can be exercised on specific dates before expiration.

The Black-Scholes-Merton (BSM) model assumes that volatility is constant across strike prices and throughout the life of the option. However, in reality, implied volatility varies by strike and expiration, leading to volatility skews and smiles.

Additionally, BSM assumes that the underlying asset follows a lognormal price distribution, meaning that continuously compounded (log) returns are normally distributed over a fixed horizon. However, real-world market returns exhibit fatter tails (higher kurtosis) than a normal distribution, leading to a greater probability of extreme moves.

The BSM also assumes that prices follow a random walk, that the risk-free rate is constant, that the option can only be exercised at expiration (so it only works with European options), and other assumptions about the market (no transaction costs, perfect liquidity, no restrictions to short selling, and no arbitrage).

While the Black-Scholes-Merton (BSM) model has well-documented limitations, some research suggests that it can still provide reasonably accurate pricing under certain conditions. A recent empirical study on the U.S. stock market found that BSM-calculated premiums closely matched actual market premiums for call options on seven out of nine stocks analyzed. However, the model struggled with put options, where only four out of nine stocks showed no significant pricing difference (Frontiers in Applied Mathematics and Statistics). This suggests that while BSM can be useful for pricing some options, its accuracy is inconsistent across different market conditions and option types.

That said, a large body of research highlights BSM’s inherent pricing biases. One of its biggest shortcomings is its assumption of constant volatility, which does not hold in practice. Empirical studies have shown that implied volatility varies with strike price and expiration, leading to systematic biases in BSM pricing, particularly for deep in-the-money or out-of-the-money options (Columbia University).

Additionally, BSM assumes that prices follow a lognormal distribution, but financial markets exhibit fat tails (higher kurtosis), meaning extreme price moves occur more frequently than the model predicts. A single constant-volatility Black–Scholes calibration cannot fit observed option prices across all strikes; market prices often imply higher volatilities in the wings than near the money.

BSM also struggles with long-maturity options, where its simplifying assumptions become less reliable over extended time horizons. Research has shown that factors like stochastic volatility, jumps in asset prices, and changing interest rates make BSM less effective for pricing long-dated options, leading to the adoption of alternative models like Heston’s stochastic volatility model and Merton’s jump-diffusion model (Diva Portal).

Despite these issues, BSM remains widely used in the industry. Traders often adjust for its flaws using implied volatility surfaces, risk-neutral adjustments, and numerical techniques rather than abandoning the model altogether. While BSM is not perfect, it still serves as the foundation for modern derivatives pricing, with modifications and extensions making it more applicable to real-world markets.

The research companion at the end of this post includes the Black–Scholes–Merton model with dividends, the primary Greeks, and a guided Jupyter notebook.