Limitations of the Continuous-Time Lognormal Model

The Myth of Constant Volatility: The Volatility Smile

The first and most significant assumption of the GBM model that we will challenge is that volatility, $\sigma$, is constant. The model assumes that the stock's volatility is a fixed number, regardless of the stock's price, the time to expiry, or overall market conditions.

If this assumption were true, and if the lognormal model perfectly described market prices, then the implied volatility for options on the same stock should be the same for all strike prices. Implied volatility is the value of $\sigma$ that, when plugged into an option pricing model like Black-Scholes (which is built on GBM), returns the observed market price of the option. A plot of implied volatility against the strike price should therefore be a flat line.

In reality, this is not what we observe. For most asset classes, this plot forms a curve, famously known as the volatility smile or volatility skew.

As the video illustrates, the observed pattern is typically not flat:

• Volatility Smile: Implied volatility is lowest for at-the-money (ATM) options and increases for both out-of-the-money (OTM) and in-the-money (ITM) options. This creates a 'U' shape, or a smile. This is common in foreign exchange markets.
• Volatility Skew (or Smirk): This is an asymmetric smile, which is the typical pattern for equity markets. Implied volatility increases significantly for low-strike options (i.e., OTM puts) and is flatter for high-strike options (OTM calls).

The very existence of these patterns is direct market evidence that the constant volatility assumption of the lognormal model is flawed.

What the Smile Is Telling Us: Fat Tails and Skewness

The volatility smile is not just a curious anomaly; it is the market's way of telling us about the true shape of the asset return distribution. As a statistician, you know that a probability distribution is defined by its moments (mean, variance, skewness, kurtosis). The lognormal model assumes that continuously compounded returns are normally distributed, which implies zero skewness and zero excess kurtosis. The volatility smile tells us this is wrong.

1. Fat Tails (Leptokurtosis): A smile shape (higher implied volatility for options far from the current price) means that the market is assigning a higher probability to large price movements—both up and down—than the normal distribution would suggest. The distribution has "fatter tails," a property measured by kurtosis. Investors are willing to pay more for options that protect against extreme events, driving up their implied volatility.

2. Negative Skew (Crash-o-phobia): The more common smirk or skew in equity markets, where implied volatility is highest for low-strike puts, indicates that the market assigns a much higher probability to large downside moves (market crashes) than to large upside moves. This implies that the true distribution of returns is negatively skewed. This phenomenon became particularly pronounced after the stock market crash of 1987.

The following segments of the video you just watched explain this connection between the smile/skew and the higher moments of the return distribution.

The Problem of Continuity: Markets Can Jump

Another fundamental assumption of Geometric Brownian Motion is that the sample path of the stock price is continuous. This means the price changes smoothly over time without any sudden gaps or jumps. While this is a mathematically convenient property, it is not a realistic depiction of how markets behave.

Major, unexpected news—such as a surprise earnings announcement, a central bank decision, a geopolitical event, or a sudden market panic—can cause prices to gap up or down instantaneously. These are known as jumps. A pure GBM diffusion process cannot account for such events.

This limitation is vividly illustrated when comparing a simulated GBM path against actual historical market data, which includes periods of crisis.

As the paper demonstrates, while GBM might capture the general long-term trend, it is structurally incapable of reproducing the sudden, sharp losses characteristic of a market crash. This is a severe limitation, especially for risk management applications where understanding and preparing for extreme events is paramount.

A Summary of Core Limitations

We have now identified the primary weaknesses of the continuous-time lognormal model. These are frequently tested in the CM2 exam and are critical for any practitioner to understand.

The key limitations are:

• Stochastic Volatility: Volatility is not constant. It varies over time and depends on the asset price level, leading to the volatility smile/skew.
• Non-Normal Returns: Real asset returns exhibit "fat tails" (leptokurtosis) and skewness. The lognormal model, which assumes normally distributed log-returns, systematically underestimates the probability of extreme events.
• Price Jumps: Real asset prices can experience sudden, discontinuous jumps, which are not captured by the continuous GBM process.

These points are neatly summarized in the following resource.

While we have focused on the three most significant limitations, other assumptions of the model, such as a constant drift rate, no transaction costs, and perfect liquidity, are also simplifications of reality. However, the issues of volatility, fat tails, and jumps are the most profound and have spurred the development of more advanced models (e.g., stochastic volatility models, jump-diffusion models) that you will encounter later in your studies.

Conclusion

Today, we have taken a necessary step back from applying the lognormal model to critically evaluate its foundations. While GBM provides an indispensable introduction to stochastic modeling in finance, it is built on assumptions that do not hold up to empirical scrutiny.

Key Takeaways:

• The assumption of constant volatility is violated in practice, as evidenced by the existence of the volatility smile and skew.
• The smile and skew indicate that the true distribution of returns has fat tails (leptokurtosis) and, for equities, negative skew, meaning extreme events and crashes are more likely than the model predicts.
• The model assumes a continuous price path, failing to account for the sudden jumps and gaps that occur in real markets during major news events or crises.

Understanding these limitations is essential. It explains why the basic model can be dangerous if used naively for risk management and provides the motivation for the more sophisticated models used by professionals.

Having now explored a foundational single-asset model and its weaknesses, our next lesson will broaden our perspective. We will examine the Wilkie model, a multi-asset stochastic investment model specifically designed for long-term actuarial forecasting, which attempts to build a more holistic and economically interconnected view of asset returns.