Breaking Down the Black-Scholes-Merton Model Assumptions
Let's break down the issues with the Black-Scholes-Merton (BSM) model's assumptions, but before that I will be assuming that you know the following. If you don't then here are links to the prerequisites:
Prerequisites
- Introduction to the Black-Scholes
- Option Greeks: A Deep Dive
- The Math Behind BSM
- Popular Math Notations in Finance
- Volatility: A Primer
I. Returns being Log-Normally Distributed
What it means in BSM: The model assumes that the percentage changes (returns) in the underlying asset's price, over a short period, follow a normal distribution. Mathematically, this translates to the price itself following a log-normal distribution. This means the price is always positive (since the exponential of a normal variable is always positive) and that large upward movements are more likely than equally large downward movements.
Why it's a problem:
- Fat Tails (Kurtosis): Real-world financial data often exhibits "fat tails" meaning that extreme price movements (large gains or losses) happen more frequently than predicted by a normal distribution. BSM underestimates the probability of large price swings. This is very important for options pricing, as options benefit (or suffer) disproportionately from big moves.
- Skewness: Actual returns are often skewed, meaning the distribution is not symmetrical. For example, a distribution might have a longer tail on the negative side (more frequent large losses). BSM assumes a symmetrical distribution.
- Market Crashes/Jumps: The BSM model doesn't adequately account for sudden jumps in price, often caused by unexpected news or events. A log-normal distribution assumes price movements are continuous and gradual.
II. Volatility being Constant
What it means in BSM: The model assumes the volatility of the underlying asset (the standard deviation of its returns) remains constant over the entire life of the option.
Why it's a problem:
- Volatility Smile/Skew: In reality, implied volatility (the volatility implied by market option prices) varies depending on the strike price of the option. When you plot implied volatility against strike price, you often see a U-shaped "smile" or a slanted "skew." This contradicts the BSM assumption of a single, constant volatility for all options on the same underlying asset with the same expiration date.
- Volatility Clustering: Volatility tends to cluster. Periods of high volatility are often followed by periods of high volatility, and periods of low volatility by periods of low volatility. BSM doesn't capture this time-varying nature of volatility.
- Volatility as a Function of Price: Volatility sometimes moves inversely with price. When prices fall quickly, volatility often rises (panic selling).
III. Having a Closed-Form Solution
What it means in BSM: The BSM model provides a direct formula (a closed-form solution) to calculate the theoretical price of an option. You simply plug in the inputs (stock price, strike price, time to expiration, risk-free rate, volatility) and get the option price.
The Black-Scholes formula is as follows:
$$ \begin{aligned} C &= S_0 N(d_1) - K e^{-rT} N(d_2) \\ \text{where: } \\ d_1 &= \frac{\ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} \\ d_2 &= d_1 - \sigma\sqrt{T} \end{aligned} $$Where:
- \(C\) = Call option price
- \(S_0\) = Current stock price
- \(K\) = Strike price
- \(r\) = Risk-free interest rate
- \(T\) = Time to expiration
- \(N(x)\) = Cumulative standard normal distribution function
- \(\sigma\) = Volatility of the stock
Why it's related to the other problems:
While having a closed-form solution is computationally efficient, it's a consequence of the simplifying assumptions. If you try to relax those assumptions (e.g., allow for stochastic volatility, jumps in the price process), you generally lose the closed-form solution. You then have to rely on numerical methods (e.g., Monte Carlo simulations, binomial trees) to approximate the option price, which are more computationally intensive. The desire for a simple, easy-to-use formula forced the model to make unrealistic assumptions.
IV. Assuming Events are Independent
What it means in BSM: The model implicitly assumes that price movements in the underlying asset are independent of each other. The past price history has no bearing on future price movements, beyond its impact on estimated volatility.
Why it's a problem:
- Correlations: Financial markets are interconnected. The price of one asset can be affected by the price of other assets or by broader market trends. The BSM model doesn't account for these correlations.
- Mean Reversion: Some assets exhibit mean reversion, meaning that prices tend to revert to their average levels over time. This violates the independence assumption.
- Market Psychology: Human behavior plays a role in markets. Sentiment, herd behavior, and other psychological factors can lead to dependencies in price movements.
V. In Summary: Why These Assumptions Matter
The flawed assumptions of the BSM model mean that:
- Option prices calculated using BSM can be inaccurate, particularly for options that are far out-of-the-money (very high or very low strike prices). These options are more sensitive to the "fat tails" problem.
- Hedging strategies based on BSM may not be as effective in practice, especially during periods of high volatility or market stress. The hedge ratios (delta, gamma, etc.) calculated by the model are based on the constant volatility assumption, and if volatility changes unexpectedly, the hedges can become ineffective.
- Risk management decisions based solely on BSM can underestimate the true level of risk. The model doesn't capture the full range of possible outcomes, particularly the rare but potentially devastating extreme events.
VI. What's Used Instead?
Because of these limitations, practitioners have developed more sophisticated models that try to address these issues, including:
- Stochastic Volatility Models (Heston model, SABR model): These models allow volatility to vary randomly over time.
- Jump Diffusion Models (Merton jump-diffusion model, Kou model): These models incorporate the possibility of sudden jumps in asset prices.
- Local Volatility Models: These models allow volatility to be a function of both time and the underlying asset's price.
- Implied Volatility Surfaces: Rather than relying on a single volatility input, using the entire implied volatility surface (as a function of strike and maturity) to price and hedge options.
- Numerical Methods (Monte Carlo Simulation, Binomial Trees): Used when closed-form solutions are unavailable.
While BSM has its limitations, it is still a valuable tool for understanding options pricing and hedging, and it serves as a foundational model for more advanced techniques. It's crucial to be aware of its shortcomings and to use it judiciously.