Alternatives to Black–Scholes — HTML

Alternatives to Black–Scholes and What Happens When Gaussian Assumptions Fail

Below is a structured HTML version of the explanation covering: Black–Scholes assumptions, consequences of incorrect Gaussian assumptions, major alternative models, and a compact summary table.

1. Background: What Black–Scholes Assumes

Asset returns follow a log-normal distribution — equivalently, log-returns are Gaussian with constant volatility.
Price follows geometric Brownian motion:

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$
Volatility is constant (not stochastic).

2. What Happens if the Gaussian Assumption Is Incorrect?

Real markets deviate from Gaussian assumptions in several important ways:

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Fat tails

Large moves (jumps, crashes) occur much more frequently than a Gaussian would predict. Effect: Black–Scholes tends to underprice deep out-of-the-money options (puts and sometimes calls).

Skew / Smile

Empirically implied volatilities vary with strike; if Black–Scholes held perfectly the implied-volatility surface would be flat. Markets show a smile or skew.

Volatility is not constant

Volatility varies with time (stochastic volatility), exhibits clustering, and is affected by market events. Black–Scholes underestimates convexity and time-value effects when volatility is not constant.

Returns may have memory

Autocorrelation, volatility clustering, and other dependencies break the independent-increments assumption of geometric Brownian motion.

In short: When Gaussian/log-normal assumptions fail, Black–Scholes produces systematic pricing biases and practitioners typically back out an implied volatility surface to force the model to match observed prices.

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3. Major Alternatives to Black–Scholes

Below are widely used model families that address various weaknesses of Black–Scholes.

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A. Stochastic Volatility Models

Heston model — volatility follows a mean-reverting square-root process:

$\begin{aligned} dS_t &= \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_1\\ dv_t &= \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_2 \end{aligned}$

Pros: produces skew/smile and stochastic volatility; widely used for equities and FX.
SABR — popular in interest-rate derivatives (produces analytic approximations for the smile).
GARCH family — volatility driven by past returns (volatility clustering), useful in econometric and risk settings.

B. Jump–Diffusion Models

Merton jump–diffusion — adds Poisson jumps to the diffusion:

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t + J S_t\,dq_t$

Captures sudden large moves, produces fatter tails and helps explain the smile.
Kou double-exponential jump model — asymmetric jumps with heavier tails on one side; better fit for crash/rally asymmetry.

C. Lévy and Heavy-Tailed Models

Variance Gamma — replaces Brownian motion with a pure-jump process driven by a Gamma time-change.
CGMY / KoBoL — flexible infinite-activity Lévy models with tunable tail behavior.

These model families greatly increase tail mass and skew relative to Gaussian models.

D. Local Volatility Models

Dupire local volatility — volatility is a deterministic function of spot and time, $\sigma=\sigma(S,t)$ , calibrated to match the entire implied-volatility surface exactly.
Pros: exact calibration to market prices; Cons: questionable realistic dynamics for forward evolution (may misrepresent pathwise behavior).

E. Machine Learning & Data-Driven Models

Neural networks to model implied volatility surfaces or directly price options.
Reinforcement-learning based hedging strategies that learn from historical data rather than relying on closed-form Greeks.
Nonparametric density estimation for risk and pricing.

These methods are increasingly used in practice where rich data and computational power are available.

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4. Practical Consequences (Pricing & Hedging)

Systematic mispricing: deep OTM options (especially puts) often become underpriced by Black–Scholes.
Hedging failures: Greeks computed under wrong assumptions cause under- or over-hedging, especially around jumps — P/L surprises often occur.
Implied volatility surface: traders use implied vol as a corrective plug-in; better models attempt to explain and predict that surface rather than merely fit it.

5. Summary Table

Model Type	Fixes Gaussian Weakness?	Captures Jumps?	Captures Stochastic Vol?	Common Use
Black–Scholes	No	No	No	Baseline / pedagogical
Heston	Yes	No	Yes	Equity, FX
SABR	Yes	No	Yes	Rates (swaptions, caps)
Merton Jump	Yes	Yes	No	Modeling crashes / large moves
Kou Jump	Yes	Yes	No	Asymmetric jump fits
Variance Gamma / CGMY	Yes	Yes	Implicit	Exotic options, fat tails
Dupire Local Vol	Yes	No	Implicit	Exact calibration to IV surface
ML / Data-driven	Yes	Yes	Yes	Quant shops, empirical pricing

6. Want More?

If you’d like, I can provide any of the following right away as HTML/attachments:

A visual comparison of Gaussian vs heavy-tailed distributions (plot).
Diagrams contrasting Black–Scholes, Heston, and Jump-Diffusion dynamics.
Concrete pricing code (Python) for Heston or Merton jump model with sample calibration.
An explainer on how implied-volatility surfaces are built and used in practice.

Black Scholes – and non Gaussian Price Distributions