Why Black–Scholes Isn't Enough
The Black–Scholes world
Black–Scholes models the stock price as geometric Brownian motion:
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Black–Scholes models the stock price as geometric Brownian motion:
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Lesson close · From Diffusion to Jumps: Lévy Models in Finance
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Sequence
Continue to Lévy Processes & the Characteristic Triplet, the next prerequisite step in this course.
Its paths are continuous — between any two prices the process passes through every value in between. Mathematically convenient, empirically false.
Extreme moves occur far more often than a normal law predicts. The empirical distribution of daily returns has fat tails — its kurtosis sits well above the Gaussian value of 3.
Down-moves are sharper and faster than up-moves. Equity returns are negatively skewed: crashes happen in days, recoveries take months.
Prices move discontinuously on news — an earnings miss, a rate decision, a default — with no path between the old price and the new one. A continuous model cannot represent this at all.
We replace Brownian motion with a Lévy process — independent, stationary increments like Brownian motion, but allowed to be discontinuous. The price becomes
where can drift, diffuse, and jump. Every model in this course — tail risk (VaR/ES), credit default, insurance ruin — is just a choice of the jump structure of .
Interactive · GBM vs jump-diffusion path
secondary=GBM · primary=jumps
Free preview. Subscribe to continue to the formal definition and the characteristic triplet.
Based on “From Diffusion to Jumps: Lévy Model in Finance” by Ibrahim Lanre Adedimeji. Watch the guest lecture and read the original article.
Knowledge check
Q1. Which single property of geometric Brownian motion is the root cause of its tail-risk failure?
Q2. An asset shows frequent overnight gaps and crashes sharper than its rallies. Which two features are these, respectively?
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