MatheLinuxMatheLinux
Curriculum
Sign inStart free
From Diffusion to Jumps: Lévy Models in Finance
0%

0 of 5 lessons complete

Lessons

  • Why Black–Scholes Isn't Enough8m
  • Lévy Processes & the Characteristic Triplet8m
  • Poisson & Compound Poisson Jumps8m
  • Tail Risk: VaR & Expected Shortfall under Lévy8m
  • Default as a Jump: Credit Risk & Ruin Theory8m

MatheLinux — quantitative finance, taught rigorously.

Course content credited to Ibrahim Lanre Adedimeji.

← From Diffusion to Jumps: Lévy Models in Finance

Why Black–Scholes Isn't Enough

In this lesson

  • Why the continuity of geometric Brownian motion is its fatal modelling flaw
  • The three empirical features of returns a Gaussian model cannot produce
  • How replacing Brownian motion with a Lévy process sets up the rest of the course

The Black–Scholes world

Black–Scholes models the stock price as geometric Brownian motion:

dSt=μSt dt+σSt dBt.dS_t = \mu S_t\, dt + \sigma S_t\, dB_t .dSt​=μSt​dt+

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?

Mark this lesson complete to track progress
σSt​dBt​.

Its paths are continuous — between any two prices the process passes through every value in between. Mathematically convenient, empirically false.

Intuition

A continuous path means the price can never gap. But markets gap constantly: an earnings miss, a central-bank surprise, a default. Overnight the price simply opens somewhere new, with no trades in between. No continuous model can produce that.

Three things real returns do that Gaussians can't

Heavy tails

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.

Asymmetry (skew)

Down-moves are sharper and faster than up-moves. Equity returns are negatively skewed: crashes happen in days, recoveries take months.

Jumps

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.

Gaussian (thin tails)Heavy-tailed (Lévy)large losslarge gain
Heavy-tailed returns place far more probability on extreme moves than a Gaussian with the same variance — identical standard deviation, dramatically more tail risk. · Source: Cont (2001), Empirical properties of asset returns, Quantitative Finance

Common pitfall

A normal distribution assigns a probability near 10−410^{-4}10−4 or smaller to a −20%-20\%−20% day. Markets deliver them every few years. If your risk engine is Gaussian, your tail risk is silently underestimated — the model will look calm right up until it breaks.

The fix: let the process jump

We replace Brownian motion with a Lévy process LtL_tLt​ — independent, stationary increments like Brownian motion, but allowed to be discontinuous. The price becomes

St=S0 eLt,S_t = S_0\, e^{L_t},St​=S0​eLt​,
diffusion (continuous)jump-diffusionjump
A jump-diffusion sample path: continuous diffusion punctuated by a discontinuous jump — the gap a pure Brownian model can never produce.

where LtL_tLt​ 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 LtL_tLt​.

Key takeaways

  • Black–Scholes fails because GBM paths are continuous and thin-tailed.
  • Real returns show heavy tails, skew, and genuine jumps.
  • Lévy processes keep BM's nice increment structure but add jumps — the single idea the whole course builds on.

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.