Concept
Differential equations with a random noise term. Foundation of Black-Scholes option pricing (1973) and modern finance.
Understand it in one breath
Add random fluctuations such as Brownian motion to a deterministic rate of change. SDEs model finance, particle diffusion, and biology, but real stock prices do not exactly follow one simple geometric-Brownian model. The 1997 economics prize recognized Merton and Scholes for a new method of valuing derivatives; collaborator Black had died and Nobel prizes are not awarded posthumously.
At a glance
Model | SDE | Application |
|---|---|---|
Geometric Brownian motion (GBM) | dS = μS dt + σS dW | Stock prices; Black–Scholes |
Vasicek interest-rate model | dr = a(b−r) dt + σ dW | Mean-reverting interest-rate model |
Heston volatility model | dv = κ(θ−v) dt + ξ√v dW | Volatility is stochastic too |
Langevin equation | dX = −∇U dt + √(2D) dW | Particle physics and MCMC |
Neural SDE | dh = f(h,t) dt + g(h,t) dW | Stochastic neural networks (2018+) |
The Black–Scholes–Merton model supplies a benchmark option price under specific assumptions. Real markets require richer models and risk management for volatility smiles, jumps, transaction costs, and other effects.
Key formula
Modern applications
Option pricing, interest-rate models, algorithmic trading, and policy-gradient methods in reinforcement learning.
Beyond MathVoyage
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