Structure in Motion
dSt=μ(St,t)dt+σ(St,t)dWt
On (Ω,F,(Ft)t≥0,P)(Ω,F,(Ft)t≥0,P), with WtWt a Brownian motion, the asset price StSt is an FtFt-adapted semimartingale with drift μ(St,t)μ(St,t) and diffusion σ(St,t)σ(St,t). Under an equivalent martingale measure QQ, the discounted price is a martingale, yielding arbitrage-free valuation, pricing consistency, and tractability under both physical and risk-neutral measures. This forms the baseline from which broader and more refined modeling frameworks are developed.
Notation
-
St: asset price
-
μ(St,t)μ(St,t): drift (expected return)
-
σ(St,t)σ(St,t): volatility (diffusion coefficient)
-
WtWt: Wiener process (Brownian motion)
Extensions
Brownian models omit fat tails, clustering, and jumps; modeled through:
-
Stochastic volatility: Heston, CIR/OU factors
-
Rough volatility: fractional Brownian components
-
Jump–diffusion: Poisson, Merton, Bates
-
Lévy processes: infinite-activity jumps
-
Regime switching: Markov state models
-
Multi-asset systems: correlated vector SDEs
-
Control formulations: HJB-based optimal control
Methodology
-
Measures: Girsanov, Esscher, numeraire change
-
Filtering: Kalman, particle, Bayesian
-
Simulation: Monte Carlo, quasi, variance reduction
-
Estimation: MLE, GMM, Bayesian
-
Numerics: PDE, FFT, BSDE
-
Control: HJB, dynamic programming
Calibration
-
Surfaces: skew, smile, term-structure
-
High-freq: realized variance, bipower
-
Estimation: MLE, Bayesian, spectral
-
Robustness: stability, regime sensitivity
-
Microstructure: order book, liquidity
Applications
-
Option pricing: Heston, jumps, Lévy
-
Portfolio: HJB, BSDEs
-
Risk: tails, jumps, contagion
-
Execution: trading, liquidity
-
Digital assets: microstructure, fragmentation