GBM under physical and pricing measures
Separate forecast drift from no-arbitrage drift before pricing a payoff.
01Solve the GBM SDE and identify its lognormal law.
02Distinguish μ estimated under P from r−q imposed under Q.
03Verify the discounted total-return martingale condition.
04Explain which GBM assumptions fail in option markets.
Identify the state variables and the behavior they add.
GBM is one process written in different probability coordinates. Historical estimation targets the physical law; arbitrage pricing chooses a measure under which discounted tradables are martingales.
The diffusion coefficient controls instantaneous return variance under both measures in the base model.
Changing μ to r−q is a measure statement, not a forecast revision.
The exact exponential solution preserves positivity; a naive Euler step need not.
Ask which instruments can identify the dynamics.
Equity forwards, vanilla options, implied volatility and delta hedging all inherit GBM's carry, positivity and lognormal assumptions.
Equity forwards
European options
delta-one hedges
volatility quotes
S is a currency price, q is continuous proportional dividend yield, r is the continuously compounded funding rate, σ is annualized decimal volatility, and T is years.
Write the dynamics before interpreting parameters.
GBM dynamics
The same diffusion is paired with different drifts after a measure change.
Exact lognormal solution
The −σ²/2 term comes from Itô's lemma and makes E[S_T|S_t]=S_te^{bτ}.
Full derivation
Solve GBM by transforming log-price
Apply Itô's lemma to log S, integrate the resulting arithmetic Brownian motion, then exponentiate.
- 01
Choose the transformation
Logarithms turn proportional shocks into additive shocks while preserving positivity after inversion.
- 02
Apply Itô's lemma
The quadratic-variation correction subtracts half the instantaneous variance from log drift.
- 03
Integrate over the horizon
Constant coefficients integrate directly and the Brownian increment is normal with variance τ.
- 04
Standardize and exponentiate
Write W_T−W_t=√τ Z and invert the log transformation.
The exact solution separates drift, convexity correction, and random shock; changing measure changes b, not the algebra of the solution.
Inputs
μ: physical expected return under Pr−q: risk-neutral ex-dividend drift under Qσ: annualized return volatilityBₜ=e^{rt}: money-market numeraire
Assumptions and limits
- Constant volatility cannot reproduce smile or stochastic variance.
- Continuous paths omit jumps and gap risk.
- Continuous trading and frictionless funding are idealizations.
Discounted total-return martingale
Dividend reinvestment and funding discounting remove predictable drift under Q.
Calibrate, compute, and challenge the dynamics.
Use the exact transition for simulation when coefficients are constant; use the P law for estimation and the Q law for arbitrage pricing.
Estimate μ and σ from historical returns only for a physical model. Calibrate pricing volatility to option prices under Q and infer carry from consistent curves/dividends.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Separate physical and pricing parameter objects
- Use exact transition as numerical reference
- Return martingale and moment diagnostics
Exact GBM transition under P and Q
Use identical shocks to isolate the effect of changing drift and verify the Q expectation.
from __future__ import annotations import numpy as np def gbm_terminal(spot: float, drift: float, vol: float, time: float, z: np.ndarray) -> np.ndarray: if min(spot, time) <= 0 or vol < 0: raise ValueError("invalid GBM inputs") return spot*np.exp((drift-0.5*vol*vol)*time+vol*np.sqrt(time)*z) rng = np.random.default_rng(19)z = rng.standard_normal(500_000)spot, rate, dividend, vol, time = 100.0, 0.04, 0.01, 0.20, 1.5terminal = gbm_terminal(spot, rate-dividend, vol, time, z)target = spot*np.exp((rate-dividend)*time)assert abs(terminal.mean()-target)/target < 2e-3assert np.all(terminal > 0)print(f"sample_mean={terminal.mean():.4f} analytical_mean={target:.4f}")Shock one parameter and trace the full response.
GBM under physical and pricing measures
Switch measure and move the physical inputs. Drift, numeraire, martingale and density update together.
Likelihood-ratio weights across the current Brownian state range.
- Zₜ(Wₜ)
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Change weight, drift and martingale together
B₀ Eᴼ[S payoff / Bₜ]
μ=8.50%Forecast and risk-premium distribution
λ=0.227Positive likelihood-ratio process
Sₜ / BₜMoney-market account Bₜ
Where the model meets the book.
“Risk-neutral drift is a pricing coordinate, not the desk's best forecast.”
spot and dividend curve
funding/discount curve
volatility convention
measure and numeraire
Calibrate Q-volatility to liquid option prices; estimate P-drift only for forecasting and scenario applications with a separate validation objective.
RISKcarry and dividend risk
volatility/model risk
measure confusion
- freeze spot and curves
- declare P or Q objective
- compute exact reference
- validate moments and forwards
Production failure modes
- mixing historical and implied volatility
- incorrect dividend carry
- Euler negative states
09MACRO CONNECTIONOpen the transmission channel.
Expected returns versus pricing carry
Risk premia affect the physical distribution; funding and dividends fix no-arbitrage carry in the pricing coordinate.
transmitstransmitsfix r−q under Q
outputidentifies pricing volatility
10COMMON PITFALLSOpen the failure checklist.
Replacing μ by r without declaring a measure change.
Forgetting dividend carry in the Q drift.
Estimating pricing drift from historical returns.
Using an Euler scheme when an exact transition exists.
11SOURCES / FURTHER READINGOpen sources and continue the track.
Ch. 2 §§2.1–2.3, printed pp. 27–49 (PDF pp. 46–67)
The formulation was checked against the cited sections; prose, examples and code are original to the Academy.
- Source
- Mathematical Modeling and Computation in Finance
- Author
- Cornelis W. Oosterlee and Lech A. Grzelak
- Ref
- First edition (2020)