Girsanov theorem & risk-neutral pricing
Move the risk premium from diffusion drift into the density.
01Compute the market price of Brownian risk
02Transform P-Brownian motion into Q-Brownian motion
03Derive the discounted pricing expectation
Observe the object before formalizing it.
Girsanov does not remove risk; it changes the probability weights so discounted tradables have zero drift.
λ=(μ−r)/σ links the two drifts.
The exponential density must be a true martingale.
No-arbitrage selects Q, not statistical accuracy.
Connect the mathematical object to a pricing question.
The theorem is the continuous-time bridge from estimated dynamics to arbitrage-free valuation.
equity options
FX options
rate derivatives
S is a tradable ex-dividend spot and B is the money-market numeraire.
Construct the definition and its invariants.
Density process
The stochastic exponential reweights Brownian paths.
Pricing identity
A claim equals its Q expected payoff in units of the numeraire.
Full derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Write physical dynamics
Start with an observed or estimated drift μ.
- 02
Define λ
Scale excess drift by diffusion exposure.
- 03
Shift Brownian motion
Under the density, Wᴼ=Wᴾ+λt is Brownian.
- 04
Discount the tradable
Substitution leaves drift r, so S/B is a Q local martingale.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
λ: market price of Brownian riskWᴾ,Wᴼ: Brownian motions
Assumptions and limits
- Local martingales can fail to be true martingales.
- Jumps require a richer change-of-measure specification.
Fit, compute, then challenge the assumptions.
Validate Novikov-like integrability, compute λ, shift the Brownian driver and price in numeraire units.
Calibrate σ to pricing instruments; μ and λ belong to a separate physical-risk exercise.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Girsanov theorem & risk-neutral pricing
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def ztexpleftlambdawtm(x: np.ndarray) -> float: x = np.asarray(x, dtype=float) assert np.isfinite(x).all() return float(np.mean(x)) sample = np.array([0.8, 1.0, 1.2])value = ztexpleftlambdawtm(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Run the thought experiment.
Girsanov theorem & risk-neutral pricing
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
Carry the abstraction into valuation.
“The equity risk premium disappears from the option price, not from the risk report.”
spot
carry curve
pricing volatility
Calibrate σ to pricing instruments; μ and λ belong to a separate physical-risk exercise.
RISKbubble/local-martingale risk
drift leakage
- Validate market state and timestamp
- Recompute the baseline
- Run a controlled perturbation
- Explain P&L and residuals
Production failure modes
- Silent convention or measure changes
- Unstable numerics hidden by plausible prices
09MACRO CONNECTIONOpen the transmission channel.
Transmission from state to valuation
The causal chain separates the economic shock from the modelling response.
transmitscontains risk premium μ−r
transmitsmoves premium into density
outputuses arbitrage-free Q dynamics
10COMMON PITFALLSOpen the failure checklist.
Assuming every exponential local martingale is normalized
Estimating μ to price a vanilla option
11SOURCES / FURTHER READINGOpen sources and continue the track.
Measure theory, simulation and computational-finance lectures
The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.
- Source
- Computational Finance Course
- Author
- L. A. Grzelak
- Ref
- main