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PART IX — MATHEMATICAL FINANCE🔴 Advanced

How Do Asset Prices Move Randomly Over Time?

Stochastic Processes and Value at Risk

Geometric Brownian motion, Monte Carlo simulation, and Value at Risk — the mathematical framework for modelling asset prices and measuring portfolio risk.

math financestochasticBrownian motionGBMMonte CarloVaRrisk management

Live Result

$16,450

ƒ: portfolio_value * z_score * sigma * sqrt(t)raw: 16450.0

ƒ(x)

Use variable names from the panel above (e.g. FV, r, n) — or type numbers directly: 10000 / (1 + 0.08)^10

portfolio_value

Total portfolio value ($)

portfolio_value = 1000000

z_score% as decimal

Standard normal quantile (1.645 = 95%, 2.326 = 99%)

z_score = 1.645

sigma% as decimal

Daily portfolio volatility (decimal, e.g. 0.01 = 1%)

sigma = 0.01

t% as decimal

Holding period in days

t = 1

💡 You can also enter values directly in the formula: 10000 / (1 + 0.08)^10

⬇ Export Calculation

Exports a plain .txt file with your expression, formula, all variable values, result, and educational notes — ready to paste into any report, Word doc, Notion, or Google Docs.

The exported file includes the formula in standard mathematical notation — you can paste it directly into Excel, Google Sheets, or back into FinanceSheep.

📖

Learn: Stochastic Processes and Value at Risk

PART IX — MATHEMATICAL FINANCE · Educational Guide

The Core Idea

Stock prices don't move in predictable straight lines — they follow a random walk with a drift. The mathematical language for this is stochastic calculus, and it underpins everything from Black-Scholes to bank capital requirements. Even if you never derive Ito's lemma on the job, understanding GBM, Monte Carlo, and VaR puts you in the same conceptual framework as every quant, risk manager, and derivatives desk on Wall Street.

How It Works

Geometric Brownian Motion: dS = μS dt + σS dW. S stays positive (can't have negative stock prices). Log-returns are normally distributed. Monte Carlo simulates thousands of GBM paths to estimate price distributions, option values, and risk metrics. Value at Risk (VaR): maximum loss at a given confidence level over a given holding period. Parametric VaR = Portfolio × z × σ × √t. At 95% confidence, z = 1.645. '95% VaR of $16,450' means: 'we are 95% confident we will NOT lose more than $16,450 in one day.' VaR says nothing about losses BEYOND this threshold — the key limitation revealed in 2008.

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Real-World Example: $1M portfolio, daily σ = 1%, 1-day 95% VaR = $1,000,000 × 1.645 × 0.01 × 1 = $16,450. Scale to 10 days: $16,450 × √10 = $52,015 (10-day 95% VaR). Basel bank capital rules require banks to hold capital ≥ 3 × 10-day 99% VaR. GBM step: S = $100, μ = 10%/yr, σ = 25%, dt = 1/252, z = 0.8: S(t+dt) = $100 × exp[(0.10 − 0.03125)/252 + 0.25/√252 × 0.8] ≈ $101.29. Run 10,000 such paths → full distribution of outcomes → price options, measure tail risk.

How Do Asset Prices Move Randomly Over Time?

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