Investment Growth Models Explained: Which Strategy Fits Your Goals?

Core Concepts: Continuous vs Discrete Compounding

Introduction

In financial mathematics, compounding models form the bedrock of predictive analytics. This article examines the fundamental dichotomy between continuous and discrete compounding through the lens of stochastic calculus and discrete mathematics, providing a rigorous framework for model selection in various financial contexts.

Mathematical Foundations of Financial Modeling

Formal Definitions and Governing Equations

Model	Domain	Canonical Formula	Differential Form
Continuous	Stochastic Systems	P(t) = P₀e^rt	dP/dt = rP
Discrete	Periodic Systems	P(t) = P₀(1 + r)^t	ΔP = rPₙΔt

Convergence Analysis and Error Margins

Through Taylor series expansion, we demonstrate the asymptotic relationship between models:

Continuous: e^r = lim_n→∞(1 + r/n)ⁿ
Discrete approximation error: |e^r - (1 + r)| ≈ r²/2 for small r

Empirical validation (7% annual growth):

Discrete: $100 × 1.07 = $107.00
Continuous: $100 × e^0.07 ≈ $107.25
Relative error: 0.23% (compounding period Δt → 0)

Asset-Class Specific Modeling Paradigms

1. Inflation Dynamics: Ornstein-Uhlenbeck Process

Stochastic inflation model — Inflation paths!

Continuous Model Justification:

This stochastic model (dπ(t) = θ(μ - π(t))dt + σdW(t)) captures inflation’s mean-reverting nature, where π(t) fluctuates around a long-term average (μ) at speed θ, while σdW(t) injects market volatility through Wiener-process shocks 💥.

By blending predictable drift (θ(μ - π(t))) with random turbulence (σdW(t)), it mirrors real-world inflation’s push-pull between stability and chaos 🎯.

The continuous framework elegantly balances persistent price trends with sudden, unpredictable jolts—perfect for simulating central bank targets amid real economic noise!

2. Equity Pricing: Geometric Brownian Motion

Itô Calculus Formulation:

The iconic dS(t)/S(t) = μdt + σdW(t) models stock dynamics via drift (μ) and volatility (σ), where Wiener process W(t) injects randomness, mimicking market turbulence.

🔄⚡ Its closed-form solution S(t) = S₀exp[(μ - σ²/2)t + σW(t)] reveals log-normal returns and fat tails (leptokurtic distribution) through [W]ₜ = t, reflecting real-world price jumps!

🎢📉 Despite assuming constant volatility, it remains finance’s cornerstone for valuing options and simulating market chaos. 🏛️💥

Stock price simulation — Multiple Monte Carlo paths of stock prices with volatility clustering and drift

3. Cryptocurrency Valuation: Lévy Jump-Diffusion

Crypto price jumps — Fractal-like cryptocurrency price chart with discontinuous jumps and heavy tails"

Modified SDE:

The model dP(t)/P(t) = μdt + σdW(t) + J(t)dN(t) fuses continuous volatility (σdW) with discontinuous jumps (Poisson-driven N(t) and J(t) ~ Normal jumps), capturing crypto's wild price spikes and crashes!

🌪️📉 Unlike traditional models, it quantifies extreme >4σ events (common in BTC/USD) through sudden, fat-tailed shocks.

🔮💰 Perfect for pricing crypto derivatives or stress-testing portfolios against Black Swan moments! 🦢⚡

Model Selection Heuristics

Discrete Compounding Preference When:
- Contractual periodicity exists (e.g., annual salary contracts)
- Δt between observations > 1/252 (daily resolution)
- Transaction costs dominate microstructure effects
Continuous Compounding Required When:
- Markov property holds (memoryless price process)
- High-frequency data (Δt → 0)
- Risk-neutral valuation (Q-measure)

"The fundamental theorem of asset pricing forces continuous models in complete markets through the martingale representation theorem." - Delbaen & Schachermayer (1994)

Quick Guide: Best Model for Each Market

Market	Recommended Model	Why?
Stock Market	Continuous (GBM)	Constant price fluctuations during trading hours
Real Estate	Discrete Annual	Quarterly/annual appraisal cycles
Crypto	Continuous with Jumps	24/7 trading with sudden price spikes
Bonds	Discrete Periodic	Fixed coupon payment schedules
Commodities	Continuous Mean-Reverting	Seasonal patterns with daily adjustments

Academic References & Further Reading

1. Hull, J.C. (2022). Options, Futures and Other Derivatives. Pearson.
- Standard textbook for derivative pricing models
2. Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities".
- Seminal paper on continuous-time modeling
3. Cont, R., & Tankov, P. (2003). Financial Modelling with Jump Processes.
- Advanced treatment of Lévy processes
4. Wilmott, P. (2006). Quantitative Finance. Wiley.
- Practical guide to financial modeling techniques
5. OECD Inflation Handbook (2023).
- Official guidelines for inflation modeling

"The choice between discrete and continuous time models is not merely technical, but reflects fundamental differences in market microstructure."
- Robert C. Merton (1990)