📈 Option Pricing Engine (C++)

A modular C++ option pricing engine implementing analytical and numerical methods for derivative pricing, including Black-Scholes, Monte Carlo simulation, variance reduction techniques, and performance benchmarking.

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🚀 Features

📊 Pricing Models

• Black-Scholes (closed-form)
• Monte Carlo Simulation
  • European options
  • Asian options (arithmetic average)

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⚙️ Advanced Monte Carlo

• Standard Monte Carlo
• Antithetic Variates (variance reduction)
• Confidence Intervals (95%)
• Standard Error estimation

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📐 Greeks (Black-Scholes)

• Delta
• Gamma
• Vega
• Theta
• Rho

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🔍 Validation & Analysis

• Call-Put Parity (Black-Scholes & Monte Carlo)
• Black-Scholes vs Monte Carlo comparison
• Monte Carlo convergence analysis

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⚡ Performance Benchmark

• Runtime comparison:
  • Black-Scholes (μs)
  • Monte Carlo Standard (ms)
  • Monte Carlo Antithetic (ms)

🧑‍💻 Interactive CLI

• User inputs all parameters dynamically:
  • S0, K, r, y, sigma, T
  • pricing method (Black-Scholes / Monte Carlo)
  • payoff style (European / Asian)
  • option type (Call / Put)
  • simulation count
  • antithetic variates toggle

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🧠 Mathematical Models

Black-Scholes Formula

[ C = S_0 e^{-yT} N(d_1) - K e^{-rT} N(d_2) ]

[ d_1 = \frac{\ln(S_0/K) + (r - y + 0.5\sigma^2)T}{\sigma \sqrt{T}} ]

[ d_2 = d_1 - \sigma \sqrt{T} ]

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Geometric Brownian Motion (GBM)

[ S_T = S_0 \exp\left((r - y - 0.5\sigma^2)T + \sigma \sqrt{T} Z \right) ]

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Monte Carlo Estimator

[ V = e^{-rT} \cdot \mathbb{E}[\text{Payoff}] ]

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Antithetic Variates

Use paired samples: [ Z \quad \text{and} \quad -Z ]

To reduce variance in Monte Carlo estimation.

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📁 Project Structure

Input

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Example Output

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Compile

g++ -std=c++17 main.cpp src/math_utils.cpp src/black_scholes.cpp src/monte_carlo.cpp src/analysis.cpp -o option_engine ./option_engine

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🎯 Key Learnings

•	Built a modular C++ derivatives pricing engine
•	Implemented stochastic simulation with GBM
•	Applied Monte Carlo variance reduction
•	Validated numerical methods against analytical pricing
•	Performed convergence and runtime analysis
•	Structured the codebase using reusable headers and source files

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🧩 Model Interpretation

Black-Scholes Assumptions

- Asset follows Geometric Brownian Motion (GBM)
- Constant volatility
- Constant risk-free rate
- Continuous dividend yield
- No arbitrage opportunities
- Frictionless markets (no transaction cost, perfect liquidity)
- Continuous trading
- Lognormal return distribution

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When Black-Scholes Breaks

- Volatility is not constant (volatility smile/skew)
- Sudden jumps (earnings, macro shocks) violate GBM
- Interest rates and dividends change over time
- Transaction costs and liquidity constraints exist
- Discrete hedging introduces errors
- Not suitable for path-dependent options like arithmetic Asian

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Monte Carlo Assumptions

- Underlying follows GBM
- Random variables are normally distributed
- Large number of simulations ensures convergence
- Constant discount rate

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Monte Carlo Limitations

- Model risk persists (depends on GBM assumption)
- Slow convergence for high accuracy
- High computational cost
- Tail risks may be underrepresented
- No closed-form benchmark for some exotic options

🚀 Future Improvements

Pricing Models

- Binomial Tree
- Trinomial Tree
- Barrier Options
- Lookback Options
- Digital Options

This project is designed as a foundational quantitative finance engine for:

- Understanding derivatives pricing
- Comparing analytical vs numerical methods
- Studying Monte Carlo convergence
- Building reusable C++ quant infrastructure

👤 Author Wielly Halim Aspiring Quant Researcher | C++ & Python | Derivatives Pricing

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