Overview
Production-grade pricing engine for multi-asset derivatives combining analytical moment-matching techniques (Brigo et al.) and Monte Carlo simulation with sophisticated variance reduction methods.
Type: Personal Project
Language: C#
Focus: Quantitative Finance, Derivatives Pricing, Numerical Methods
Project Goals
Develop a comprehensive, production-ready pricing engine for multi-asset basket options that:
- Combines analytical and numerical pricing methods
- Implements state-of-the-art variance reduction techniques
- Supports realistic term structure modeling
- Integrates with real market data sources
Key Features
Pricing Methodologies
Analytical Approach
- Moment-matching techniques based on Brigo et al. (2004)
- Efficient for European-style basket options
- Provides fast approximate pricing for calibration and risk management
Monte Carlo Simulation
- Full path simulation for complex payoffs
- Support for path-dependent features
- Flexible framework for exotic option structures
Variance Reduction
- Control variates for improved convergence
- Reduced standard errors in Monte Carlo estimates
- Significantly improved computational efficiency
- Antithetic variates and importance sampling options
Market Data Integration
- ECB €STR (Euro Short-Term Rate) curve integration
- Bloomberg volatility surfaces for realistic vol modeling
- Term structure modeling with:
- Zero curves
- Forward curves
- Volatility term structures
- Full correlation matrices for multi-asset modeling
Advanced Features
- Support for multiple underlying assets
- Flexible correlation structure
- Time-dependent parameters
- Real-time market data updates
- Greeks computation (Delta, Gamma, Vega, etc.)
Technical Implementation
Architecture
// Core components:
- PricingEngine: Main orchestrator
- MonteCarloSimulator: Path generation and pricing
- AnalyticalPricer: Moment-matching approximations
- VarianceReducer: Control variate implementation
- MarketDataProvider: Real-time data integration
- TermStructureModel: Curve and surface modeling
Key Classes
BasketOption: Multi-asset option representationUnderlyingAsset: Individual asset modelingCorrelationMatrix: Asset correlation handlingVolatilitySurface: Implied volatility interpolationRateCurve: Interest rate term structure
Mathematical Foundation
Basket Option Pricing
The basket option payoff depends on a weighted combination of underlying assets:
\[\text{Payoff} = \max\left(\sum_{i=1}^{n} w_i S_i(T) - K, 0\right)\]
Where:
- $w_i$ = weight of asset $i$
- $S_i(T)$ = price of asset $i$ at maturity
- $K$ = strike price
- $n$ = number of assets
Control Variate Variance Reduction
Using a control variate $Y$ with known expectation $\mathbb{E}[Y]$:
\[\hat{\theta}_{\text{CV}} = \hat{\theta}_{\text{MC}} + \beta(\mathbb{E}[Y] - \hat{Y})\]
This reduces variance when $Y$ is correlated with the target payoff.
Technical Stack
- Language: C# (.NET Framework/Core)
- Numerical Libraries: Math.NET Numerics
- Data Sources: ECB API, Bloomberg API
- Testing: Unit tests, integration tests, convergence validation
- Performance: Parallelized Monte Carlo paths
Validation & Testing
- Black-Scholes Convergence: Validated against analytical solutions for single-asset options
- Multi-asset Benchmarks: Compared with industry-standard pricing libraries
- Greeks Accuracy: Numerical differentiation vs. analytical Greeks
- Variance Reduction Efficiency: Measured improvement in standard errors
Results
- Successfully prices complex multi-asset derivatives
- Variance reduction achieves 50-70% reduction in Monte Carlo standard errors
- Real-time market data integration for production use
- Comprehensive Greeks calculation for risk management
- Production-ready code quality with extensive testing
Applications
- Portfolio Hedging: Basket options on equity portfolios
- Index Options: Pricing custom index derivatives
- Risk Management: Greeks for multi-asset exposure analysis
- Trading Strategies: Structured products and exotic derivatives
Key Learnings
- Variance Reduction is Critical: Control variates dramatically improve Monte Carlo efficiency
- Market Data Integration: Real-world pricing requires robust data pipelines
- Correlation Modeling: Accurate correlation estimation is essential for multi-asset pricing
- Numerical Stability: Careful implementation needed for robust production systems
Future Enhancements
- Local volatility models
- Stochastic volatility (Heston, SABR)
- Jump-diffusion models
- GPU acceleration for Monte Carlo
- Machine learning for calibration
Links
References
- Brigo, D., Mercurio, F., Rapisarda, F., & Scotti, R. (2004). “Approximated moment-matching dynamics for basket-options pricing”
- Glasserman, P. (2003). “Monte Carlo Methods in Financial Engineering”
- Hull, J. C. (2018). “Options, Futures, and Other Derivatives”