Derivatives Pricing Models

Core principles of derivatives pricing

The foundation of modern derivatives pricing rests on several key principles:

1. No-arbitrage principle - Prices must be consistent across related instruments to prevent risk-free profits
2. Risk-Neutral Valuation - Future payoffs are discounted at the risk-free rate
3. Replication - Derivative payoffs can be replicated using simpler instruments
4. Market completeness - All relevant risks can be hedged

These principles enable the construction of mathematical models that capture market dynamics and price derivatives consistently.

Common pricing models

Black-Scholes Model

The Black-Scholes Model serves as the foundation for options pricing. It assumes:

• Log-normal distribution of asset prices
• Constant volatility
• No transaction costs
• Risk-free borrowing and lending

While these assumptions are simplified, the model provides a framework for more sophisticated approaches.

Stochastic Volatility Models

These models extend Black-Scholes by allowing volatility to vary over time:

• Heston Model - Introduces correlation between asset returns and volatility
• SABR Model - Popular for interest rate derivatives
• Local volatility models - Allow volatility to depend on price and time

Model risk and calibration

Model risk arises from:

• Parameter uncertainty
• Model misspecification
• Numerical implementation errors
• Market conditions violating assumptions

Proper model calibration requires:

Real-time pricing considerations

Modern trading systems require efficient implementation:

1. Parallel computation of prices and Greeks
2. Optimization of numerical methods
3. Caching of intermediate calculations
4. Hardware acceleration when needed

Trading firms must balance accuracy with computational speed for Real-Time Risk Assessment.

Market microstructure impact

Derivatives pricing models must account for:

• Bid-ask spreads
• Market Impact Cost
• Liquidity Risk Premium
• Trading costs

These factors affect both pricing and risk management decisions.

Regulatory considerations

Models must comply with:

• Capital requirements
• Risk reporting standards
• Model validation requirements
• Stress testing frameworks

Firms need robust model governance and documentation processes.

Future developments

Emerging trends include:

1. Machine learning for parameter estimation
2. Quantum computing applications
3. Integration of alternative data
4. Real-time model adaptation

These advances aim to improve pricing accuracy and computational efficiency while managing increasing market complexity.