Black-Scholes Model for Option Pricing

Core equation and assumptions

The Black-Scholes partial differential equation (PDE) for option pricing is:

$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$

Where:

• $V$ is the option value
• $S$ is the underlying asset price
• $t$ is time
• $r$ is the risk-free rate
• $\sigma$ is volatility

The model makes several key assumptions:

• European-style options (no early exercise)
• Log-normal distribution of underlying returns
• Constant volatility and risk-free rate
• No dividends
• No transaction costs or taxes
• Continuous trading

Closed-form solution

For a European call option, the Black-Scholes formula is:

$C = SN(d_1) - Ke^{-rT}N(d_2)$

Where:

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

And:

• $C$ is the call option price
• $S$ is the current stock price
• $K$ is the strike price
• $T$ is time to expiration
• $N()$ is the cumulative normal distribution function

For put options, we use put-call parity:

$P = C - S + Ke^{-rT}$

Applications and limitations

The model provides the foundation for modern derivatives pricing and risk management, enabling:

• Quick theoretical pricing of options
• Calculation of Greeks for risk management
• Development of trading strategies
• Basis for more complex models

However, key limitations include:

• Assumption of constant volatility contradicts observed volatility surface
• No consideration of liquidity risk
• Idealized market assumptions
• Poor performance during extreme market conditions

Extensions and modern usage

Several extensions address the model's limitations:

1. Stochastic volatility models

   • Heston Model
   • SABR Model
   • Local volatility models

2. Jump diffusion models

   • Merton jump-diffusion
   • Variance Gamma Model

3. Practical adjustments

   • Dividend adjustments
   • Early exercise premium
   • Volatility skew corrections

Despite its limitations, the Black-Scholes model remains fundamental in:

• Option market making
• Risk management systems
• Structured product design
• Academic research and education

Model risk and governance

Financial institutions must carefully manage Black-Scholes model risk through:

1. Validation procedures

   • Backtesting against market prices
   • Stress testing under various scenarios
   • Regular calibration of parameters

2. Risk controls

   • Model parameter limits
   • Trading limits based on Greeks
   • Regular review of assumptions

3. Documentation

   • Model methodology
   • Validation results
   • Usage guidelines
   • Known limitations

Modern trading systems incorporate these controls within their algorithmic risk controls framework.