Derivation of the Black-Scholes-Merton PDE › Limitations and Practical Implications

The Black-Scholes-Merton framework transformed derivatives pricing and risk management, but its assumptions impose significant limitations that you must understand. The assumption of constant volatility is perhaps the most problematic: in real markets, volatility changes over time and varies with strike price and maturity. This discrepancy manifests as the volatility smile and volatility surface, which we'll explore in detail in a later chapter. When market-implied volatilities differ systematically from a single constant value, the BSM model cannot perfectly match observed option prices across all strikes.

The continuous trading assumption is also unrealistic. Real markets have discrete trading opportunities, transaction costs, and bid-ask spreads. These frictions make perfect delta hedging impossible: you cannot adjust your hedge infinitely often, and each adjustment costs money. The gap between theoretical continuous hedging and practical discrete rebalancing creates residual risk that the basic BSM framework ignores. You can address this through discrete hedging strategies and by accounting for the costs of hedging in your pricing.

Despite these limitations, the BSM PDE remains the foundation of modern derivatives pricing. Its power lies not in its literal accuracy but in providing a systematic framework for thinking about derivative valuation. Extensions and modifications build on this foundation: stochastic volatility models modify the volatility dynamics, jump-diffusion models add discontinuous price movements, and local volatility models allow volatility to depend on price and time. All these approaches either modify the PDE or reframe the problem while retaining the no-arbitrage core insight.

The equation also established delta hedging as the standard approach to managing derivative risk. Even when the assumptions don't hold exactly, the hedge ratios derived from BSM-type models provide useful starting points. The Greeks, which we'll derive from the Black-Scholes formula in an upcoming chapter, give you a systematic way to decompose and manage the various risk exposures embedded in derivative positions.