Black-Scholes PDE Derivation via Delta Hedging

The pricing of derivatives follows a systematic framework: (1) Define the payoff structure, (2) Specify the underlying model, (3) Construct a replicating portfolio, (4) Derive the pricing PDE, (5) Solve using PDE or expectation methods. The Feynman-Kac theorem connects PDEs to stochastic processes, enabling conversion to risk-neutral expectations. Under risk-neutral measure, the discounted stock process becomes a martingale, requiring drift adjustment from μ to r. This framework applies to any SDE-governed stock process, forming the theoretical foundation for derivative pricing that extends beyond simple European options.

This section establishes the profound connection between partial differential equations and stochastic processes through the Feynman-Kac theorem. It demonstrates how to prove that solutions to the Black-Scholes PDE equal risk-neutral expectations by applying Ito's lemma and leveraging the zero-drift property. The theorem provides a powerful numerical method: instead of solving PDEs analytically, we can simulate many paths under the risk-neutral measure, compute payoffs at maturity, average results, and discount back to present value. The section then derives the Black-Scholes formula explicitly, showing how the log-normal distribution of stock prices enables analytical computation of the risk-neutral expectation. This yields the famous closed-form solution for European call options. The section emphasizes that the Feynman-Kac framework unifies deterministic PDE methods with probabilistic simulation techniques, providing flexibility for complex financial instruments beyond simple European options.

The Feynman-Kac formula establishes a fundamental connection between partial differential equations (PDEs) and stochastic processes, allowing solutions to certain PDEs to be expressed as expectations of terminal conditions evaluated along stochastic processes. Specifically, for a PDE of the form ∂V/∂t + μ∂V/∂x + (σ²/2)∂²V/∂x² = 0 with terminal condition V(T,x) = η(x), the solution V(t,x) equals the expectation of η evaluated at the stochastic process X_t, where dX_t = μdt + σdW_t. This powerful framework enables solving complex pricing problems in finance by transforming deterministic PDEs into probabilistic expectation calculations, bridging the gap between portfolio replication approaches and Monte Carlo simulation methods.

The Feynman-Kac theorem establishes that the solution to the Black-Scholes PDE equals the discounted expectation under the risk-neutral measure: V(t,S) = e^(-r(T-t))E_Q[V(T,S_T)|S_t=S]. This connection applies to the Heston PDE as well, enabling Monte Carlo simulation approaches. The theorem reveals that option pricing fundamentally relates to expected payoffs, providing the theoretical foundation for numerical methods. This relationship justifies both PDE-based and Monte Carlo-based pricing approaches, with the choice depending on problem dimensionality and computational requirements.

The Feynman-Kac theorem establishes a fundamental connection between stochastic differential equations (SDEs) and partial differential equations (PDEs), showing that the conditional expectation E[H(X(T))|X(τ)=x] can be computed by solving either the SDE numerically using Euler's method or by solving the corresponding PDE ∂u/∂t + β(t,x)∂u/∂x + (1/2)γ²(t,x)∂²u/∂x² = 0 with terminal condition u(T,x) = H(x); this equivalence allows practitioners to choose the most computationally efficient approach for pricing financial derivatives and other applications involving diffusion processes.

The Greeks are mathematical measures for options behavior. Delta shows profit per dollar of stock movement—for example, delta 25 means $25 profit per $1 stock move. Gamma shows how delta changes per dollar movement. Theta shows daily time decay loss regardless of market hours. Vega shows sensitivity to implied volatility changes. Rho shows interest rate sensitivity. The main Greeks to focus on are delta and theta for calculating potential profits and understanding time decay costs.

There are five option Greeks: Delta, Gamma, Theta, Vega, and Rho. Delta measures the rate of change in option price with respect to underlying price movement. For example, if delta is 0.5, a 1-point move in the underlying causes a 0.5-point change in option price. Gamma measures the rate of change in option delta with respect to underlying price movement. For example, if gamma is 0.02, a 1-point move in the underlying changes delta by 0.02. Gamma affects option pricing indirectly through its impact on delta, causing acceleration or deceleration of delta changes. Theta measures the rate of change in option price with respect to time decay. For example, if theta is -10, a 1-day change in time causes a 10-point decrease in option price. Vega (Kappa) measures the rate of change in option price with respect to implied volatility changes. For example, if vega is 10, a 1% change in implied volatility causes a 10-point change in option price.

The five Option Greeks measure different sensitivities affecting option prices: (1) Delta (0-1 for calls, 0-(-1) for puts) - measures how much the option price changes for a ₹1 change in the underlying asset; (2) Gamma - measures how much delta changes as the underlying price moves; (3) Vega - measures how much the option price changes for a 1% change in implied volatility; (4) Theta (always negative) - measures time decay impact on option price; (5) Rho - measures how much the option price changes for a 1% change in interest rates. These Greeks help traders understand and manage risk exposure.

The Option Greeks (Delta, Gamma, Theta, Vega, and Rho) are theoretical measurements that help traders understand how option contracts react to various market factors: Delta measures the expected price change in an option for a $1 move in the underlying stock (positive for calls, negative for puts); Gamma measures the rate of change in Delta; Theta represents daily time decay (value loss) of an option; Vega indicates sensitivity to volatility changes; and Rho shows how interest rate changes affect option value. These metrics explain why option prices don't always move in sync with stock prices and help traders make informed decisions about position management and timing.

Each Greek can be summarized with one key concept: Delta represents directionality (bullish or bearish position), Gamma represents acceleration (how fast delta changes), Theta represents time decay (rate of price decay over time), and Vega represents volatility exposure (sensitivity to implied volatility changes). These single-word summaries help traders quickly understand what each Greek measures when analyzing their positions.

The implied volatility smile is a plot showing how implied volatility varies with strike price, revealing that the Black-Scholes-Merton model's assumption of constant volatility is incorrect; the smile indicates that actual asset return distributions have heavier tails than the log-normal distribution assumed by the model, and the post-1987 'crash of phobia' hypothesis explains why out-of-the-money puts show higher implied volatility due to increased trader concern about tail risk.

The Black-Scholes model's assumption of constant volatility is a major limitation. In practice, implied volatility varies with strike price and maturity, creating a 'volatility smile' where at-the-money options have the lowest volatility and out-of-money options have higher volatility. The smile is often asymmetric, with puts showing higher volatility than calls. This reflects market participants' greater sensitivity to downside risk. The model also cannot handle non-constant interest rates, non-constant volatility, or options on dividend-paying assets. The 1987 market crash highlighted these limitations, leading to numerous academic extensions and alternative models.

The volatility smile phenomenon demonstrates that options on the same underlying with different strikes and maturities show different implied volatilities. According to Black-Scholes, volatility should be constant as it's a property of the underlying, not the option. However, market data shows varying implied volatilities, indicating the model's limitations. This discrepancy arises because Black-Scholes assumes constant volatility while real markets exhibit state-dependent volatility, necessitating more sophisticated models like local volatility approaches.

Implied volatility is the volatility parameter in the Black-Scholes model that, when plugged into the pricing formula, produces the market price of an option; it is calculated using iterative methods like Newton-Raphson because the Black-Scholes formula is monotonic in volatility. However, the Black-Scholes model has significant limitations: it cannot capture the implied volatility smile/skew observed in markets (where volatility varies with strike price), assumes continuous hedging which is impractical due to transaction costs, and relies on the log-normal distribution assumption which underestimates extreme market events (heavy tails). To address these limitations, more sophisticated models such as local volatility models, stochastic volatility models, and jump-diffusion models have been developed.

Black-Scholes assumes constant volatility across all strike prices and time periods, predicting a flat volatility smile. However, real-world smiles exist because: (1) Returns follow fat-tailed distributions rather than normal distribution, causing extreme moves to happen more frequently than predicted, making out-of-the-money options worth more and thus having higher implied volatility; (2) Equity returns are negatively skewed (large negative returns are more common than large positive returns of same magnitude), creating the leftward skew in equity smiles; (3) Volatility is not constant but clusters and changes with price level (volatility increases when stocks fall). The smile represents the market's correction of Black-Scholes assumptions about return distributions.

Black-Scholes assumptions—constant volatility, continuous trading, log-normal returns—fail in practice. The 1987 crash revealed extreme events occur more frequently than predicted. Volatility smiles show implied volatility varies across strike prices, contradicting constant-volatility assumptions. Modern extensions address these limitations: stochastic volatility models treat volatility as a random process, while jump-diffusion models incorporate sudden price jumps. These advanced models better capture real-world market dynamics at the cost of increased complexity.

The Heston model extends Black-Scholes by making volatility itself a stochastic process following square-root diffusion with mean reversion, similar to the CIR model. This addresses Black-Scholes limitations: empirical evidence shows return distributions have fatter tails and taller peaks than normal, and implied volatility varies by strike (volatility smile) and maturity. Unlike Black-Scholes with one source of randomness requiring only one hedging instrument, Heston has two sources requiring two hedging instruments. Since volatility is not traded, the market is incomplete with multiple equivalent martingale measures. A portfolio of Δ units of stock, Σ units of a longer-maturity option, and α units of the bank account is constructed. Applying Ito's lemma to this two-dimensional system yields expressions for Δ and Σ that eliminate stochastic terms, forming the foundation for deriving the Heston PDE.

The Heston Model, developed by finance professor Steven Heston in 1993, is a stochastic volatility model used to price European options that addresses a key limitation of the Black-Scholes model by assuming volatility is arbitrary (randomly varying) rather than constant; unlike the Black-Scholes model which holds volatility constant, the Heston Model incorporates stochastic variables to provide more realistic option pricing estimates, and it also functions as a volatility smile model that graphically represents how volatility increases for options that are increasingly in-the-money or out-of-the-money relative to at-the-money options.

The Heston model addresses critical limitations of the Black-Scholes model by treating both asset prices and their variances as stochastic processes. While Black-Scholes assumes constant volatility, empirical evidence shows volatility changes over time, clusters (large movements follow large movements), and mean-reverts to a long-term average. The leverage effect explains negative correlation between stock returns and volatility: falling prices increase financial leverage, amplifying volatility. Additionally, risk-averse investors flee to quality during high-volatility periods, further depressing prices. Heston models these phenomena by having variance follow a mean-reverting Cox-Ingersoll-Ross process, with correlation between price and volatility dynamics.

This concluding section presents three major extensions to the Black-Scholes framework. Local volatility models treat volatility as a function of time and asset price, theoretically capable of reproducing any arbitrage-free surface but requiring complex calibration procedures. Stochastic volatility models treat volatility as a separate stochastic process, introducing multiple dimensions and capturing volatility clustering effects. Jump-diffusion models add discrete jump components to the diffusion process, enabling heavy-tailed return distributions that better capture market crash probabilities. The section emphasizes that while these models provide improved realism for pricing complex derivatives, they introduce additional complexity and parameters. The Black-Scholes model remains valuable as a fundamental framework for understanding derivative pricing principles before advancing to these more sophisticated approaches.