Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Derivation of the Black-Scholes-Merton PDE › Visualizing the PDE Residual

Michael BrenndoerferNovember 26, 20251 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 29Derivation of the Black-Scholes-Merton PDE › Visualizing the PDE Residual

Reading section 10 of 16 · Visualizing the PDE Residual · 1 minSection 10 of 16 · 1 min

Section 10 of 16 · 1 min read

Derivation of the Black-Scholes-Merton PDE › Visualizing the PDE ResidualOne section · Sections in this chapter10 / 16 ↓

Sections in this chapter16

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Let's create a visualization showing the PDE residual across different stock prices for both valid and invalid pricing functions.

In[16]:

Code

S_range = np.linspace(50, 150, 100)
residuals_valid = [
    bs_pde_residual(
        power_derivative_price, S, t_test, r_test, sigma_test, T_test
    )
    for S in S_range
]
residuals_invalid = [
    bs_pde_residual(
        invalid_pricing_function, S, t_test, r_test, sigma_test, T_test
    )
    for S in S_range
]

Out[17]:

Visualization

Line plot comparing PDE residuals showing valid solution near zero and invalid solution with large non-zero values. — Black-Scholes PDE residual across stock prices. The valid solution (power derivative) has near-zero residual everywhere, while the invalid function shows substantial residual, confirming it does not satisfy the PDE.