Black-Scholes Gamma-Theta Trade-off and Delta-Neutral Portfolio

Scenario

An options market maker maintains a continuously delta-hedged portfolio of European options and shares of the underlying stock: $\Pi = V(S, t) - \Delta S, \quad \Delta = \frac{\partial V}{\partial S}$ The underlying stock follows geometric Brownian motion with continuous volatility $\sigma$ and risk-free interest rate $r$.

Your Task

1. Using the Black-Scholes partial differential equation, derive the exact algebraic trade-off between Theta ($\Theta = \frac{\partial V}{\partial t}$) and Gamma ($\Gamma = \frac{\partial^2 V}{\partial S^2}$) for the delta-hedged portfolio.
2. Specialize the result to the zero interest rate regime ($r = 0$) and explain why a long options position must experience time decay to pay for convexity.
3. Given $S = 100, K = 100, r = 0.04, \sigma = 0.20, T = 0.5$, calculate $V, \Delta, \Gamma, \Theta$, and numerically verify the PDE identity $\Theta + r S \Delta + \frac{1}{2}\sigma^2 S^2 \Gamma = r V$.

Constraints

• Black-Scholes pricing dynamics.
• Theta defined as $\frac{\partial V}{\partial t} = -\frac{\partial V}{\partial \tau}$ where $\tau = T - t$.

Examples When $r = 0$, $\Theta = -\frac{1}{2}\sigma^2 S^2 \Gamma$. For positive Gamma $\Gamma > 0$, time decay is strictly negative ($\Theta < 0$).