Derive the Black-Scholes PDE from a delta hedged portfolio. Say exactly where mu disappears, which assumption forces the portfolio return to equal r, and what the argument needs that a jump model would break.

Derive the Black-Scholes PDE from a delta hedged portfolio. Say exactly where mu disappears, which assumption forces the portfolio return to equal r, and what the argument needs that a jump model would break.

Approach: Form the portfolio long the option and short delta shares, apply Ito, cancel the stochastic term by the choice of delta, then impose that a riskless portfolio earns the short rate.

V_t + (1/2) sigma^2 S^2 V_SS + r S V_S - r V = 0. Take Pi = V - Delta S. Ito's lemma gives dV = (V_t + mu S V_S + (1/2) sigma^2 S^2 V_SS) dt + sigma S V_S dW, so dPi = (V_t + (1/2) sigma^2 S^2 V_SS) dt + sigma S (V_S - Delta) dW once the stock term is subtracted. Choosing Delta = V_S removes the dW term, and the mu S V_S contribution cancels against mu S Delta at the same time, which is why the drift of the stock never enters the price. The portfolio is now locally riskless, so no arbitrage forces dPi = r Pi dt = r(V - S V_S) dt, and equating the two expressions gives the equation, in which the surviving risk-neutral drift is r. The argument needs continuous paths and continuous rebalancing: with jumps the residual after delta hedging is of first order in the jump size and cannot be removed by any single hedge ratio, so the market ceases to be complete and the PDE argument fails.

Follow-up: How large is the hedging error over a discrete rebalancing interval of length h, and how does its standard deviation scale with h?

Key concepts: delta hedging, no arbitrage, Ito's lemma, risk-neutral drift.