medium · Quantitative Finance

In the derivation of the Black-Scholes PDE, a delta-hedged portfolio Π = V - Δ S is constructed.

Why does the term involving (dS)^2 appear in the change dΠ?

  1. Because the derivative value V is a non-linear function of S, making its second-order sensitivity significant due to the properties of Brownian motion.
  2. Because transaction costs require an extra term to account for the frequency of rebalancing.
  3. Because the stock price S has a non-zero drift μ, which makes the second-order time sensitivity relevant.
  4. Because the risk-free rate r is continuously compounded, requiring a quadratic adjustment.

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