medium · Quantitative Finance stochastic
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?
- 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.
- Because transaction costs require an extra term accounting for how frequently the hedge position must be rebalanced continuously through time.
- Because the stock price S has a non-zero drift mu, which makes the second-order time sensitivity of the hedged portfolio value especially relevant here.
- Because the risk-free rate r is continuously compounded, requiring an additional quadratic adjustment term to capture growth of the riskless bond position correctly.
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