hard · Quantitative Finance

When reducing the Black-Scholes PDE to the heat equation u_τ = u_xx, which of the following substitutions is primarily responsible for converting the variable-coefficient derivatives (due to S) into constant-coefficient derivatives?

1. V = e^α x + β τ u
2. Τ = T - t
3. X = ln S
4. d_1 = fracln(S/K) + (r + 0.5σ^2)Tσ√(T)

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