Black–Scholes–Merton 方程

BSM PDE：

$$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac12\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV.$$

推导要点：

• 假设标的满足 GBM：$dS=\mu Sdt+\sigma SdW(t)$。
• 令期权价值 $V=V(S,t)$，用 Ito 引理写 $dV$。
• 构造无风险组合：多 1 份期权、空 $\frac{\partial V}{\partial S}$ 股标的，消去随机项。
• 无风险组合必须以无风险利率增长，得到 PDE。

也可用风险中性测度或 Feynman–Kac 得到同一结果。

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Original Explanation

The Black-Scholes PDE is: $$\frac{\partial V}{\partial t} \;+\; r\,S\,\frac{\partial V}{\partial S} \;+\; \frac{1}{2}\,\sigma^2\,S^2\, \frac{\partial^2 V}{\partial S^2} \;=\; r\,V.$$

• Stock follows $dS=\mu S\,dt+\sigma S\,dW(t)$.
• Let $V=V(S,t)$. By Ito’s lemma, $dV=\dots$.
• Hedge by long 1 option, short $\frac{\partial V}{\partial S}$ shares => no diffusion risk => must earn risk-free rate => PDE results.

Alternatively, from risk-neutral measure or Feynman-Kac theorem bridging SDEs and PDEs, we get the same PDE with boundary conditions.