# 以股票的形式推導Black Scholes PDE

$$dS=S r dt + S \sigma dW_B \\ dB=B r dt$$ so we simply get $$\begin{eqnarray} d\frac{C}{B} &=& \frac{\partial_t C dt + \partial_S CdS + \frac{1}{2} \partial_{S,S} CdS^2 }{B}-\frac{CdB}{B^2} \\ &=& \frac{\partial_t C + r S\partial_S C + \frac{1}{2} \sigma^2 S^2\partial_{S,S} C -rC}{B} dt + \frac{\sigma S \partial_S C}{B} dW_B + \mathcal O({dt}^{3/2}) \end{eqnarray}$$ and demanding that $\frac{C}{B}$ be a Martingale requires the vanishing of the drift term and we get the Black Scholes PDE: $$\partial_t C + r S\partial_S C + \frac{1}{2} \sigma^2 S^2\partial_{S,S} C -rC=0$$

Now I try to do the same while taking the Stock as a numeraire. I will demand, as usual, that $d \frac{C}{S}$ is a Martingale under this measure. Under this measure we have $$dS = S(r+\sigma^2) dt + S \sigma dW_S$$ so we get $$\begin{eqnarray} d\frac{C}{S} &=& \frac{\partial_t C dt + \partial_S CdS + \frac{1}{2} \partial_{S,S} CdS^2 }{S}-\frac{CdS}{S^2} + \frac{CdS^2}{S^3} \\ &=& \frac{\partial_t C + (r+\sigma^2) S\partial_S C + \frac{1}{2} \sigma^2 S^2\partial_{S,S} C}{S} dt + \frac{\sigma S\partial_S C dW_S}{S} - \frac{C}{S}\big((r+\sigma^2) dt + \sigma dW_S \big)+\frac{C}{S}\sigma^2 dt +\mathcal O(dt^{3/2}) \\ &=& \frac{\partial_t C + (r+\sigma^2) S\partial_S C + \frac{1}{2} \sigma^2 S^2\partial_{S,S} C -rC}{S} dt + \frac{\sigma S \partial_S C-C}{S} dW_S +\mathcal O({dt}^{3/2}) \end{eqnarray}$$

## 最佳答案

$$d \ left（\ frac {C_t} {S_t} \ right）= \ frac {1} {S_t} dC_t - \ frac {C_t} {S_t ^ 2} dS_t + \ frac {C_t} {S_t ^ 3} d \ langle S_t，S_t \ rangle {\ color {green} { - \ frac {1} {S_t ^ 2} d \ langle C_t，S_t \ rangle}}$$

$$d \ left（\ frac {C_t} {S_t} \ right）= \ frac {1} {S_t}（\ partial_t C_t dt + \ partial_S C_t dS_t + \ frac {1} {2} \ partial_ {SS} C_t \ sigma ^ 2 S_t ^ 2 dt） - \ frac {1} {S_t} \ left（（r + \ sigma ^ 2）C_t dt + \ sigma C_t dW_t \ right）+ \ frac {1} {S_t} \ sigma ^ 2 C_t dt - \ frac {1} {S_t} \ partial_S_t \ sigma ^ 2 S_t dt$$ 或等同於重新安排一些條款 $$d \ left（\ frac {C_t} {S_t} \ right）= \ frac {1} {S_t}（\ partial_t C_t + r S_t \ partial_S C_t + \ frac {1} {2} \ partial_ {SS} C_t \ sigma ^ 2 S_t ^ 2 - rC_t）dt +（。）dW_t$$

[Remark] This result simply comes from applying the bidimensional version of Ito's lemma $$df = (\partial_t f) dt + (\partial_X f) dX_t + \frac {1}{2} (\partial_{XX} f) d\langle X_t \rangle + (\partial_Y f) dY_t + \frac {1}{2} (\partial_{YY} f) d\langle Y_t \rangle + (\partial_{XY} f) d\langle X_t, Y_t \rangle$$