# Finding the replicating portfolio a European T-claim (put)

I have

$$dX_0(t) = ρX_0(t)dt ; \qquad X_0(0) = 1\\ dX_1(t) = αX_1(t)dt + βX_1(t)dB(t) ; \qquad X_1(0) = x_1 > 0$$

as the classical Black-Scholes market. I a trying to look for the replicating portfolio $\theta(t) = (\theta_0(t),\theta_1(t))$ for the following European T-claim:

$F(\omega) = (K -X_1(T,\omega))^+$ $\qquad$ (the European put)

$$dX_0(t)=\rho\,X_0(t)\,dt$$ thus $$X_0(t)=e^{\rho t}$$ To replicate the derivative $V=F(t,X_1(t))$ we form a self-financing portfolio with the stochastic process $X_1$ and deterministic process $X_0$ in the right proportion.
Hence we need to use the replicating $(\theta_0(t),\theta_1(t))$. The self-financing assumption means that $$V=\theta_1(t)X_1(t)+\theta_0(t)X_0(t)\tag 0$$ so
$$dV=\theta_1(t)dX_1(t)+\theta_0(t)dX_0(t)\tag 1$$ by application of Ito's lemma, we have $$dV=\left(\frac{\partial V}{\partial t}+\alpha X_1(t)\frac{\partial V}{\partial X_1}+\frac{1}{2}\beta^2X_1^2(t)\frac{\partial^2 V}{\partial X_1^2}\right)dt+\beta X_1(t)\frac{\partial V}{\partial X_1}dB_t\tag 2$$ We assume the portfolios are self-financing, which implies that changes in portfolio value are due to changes in the value of the three instruments, and nothing else.
$(1)$ and $(2)$ $$\left(\frac{\partial V}{\partial t}+\alpha X_1(t)\frac{\partial V}{\partial X_1}+\frac{1}{2}\beta^2X_1^2(t)\frac{\partial^2 V}{\partial X_1^2}\right)dt+\beta X_1(t)\frac{\partial V}{\partial X_1}dB_t=\\(\alpha\theta_1(t) X_1(t)+\rho\theta_0(t) X_0(t))dt+\beta\theta_1(t) X_1(t)dB_t\tag 3$$ so $$\theta_1(t)=\frac{\partial V}{\partial X_1}\tag 4$$ Substituting in Equation $(4)$, we have $$\left(\frac{\partial V}{\partial t}+\frac{1}{2}\beta^2X_1^2(t)\frac{\partial^2 V}{\partial X_1^2}\right)=\rho\theta_0(t) X_0(t)\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad=\rho\theta_0(t)\left(\frac{V-\theta_1(t)X_1(t)}{\theta_0(t)}\right)\\ \\ \qquad\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,=\rho V-\rho\theta_1(t)X_1(t)\\ \\ \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad=\rho V-\rho\frac{\partial V}{\partial X_1}X_1(t)$$ in other words $$\frac{\partial V}{\partial t}+\rho X_1(t)\frac{\partial V}{\partial X_1}+\frac{1}{2}\beta^2X_1^2(t)\frac{\partial^2 V}{\partial X_1^2}-\rho V=0\tag 5$$ Indeed
$$\theta_0(t)=\frac{V-\theta_1(t)X_1(t)}{X_0(t)}\\ \\ \theta_1(t)=\frac{\partial V}{\partial X_1}$$