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Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. Since there, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, by the results for the Black-Scholes model it can be replicated. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $S^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. There, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, so it can be replicated by the results for the Black-Scholes model. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $S^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. Since there, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, by the results for the Black-Scholes model it can be replicated. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $\tilde{S}^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. Since there, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, by the results for the Black-Scholes model it can be replicated. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $S^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\tilde{\mathcal{X}}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. Since there, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, by the results for the Black-Scholes model it can be replicated. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $\tilde{S}^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. Since there, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, by the results for the Black-Scholes model it can be replicated. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $\tilde{S}^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.