# Quanto put hedge\ replication with a brownian motion

Consider $$d B_{us}(t)=r_{us} B_{us}(t) dt\\dX(t)=X(t)(r_{us}-r_J)dt+X(t)\sigma^T_J dW(t)\\d B_J(t)=r_{J} B_{J}(t) dt\\dS_J(t)=S_J(t)(r_J-\sigma^T_X\sigma_J)dt+S_J(t)\sigma^T_J dW(t)$$

where the $$\sigma$$'s are 2-dimensional vectors, but the Brownian motion is the same.

Consider a quanto put, whose payoff function is $$Y_0(K-S_J(T))^+$$

where Y_0 is some agreed-upon-in-advance exchange rate.

Replicate the quanto put.

I started by crating the following portfolio:

$$V=h_1 X(t)S_J(t) + h_2 X(t)B_J + h_3(t)B_{us}$$

Applying Ito (after some computations)

$$dV(t,S_J)=(h_1X(t)S_J(r_{us}-\sigma^T_X\sigma_J)+h_2X(t)B_Jr_{us}+h_3r_{us}B_{us})dt+(h_1X(t)S_J(\sigma_J^T+\sigma_X^T)+h_2B_JX(t)\sigma_X^T))dW(t)$$

Applying Ito to $$P(t,S_J)$$ \begin{align*} dP &= \frac{\partial P}{\partial t}dt + \frac{\partial P}{\partial S_J}S_J \left(r_J- \sigma^T_X\sigma_J\right) + \frac{1}{2}\frac{\partial^2 P}{\partial S^2}S_t^2 \sigma_J^T\sigma_J dt+\frac{\partial P}{\partial S_J}S_J\sigma_J^T dW(t). \end{align*}

Now equaling the dt and dw terms on both equations I get:

1. $$h_1X(t)S_J(\sigma_J^T+\sigma_X^T)+h_2B_JX(t)\sigma_X^T=\frac{\partial P}{\partial S_J}S_J\sigma_J^T$$

2. $$h_1X(t)S_J(r_{us}-\sigma^T_X\sigma_J)+h_2X(t)B_Jr_{us}+h_3r_{us}B_{us}=\frac{\partial P}{\partial t}dt + \frac{\partial P}{\partial S_J}S_J \left(r_J- \sigma^T_X\sigma_J\right) + \frac{1}{2}\frac{\partial^2 P}{\partial S^2}S_t^2 \sigma_J^T\sigma_J$$

If I had two independent or even correlated distinct wiener processes or Brownian motion , it would be easy to solve for the equation 1) as it was done on this previous thread answer Dynamic Hedge of Quanto Options

Questions:

If I am on the right track. How should I finish this problem?

If not. How should I find the hedging portfolio?

• Your question is not clear: where is $\sigma_S$ defined? If it is two-dimensional, while the Brown motion is the same, then you can combine them together. Commented Mar 2, 2021 at 17:59
• @Gordon There is no $\sigma_s$ I was trying to write the plural of sigma. I mean both $\sigma_X$ and $\sigma_J$ are two dimensional. The question is how to finish the proof by computing the h_1, h_2 and h_3, getting an expression for each of them. I hope now it is clear. Commented Mar 2, 2021 at 18:45