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)


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


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

If not. How should I find the hedging portfolio?

Thanks in advance!

  • $\begingroup$ 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. $\endgroup$
    – Gordon
    Commented Mar 2, 2021 at 17:59
  • $\begingroup$ @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. $\endgroup$ Commented Mar 2, 2021 at 18:45
  • $\begingroup$ @Gordon How so? Should I not consider the foreign bond? $\endgroup$ Commented Mar 2, 2021 at 18:48
  • $\begingroup$ @Gordon What is the financial argument behind that? $\endgroup$ Commented Mar 2, 2021 at 19:01
  • $\begingroup$ @Gordon The problem is that on a hint it says "do a full-fledged three holdings continous-time argument", so I suppose I need three assets. So I cannot let h_2 or h_3 equal 0. $\endgroup$ Commented Mar 2, 2021 at 19:40


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