1
$\begingroup$

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?

Thanks in advance!

$\endgroup$
7
  • $\begingroup$ 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.