Replicating a Quanto Put with Correlated Brownian Risk
Summary
The document sets up a quanto put with a fixed exchange-rate conversion in its payoff and describes candidate hedge instruments involving the foreign stock, exchange-rate conversion, foreign cash, and domestic cash. It applies Itô’s lemma to the proposed portfolio and to the option value, then attempts to match their drift and Brownian terms. The underlying assets are driven by a shared two-dimensional Brownian motion, which makes the hedge equations vector-valued and raises the question of whether the proposed instruments span the option’s risk.
The post contains a question rather than a completed replication. It does not solve for hedge holdings or verify that the portfolio dynamics are correct. In particular, matching Brownian exposures requires checking the rank and span of the available risk exposures, while matching drift must be consistent with the chosen pricing measure and self-financing portfolio. The equations and signs therefore need independent validation before they can support a hedge. The material is useful as a setup for studying quanto hedging, but supplies no final strategy or numerical evidence.
Key ideas
- A quanto put pays a foreign asset payoff converted at a pre-agreed exchange rate.
- The proposed hedge combines foreign equity exposure, currency conversion, and cash accounts.
- With shared multidimensional Brownian risk, hedge exposures must be matched across all risk directions.
- Replication depends on whether the traded hedge instruments span the option’s stochastic exposures.
- The document leaves the hedge unsolved, and its portfolio equations require verification.
Tags
Full text
# Quanto put hedge\ replication with a brownian motion
# 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:
- $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 $
- $h_1X(t)S_J(r_{us}-\sigma^T_X\sigma_J)+h_2X(t)B_Jr_{us}+h_3r_{us}B_{us}=\frac{\partial P}{\partial t}dt + \frac{\partial P}{\partial S_J}S_J \left(r_J- \sigma^T_X\sigma_J\right) + \frac{1}{2}\frac{\partial^2 P}{\partial S^2}S_t^2 \sigma_J^T\sigma_J$
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.