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I have two assets, $S_1$ and $S_2$, and a European exchange-one-asset-for-another call option, such as those introduced by Margrabe (1978). So my payoff at expiration is the difference between the prices of the two assets (possibly adjusted with appropriate multipliers), basically their spread:

$$\Pi(T) = \max{\left( Q_1S_{1,T} - Q_2S_{2,T}, 0 \right)}$$

where $Q_1$ is the quantity of asset $S_1$ and $Q_2$ is the quantity of asset $S_2$. The closed-form price of the option is

$$c = Q_1S_1e^{\left( b_1 - r \right)T}\mathcal{N}(d_1) - Q_2S_2e^{\left( b_2 - r \right)T}\mathcal{N}(d_2)$$

where

$$d_1 = \frac{\ln{\left( \frac{Q_1S_1}{Q_2S_2}\right)}+\left(b_1 - b_2 + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$ $$\sigma = \sqrt{\sigma^2_1 + \sigma^2_2 - 2\rho\sigma_1\sigma_2}$$

where $\rho$ is the correlation between the two assets.

Questions:

  • I would like to synthetically replicate the payoff of this option. I can't quite figure out how synthetic replication works since I have two assets. I can calculate the $\Delta$ of the option, first of all, but is this with respect to $S_1$ or $S_2$?
  • Assuming I have calculated a $\Delta$ with respect to $S_1$ and want to get the payoff of the replicating portfolio, does this $\Delta$ tell me what fraction of capital I need to dynamically put on $S_1$ and then its complement of $1$ is the fraction of capital to put on $S_2$? Basically, it works like the Black-Scholes $\Delta$ on $S_1$ only that the other asset is not the money market account but $S_2$?
  • Put another way, how would you hedge the $\Delta$ of this option using the two assets?
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1 Answer 1

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You need both $S1$ and $S2$ to hedge, and deltas can be calculated separately for both (the partial derivatives of price w.r.t each). However you will not be able to hedge the correlation - as your product depends on the joint distribution of $S1$ and $S2$, and the hedging portfolio only depends on their respective marginal distributions.

Edit: It is like black scholes but the theoretical hedging portfolio is a combination of both assets and the risk free rate. The PDE now also has a correlation term between the brownian motions that drive both assets.

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