Delta Hedging and Replication of a Two-Asset Spread Option
Summary
The document explains how to think about hedging a European option that pays the positive difference between two asset values. Its pricing expression uses both assets, their carry rates, and a combined volatility that depends on their individual volatilities and correlation. The central hedging point is that the option has a separate delta with respect to each underlying, so a hedge involves positions in both assets; the theoretical setup also includes the risk-free asset.
The answer highlights a limitation: trading the two assets can hedge their individual price exposures, but it cannot directly hedge the option’s sensitivity to correlation between them. The note gives no worked delta calculations, hedge ratios, or empirical evidence, so it is an introductory conceptual explanation rather than a full replication recipe. A practical hedge would also depend on contract multipliers, financing, and the ability to trade both assets.
Key ideas
- A spread option’s payoff depends on the relative values of two assets.
- The option price has a separate partial delta with respect to each underlying asset.
- A theoretical hedge combines both underlying assets with the risk-free asset.
- The hedge cannot directly eliminate exposure to changes in correlation.
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# Synthetic replication of a spread option payoff
# Synthetic replication of a spread option payoff
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?
## Answer by Arshdeep (score 1, accepted)
https://quant.stackexchange.com/a/79141
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.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.