Why Options on Two Assets Cannot Replicate an Option on Their Ratio
Summary
The document asks whether options on two separate assets, each quoted against a common currency, can create an option on their exchange ratio. Its answer says that a portfolio with payoff equal to the sum of a function of each asset separately cannot generally reproduce a payoff that depends on the ratio of both assets. A boundary argument illustrates the mismatch: setting one asset's value to zero would require the payoff contribution from the other asset to be constant, although the ratio option still varies with that asset.
The answer distinguishes replication from valuation and hedging. Individual option prices reveal marginal distributions, while a ratio option depends on the assets' joint distribution, so modeling it requires additional assumptions such as correlation. Those assumptions introduce risks that options on the two assets alone cannot fully hedge, even if delta and vega exposures can be adjusted. No pricing formula or empirical evidence is provided, and correlation-sensitive hedges may behave unexpectedly.
Key ideas
- Separate options on each asset cannot generally replicate a payoff that depends on their ratio.
- Individual option prices reflect marginal distributions, while a ratio option depends on the joint distribution.
- Pricing a ratio option requires additional modeling inputs, such as correlation.
- Delta and vega may be hedged, but the added parameter risk is not fully hedgeable with the individual options.
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Full text
# Synthetic options between two underlyings # Synthetic options between two underlyings Given options (call and puts) for two given underlying pairs, A/USD and B/USD, is it possible to build synthetic options for the A/B pair? Concrete example: spot gold (XAU/USD) is around \$2600 and silver (XAG/USD) is around \$31. So XAU/XAG is around 83.87. Given gold and silver options markets, is it possible to build eg. a gold-to-silver call option with strike 85? ## Answer by spaceisdarkgreen (score 3, accepted) https://quant.stackexchange.com/a/80665 No. Say you have any portfolio with payoff in the form $f(A) + g(B)$. We can't write $(A/B-K)^+$ in this form. At $A=0,$ we would need $f(0) = -g(B)$ for all $B,$ so $g$ would be constant, which is absurd since $(A/B-K)^+$ clearly depends on $B.$ Which is not to say that the prices of options on $A$ and $B$ are entirely uninformative about the price of options on $A/B,$ or that they can't be used in any kind of hedging strategy. The fundamental modeling issue, though, is that the prices of options on $A$ and $B$ are only dependent on the marginal distributions of $A$ and $B$ whereas the price of an option on $A/B$ depends on the joint distribution. So inevitably, new parameters (e.g. correlations) need to be introduced and marked somehow (easier said than done... especially if there aren't market prices for options on $A/B$, but even if so there are complications in practice). And there will be risk to these new parameters that can't be hedged with options on $A$ and $B,$ though the vegas and deltas can be hedged. (Note also these greeks will of course be sensitive to the correlation... it can even flip the sign of one of the vegas.)
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