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Pricing Spread Options on Relative Asset Performance

Article Quant Q&A · Author: Fabio
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Summary

The document asks how a spread option based on the difference between two assets’ normalized terminal prices compares with a conventional option on their price difference. The payoff in question subtracts each asset’s initial price by expressing terminal values as performance ratios, then applies a strike and positive-part payoff. This framing matters because a price spread retains the scale of each underlying, while normalized performance expresses changes relative to starting levels.

The answer notes that under a model such as Black–Scholes, the distribution of each terminal-to-initial price ratio does not depend on the initial spot price. In that setting, the normalized payoff can be treated as a standard spread problem with both starting prices set to one. This provides a simple way to apply existing spread-option analysis. The exchange does not derive a pricing formula, establish tractability under other models, or discuss dependence assumptions, dividends, or calibration, so its conclusion is limited to models with the stated scale invariance.

Key ideas

  • A return-based spread option compares normalized terminal prices rather than raw asset prices.
  • Under Black–Scholes, terminal price ratios have distributions independent of initial spot levels.
  • That scale invariance allows the normalized problem to be represented with both initial prices set to one.
  • The argument does not establish the same simplification for models where return distributions depend on spot levels.

Tags

OptionsDerivatives pricingEquitiesStatistics
Full text
# Spread options on prices or returns?


# Spread options on prices or returns?












I need some clarifications regarding spread options. I have always found them characterized as paying, at maturity, the difference between the prices of two underlying assets: $$ (S_1(T)-S_2(T)-K)^+ $$ I have been presented with the task of pricing a spread option paying, at maturity, the difference between the performance of two underlying assets, as in: $$ \left(\frac{S_1(T)}{S_1(0)} - \frac{S_2(T)}{S_2(0)} - K)^+\right) $$ Since I have not found explicit references to the form above, I am wondering what are the differences in terms of tractability. How does the level of the spot price impact the problem? Does this form belong to the family of models for spread options that is analytically tractable?

## Answer by Mark Joshi (score 0, accepted)

https://quant.stackexchange.com/a/16463

If you are using a model such as BS where the distribution of $S_{j}(T)/S_{j}(0)$ does not depend on $S_{j}(0)$ it really makes very little difference. Just take the initial stock prices to be $1$ and away you go.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.