Modeling and Capping Stock-versus-Index Outperformance Options
Summary
The document discusses an option whose payoff depends on a stock’s performance relative to a dividend-reinvesting index. One response connects the contract to spread options and points to the Margrabe formula for intuition. Another gives the variance of relative returns in terms of the two asset volatilities and their correlation, and notes that an index stated in total-return terms has no separate dividend yield adjustment in the proposed setup.
The discussion flags assumptions and unresolved details rather than providing a complete pricing method. Margrabe’s framework relies on assumptions about asset dynamics and the volatility of their ratio; adding a maximum payoff would require adapting the payoff and pricing approach. The answers do not derive a capped formula, specify all contract terms, or establish that the proposed dividend treatment fits every index convention. The material is best read as an introduction to the modeling questions, not as a worked valuation or hedge.
Key ideas
- A stock-versus-index payoff can be analyzed as an option on relative performance, similar to a spread option.
- Relative-return variance depends on both assets’ volatilities and their correlation.
- A total-return index already reflects reinvested dividends, affecting how its dividend input is modeled.
- The Margrabe framework offers useful intuition but relies on assumptions about asset dynamics and relative volatility.
- A payoff cap changes the contract and requires additional pricing work beyond the discussion provided.
Tags
Full text
# Outperformance options
# Outperformance options
I am currently in a project regarding outperformance options.
I have built my monte carlo simulation but still have some questions.
I am looking at the performance of a stock versus an index. The index is an GI, i.e. includes re-invested dividend.
How should I think about the volatility, dividend and correlation for the two assets?
Also, If i were to use a Cap, i.e. a maximum payoff, how would this differ from a normal Capped option?
## Answer by Rylan (score 1)
https://quant.stackexchange.com/a/81902
What you're describing sounds to me like a spread option under a different name. You can probably gain intuition on the volatility, dividend, and correlation by studying the Margrabe formula and any related literature on spread options.
As for the use of a cap, I'm not totally sure I understand the question -- I expect it should be a fair bit more complicated to modify the Margrabe formula to accept a cap than the Black Scholes formula, but admittedly I haven't tried.
## Answer by João (score 1)
https://quant.stackexchange.com/a/81911
For volatility it´s a standard equation for outperformance options.
$$\sigma^2_{\text{out}} = \sigma^2_S + \sigma^2_I - 2\rho \sigma_S \sigma_I$$
Adjust this as needed.
In Dividends since the GI already includes dividends you should make this on adjusting the drift
$$q_I = 0$$
For the Margrabe’s Formula It´s a closed form pricing formula for European options to exchange one risky asset for another at maturity and I don´t really know if it's suitable for this example because:
1- The price of two underlying assets in this formula are assumed to follow geometric Brownian motions, the volatilities of these assets may not be necessarily constant but the volatility of the ratio of the two assets has to be constant.
Also for dividends and the introduction of the cap you'll need to rearrange the equation and the maximum payoff obviously, never tried it.
For the last question I think you're asking whats the difference between a capped outperformance option and a capped option ?
You have differences in all variables from the underlying, strike price, payoff, cap effect and correlation effect.
Here´s a good example:
Exotics Options Trading - Chapter 7 Monte Carlo and 21 Outperformance optionsShown 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.