When American Put–Call Symmetry Depends on the Price Model

Summary

The note explains why American put–call symmetry is described as model dependent. The symmetry relates an option on a price to a corresponding option involving the strike and the inverse price process. For American options, the key reasoning is that the optimal decision to trade the underlying against the strike must correspond to an optimal decision under the transformed, inverse process.

The explanation does not require volatility or interest rates to be constant: it says they may vary with time or price. The important condition is that the inverse process remains sufficiently tractable, which likely calls for dynamics with a logarithmic structure. Thus, matching terminal payoffs alone does not establish equal values for American options, because early exercise and the path of the underlying matter. The discussion gives a conceptual condition, not a general proof or a precise list of all processes for which the symmetry holds.

Key ideas

• American put–call symmetry is linked to optimal trading decisions under a transformed inverse-price process.
• The symmetry does not require strictly constant volatility or rates.
• The inverse process must remain sufficiently tractable for the argument to work.
• Matching payoffs alone does not establish equal values when early exercise is possible.

Tags

# Why is the Put-Call Symmetry model dependent?


# Why is the Put-Call Symmetry model dependent?












The put-call symmetry states that `C(S,t;X,r,q) = P(X,t;S,q,r)`, and that this works for American options. According to my notes, this is 'model dependent' because it depends on the assumption that the underlying price follows geometric brownian motion.

However, I don't understand why this would matter: Considering that the payoffs are the same, shouldn't the two options have the same value even if the stock price doesn't follow geometric brownian motion?

## Answer by Brian B (score 6, accepted)

https://quant.stackexchange.com/a/8983

American put-call symmetry relies on the observation that trading $S$ for $K$ is optimal when $\frac1K$ is optimally traded for $\frac1S$. So long as the dynamics of the inverse process $\frac1S$ are sufficiently tractable, you can derive the symmetry formula.

You don't have to have a "pure" GBM for this to work. For example, non-constant (and even price-dependent) volatility and rates are OK. But, given you need a good inverse process you probably do need a log-process of some kind.

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