Put–Call Symmetry and the Martingale Status of a Call Price

Summary

The document asks whether a call-price process expressed as C(K, S_t) is a martingale when put–call symmetry holds, and whether that property persists when the symmetry is broken. It states a symmetry relation connecting put and call prices under an assumed homogeneous model, then reasons that if the put-price process is a local martingale, the equal call-price expression should share that property under symmetry.

It provides no answer or derivation for the case where symmetry fails. In particular, it does not specify the pricing measure, discounting convention, underlying dynamics, or assumptions needed to determine whether a price process is a martingale or local martingale. The equality under symmetry is presented as an assumption, not established for a particular market or model. The material frames a useful derivatives-pricing question but is insufficient to conclude that the call price remains a martingale after the symmetry condition is removed.

Key ideas

• The question relates put–call symmetry to the martingale behavior of a call-price process.
• Under the stated symmetry, the call-price expression is equated with a put-price process.
• The document asks whether martingale behavior remains when symmetry fails but does not answer.
• Determining the property requires additional model and pricing-measure assumptions not supplied in the excerpt.

Tags

# Is $C(K,S_t)$ a (local) martingale if PCS is broken?


# Is $C(K,S_t)$ a (local) martingale if PCS is broken?












When put-call symmetry holds $$ P(S_t,K) = C(K,S_t) = \frac{K}{S_t} C \left( S_t, \frac{S_t^2}{K} \right) $$ where $P$ is the market price of a put option and $C$ is the market price of a call option. The second equality above holds for homogeneous models which I am assuming.

Since $P(S_t,K)$ is a (local) martingale, so is $C(K,S_t)$, correct?

When PCS is broken then $P(S_t,K) \neq C(K,S_t)$, but is $C(K,S_t)$ nonetheless still a martingale or not anymore?

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