Put-Call Symmetry at Any Time in Black-Scholes

Summary

The question asks how to extend a put-call symmetry known at the initial time to an arbitrary time before expiry in a Black-Scholes model with zero interest rate and zero dividend yield. With the current asset value used as the starting value, it asks whether a call with strike K equals a put with strike equal to that current value and strike reference K, as expressed in the document’s notation. The author has an initial-time identity but is unsure how to justify changing the starting value.

The proposed argument tries to combine the Markov property with the time-zero equality, but the question itself recognizes that this chain does not yet establish the result. No answer or proof is included, so the document supplies neither evidence nor a resolution. Its useful content is the identification of a conditional-pricing issue: a pointwise identity at the initial state cannot simply be carried to later states without establishing the relevant symmetry for the conditional model started at that state. The claim is limited to the stated Black-Scholes assumptions and does not address rates, dividends, or other models.

Key ideas

• The question concerns extending an initial-time call-put symmetry to arbitrary times before expiry.
• The model assumptions specified are zero interest rates and zero dividend yield.
• The suggested Markov-property argument is acknowledged as incomplete.
• No answer is provided, so the symmetry and its proof are not established in the excerpt.

Tags

# Put Call Symmetry for arbitrary $t\in [0,T]$


# Put Call Symmetry for arbitrary $t\in [0,T]$












I want to assume I am in a general Black Scholes Model with $r=0$ and $\delta=0$ and the typical filtered probability space.

I know that $Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T)$ with $x= S_0$, which is our start value, holds. I have proven this. My question is now how can I expand it to arbitrary $t \in [0, T]$ with $x = S_t$, i.e. $Call^{BS}(t, x, K, T) = Put^{BS}(t, K, x, T)$ with $x = S_t$?

It is for me of course somehow clear in a logical way but I have problems proving it. My idea was to use the markov property of our asset and get something like $Call^{BS}(t, x, K, T) = Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T) = Put^{BS}(t, K, x, T)$ but this is of course not true yet. How do I change the starting value?

Thank you for your answers!

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