I want to show the Put Call Symmetry without using the explicite Black Scholes formula. In other words I want to show
Call(t, x, K, T) = Pull(t, K, x, T)
where $S_t = x $ for $t \in [0, T]$.
I got the hint to use $\mathbb{E}_x[(S_T - K)^+] = xK \mathbb{E}_{1/x}[(\frac{1}{K} - S_T)^+]$. For that reason I wanted to use Girsanov and conclude from that $\mathbb{E}_x[(S_T - K)^+] = \mathbb{E}_{K}[(x - S_T)^+] $ holds. For simplicity I assume our interest rate is zero and consider the Black Scholes model with the filtration which is generated by the Brownian motion. I also found the paper form Peter Carr (https://www.math.uchicago.edu/~rl/PCSR22.pdf) but I didn't really understand it.
Thank you for your help!
Answer
The Black-Scholes symmetry formula is valid only under Black-Scholes as its name suggests. It works only for a lognormal $S$. For other models, you can find symmetry relations but they will be different.
Here is an interpretation that will help you link the result to the distributions:
Martingale case
The symmetry relation is:
$$Call^{BS}(S_0, K, T) = Put^{BS}(K, S_0, T) $$
Because the dynamics of a lognormal variable starting at $K$ are the same as those of lognormal variable starting at $S_0$ if we multiply it by $\frac{K}{S_0}$, we can write the put price as follows:
$$ Put^{BS} = \mathbb{E}\left[\left(S_0 - K \frac{S_T}{S_0}\right)^+\right] = \mathbb{E}\left[\frac{S_T}{S_0}\left(\frac{S_0^2}{S_T} - K\right)^+\right]$$
So, we can express the call-put symmetry as follows:
$$\mathbb{E}[(S_T - K)^+] = \mathbb{E}\left[\frac{S_T}{S_0}\left(\frac{S_0^2}{S_T} - K\right)^+\right]$$
More generally, for every positive function $f$: $$\mathbb{E}[f(S_T)] = \mathbb{E}\left[\frac{S_T}{S_0} f\left(\frac{S_0^2}{S_T}\right)\right]$$
Which could be interpreted as follows:
The law of $S_T$ under $\mathbb{Q}$ is the same as the law of $\frac{S_0^2}{S_T}$ under $\mathbb{Q}^S$ which is defined by its Radon-Nikodym derivative: $\frac{d\mathbb{Q}^S}{d\mathbb{Q}} = \frac{S_T}{S_0}$
This interpretation answers your question as to where to start with Girsanov.
General case
Just for reference, in the case where the drift $\mu$ is not zero, but rather $r$ or $r-q$, the idea is to use a power of S_T to get a martingale:
$S_t ^ \alpha$ is lognormal. With the right value of $\alpha$ you can make it a martingale. It is eas to show that this value is $\alpha_0 = 1 - \frac{2\mu}{\sigma}$
For every given positive $f$: $$\mathbb{E}[f(S_T)] = \mathbb{E}\left[\left(\frac{S_T}{S_0}\right)^{\alpha_0} f\left(\frac{S_0^2}{S_T}\right)\right]$$
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