Friday, April 6, 2018

black scholes - Put Call Symmetry


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 St=x for t[0,T].


I got the hint to use Ex[(STK)+]=xKE1/x[(1KST)+]. For that reason I wanted to use Girsanov and conclude from that Ex[(STK)+]=EK[(xST)+] 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:


CallBS(S0,K,T)=PutBS(K,S0,T)


Because the dynamics of a lognormal variable starting at K are the same as those of lognormal variable starting at S0 if we multiply it by KS0, we can write the put price as follows:


PutBS=E[(S0KSTS0)+]=E[STS0(S20STK)+]


So, we can express the call-put symmetry as follows:



E[(STK)+]=E[STS0(S20STK)+]


More generally, for every positive function f: E[f(ST)]=E[STS0f(S20ST)]


Which could be interpreted as follows:



The law of ST under Q is the same as the law of S20ST under QS which is defined by its Radon-Nikodym derivative: dQSdQ=STS0



This interpretation answers your question as to where to start with Girsanov.


General case


Just for reference, in the case where the drift μ is not zero, but rather r or rq, the idea is to use a power of S_T to get a martingale:





  1. Sαt is lognormal. With the right value of α you can make it a martingale. It is eas to show that this value is α0=12μσ




  2. For every given positive f: E[f(ST)]=E[(STS0)α0f(S20ST)]




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