Tuesday, September 12, 2017

risk neutral measure - probability that the stock price is below the strike price



How can I prove that under the risk-neutral probability:


$\mathbb{P}[S_{t}

where


St is the stock price, K is the strike price, C is the call option price


Thank you !



Answer



Your posting has an error, that is, the identity should be P(0,T)P(ST>K)=CK.

The derivation below is based on this assumption. We denote by f(x) the density function for ST. Then P(ST>K)=Kf(x)dx,
and C(K,T)=P(0,T)E((STK)+)=P(0,T)K(xK)f(x)dx=P(0,T)[Kxf(x)dxKKf(x)dx].
Then, CK=P(0,T)Kf(x)dx.
That is, P(0,T)P(ST>K)=CK.


We can additionally obtain that P(0,T)f(K)=2CK2.


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