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)=∂C∂K.
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((ST−K)+)=P(0,T)∫∞K(x−K)f(x)dx=P(0,T)[∫∞Kxf(x)dx−K∫∞Kf(x)dx].
Then, ∂C∂K=−P(0,T)∫∞Kf(x)dx.
That is, −P(0,T)P(ST>K)=∂C∂K.
We can additionally obtain that P(0,T)f(K)=∂2C∂K2.
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