i am reading a article about the option pricing.
and i got stuck with some typical statement.
$C(K)=\int (x-K)^+\mu(dx)$
is given. here, $\mu$ is implied law of asset price and C(K) is the price of call. Then if C is twice differentiable, then $\mu(dx)=C''(x)dx$. i don't know how to get this formula. Plaease explain slowly in details. Thanks in advance.
$\mu$ is the measure induced by the cdf $F$ of the corresponding rv so $\mu(-\infty,x]=F(x).$ With this integrand, the Lebesgue-Stieljes integral is equivalent to the Riemann-Stieltjes: $$C(K)=\int_K^\infty (x-K)dF(x) $$
In the special case where $F$ has density $f$: $$ C(K)=\int_K^\infty xdF(x)-K(1-F(K)) $$ $$C^\prime (K)=-Kf(K)-(1-F(K))+Kf(K)=F(K)-1 $$ So $C^{\prime\prime}(K)=f(K) $ for all $K$. (Assuming $C$ is twice differentiable seems to imply $f$ exists.)
For all $x$ we can write this as $C^{\prime\prime}(x)dx=f(x)dx=F(dx)=\mu(dx)$