= Solution
Assume the usual positive initial <stock> price and strike, so the logarithm is defined, and interpret $U$ as a classical solution of the displayed <partial differential equation>. For a fixed complex $z$, put $F_t(z)=S_t^zU(t,\sigma_t,z)$, defining $S_t^z=\exp(z\log S_t)$. The <Itô formula> for the two <Brownian motions> with <correlation coefficient> $\rho$ gives
$$
\begin{aligned}
dF_t(z)=S_t^z\bigg(&U_t+A(\sigma_t)U_\sigma+\frac12B(\sigma_t)^2U_{\sigma\sigma}+\frac12\sigma_t^2z(z-1)U+z\sigma_tB(\sigma_t)\rho U_\sigma\bigg)dt\\
&+S_t^z\bigl(z\sigma_tU\,dW_t^S+B(\sigma_t)U_\sigma\,dW_t^\sigma\bigr).
\end{aligned}
$$
The <partial differential equation> cancels the entire <drift>. By the permission to treat the resulting <local martingales> as true <martingales>, and the terminal condition, $F_t(z)=\mathbb E[S_T^z\mid\mathcal F_t]$.
The bounded <spot volatility> ensures $\mathbb ES_T^{x_0}<\infty$: the <Itô formula> for a real power and localization bound its <moment> by $S_0^{x_0}\exp(x_0(x_0-1)\|\sigma\|_\infty^2T/2)$. Rewrite the proposed integrand as
$$
\frac{S_tU(t,\sigma_t,z)}{f(z,\log(K/S_t))}=\frac{K^{1-z}F_t(z)}{z(z-1)}.
$$
Now $|F_t(x_0+iy)|\leq\mathbb E[S_T^{x_0}\mid\mathcal F_t]$, an integrable bound independent of $y$. Conditional <Fubini's theorem> and part (b) therefore imply
$$
\boxed{C_t=\mathbb E[(S_T-K)^+\mid\mathcal F_t].}
$$
This route justifies the contour exchange without assuming bounds on $U$ uniform in all complex $z$. Cash is constant, $S$ is a true <martingale> under the stated allowance, and the displayed $C$ is a true <martingale>. The original measure is thus an <equivalent martingale measure> for all three assets. The <fundamental theorem of asset pricing> gives \b[the market has no <arbitrage> under the usual admissible trading convention.]
Back to article page
