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For the Cox-Ross-Rubinstein diffusion approximation, fix calendar maturity $T$, take $h=T/n$, and choose $u=e^{\sigma \sqrt{h}}$, $d=e^{-\sigma \sqrt{h}}$, and the one-step bank factor $b=e^{\rho h}$. For sufficiently fine mesh, $d<b<u$. The risk-neutral probability expands asEach stock log-increment is $\pm \sigma \sqrt{h}$. Its mean under $Q$ is $(\rho -{\sigma }^2/2)h+O(h^2)$ and its variance is ${\sigma }^{2}h+O(h^2)$. Independent increments, their vanishing maximum size, and the Donsker invariance principle for this triangular array give the path limitExponentiation gives the geometric Brownian motion of the Black-Scholes model, and the bank account tends to $e^{\rho t}$.

The physical model is matched by taking $p_h=(e^{\mu h}-d)/(u-d)$. Its corresponding limit has log drift $\mu -{\sigma }^2/2$, so $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t^P$. The likelihood ratios from part (ii) have the matching continuous-time limit. With $\lambda =(\mu -\rho )/\sigma$, a Taylor expansion of their two possible log-multipliers giveswhere ${\epsilon }_k$ is $+1$ or $-1$. Thus the limiting Radon-Nikodym derivative is the Girsanov theorem density $\exp (-\lambda W_T^P-{\lambda }^{2}T/2)$, turning the physical stock drift into $\rho$.

The binomial risk-neutral pricing formula converges to $e^{-\rho T}{\mathbb{E}}_Q[f(S_T)]$ for bounded continuous terminal payoffs by convergence in distribution. For ordinary call and put payoffs, uniform stock second-moment bounds give the required uniform integrability and hence convergence despite unboundedness. Bounded payoffs with discontinuities only at events of zero limiting probability are handled by the same weak-convergence criterion. Corresponding continuous path functionals can be treated at the process level. Arbitrary discontinuous path claims do not follow from weak convergence without further conditions. In the smooth limiting model, the discrete hedging difference quotient becomes the delta hedge $V_x$, and its value satisfies the Black-Scholes equation.