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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.

Write $T=t_0$ and $B_t=e^{-\rho (T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$, while $d{B}_t=\rho B_{t}dt$. A self-financing portfolio with predictable stock and bond holdings $g_t,h_t$ has value $V_t=g_t{S}_t+h_t{B}_t$ and gainsRebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines self-financing.

Let $p(x,t)=xg(x,t)+B_{t}h(x,t)$. The Itô formula givesThe prescribed portfolio gains instead have diffusion coefficient $\sigma S_{t}g$ and drift $\mu S_{t}g+\rho B_{t}h$. Equality of diffusion coefficients requires $p_x=g$, equivalentlyUnder this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac{1}{2}{\sigma }^2{x}^2{g}_x=\rho B_{t}h$. Since $p_t=x{g}_t+B_t{h}_t+\rho B_{t}h$, this becomesThese conditions are also sufficient by the same Itô formula. Necessity initially holds along the stock process. Its positive transition density on $(0,\infty )$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.

For a self-financing portfolio, substitute $g=p_x$ and $B_{t}h=p-x{p}_x$ into the drift identity to obtain the Black-Scholes equationConversely, a smooth solution of this equation gives a self-financing portfolio by choosing the delta hedgeIts value is $p$, and both coefficient identities follow directly.

The printed deduction needs this holdings qualification: the Black-Scholes value equation needs delta-compatible holdings. A value function alone cannot make an arbitrary prescribed decomposition self-financing. As a counterexample, take $p=0$, $g=1$, $h=-x/B_t$. The value function solves the Black-Scholes equation, but this zero-value portfolio has gains $d{S}_t-\rho S_{t}dt$, which contain the nonzero term $\sigma S_{t}d{W}_t$. Thus for the originally specified $g,h$, the exact equivalence is the pricing equation together with $g=p_x$, not the pricing equation alone.

For a claim paying the holder at rate $k(S_t)$, let $p$ denote its remaining, ex-dividend value. A replicating portfolio pays out $k(S_t)dt$, so its net wealth equation is $dp=gdS+hdB-k(S_t)dt$. The delta hedge is still $g=p_x$, but the Black-Scholes equation becomesIf there is a terminal payoff $f$, set $p(x,T)=f(x)$; with no terminal payment set it to zero. The corresponding risk-neutral valuation isEach intervening payment is discounted from its own payment time.

Let $T=t_0$, $\Delta =T-t$ and assume the payoff has sufficient integrability for risk-neutral valuation. Under the risk-neutral measure $Q$, the Black-Scholes model has $d{S}_u=\rho S_{u}du+\sigma S_{u}d{W}_u^Q$, so conditionally on $S_t=x$,Absence of arbitrage gives the time-$t$ priceFor a regular payoff this is the solution of the Black-Scholes equation with terminal value $f$, and the delta hedge $g=V_x$ replicates it. Suitable limits or conditional-expectation representations extend the price formula beyond smooth terminal payoffs.

For the logarithmic stock payoff, put $Y=(\rho -{\sigma }^2/2)\Delta +\sigma \sqrt{\Delta }Z$. Its Gaussian moment-generating function is $\mathbb{E}{e}^Y=e^{\rho \Delta }$, while differentiating that moment gives $\mathbb{E}[Y{e}^Y]=(\rho +{\sigma }^2/2)\Delta e^{\rho \Delta }$. ThereforeThe option delta is $\log x+1+(\rho +{\sigma }^2/2)\Delta$, and the second stock derivative is $1/x$. The Black-Scholes equation can also be checked directly from $V_t=-x(\rho +{\sigma }^2/2)$.

For general convex $f$, let $Y_{+}=\exp ((\rho -{\sigma }^2/2)\Delta +\sigma \sqrt{\Delta }Z)>0$. For $0\le \lambda \le 1$, convexity givesTaking the positive discounted expectation proves that $V(\cdot ,t)$ is convex. Its derivative, where defined, is therefore nondecreasing, proving that the stock holding in the delta hedge rises weakly with the stock price. For twice differentiable $f$, differentiation gives $V_{xx}=e^{-\rho \Delta }\mathbb{E}[Y_{+}^2{f}^{\prime \prime }(x{Y}_{+})]\ge 0$. For a merely convex payoff, Gaussian smoothing at $t<T$ and the usual growth assumptions give the same conclusion; one-sided derivatives are monotone even without smoothing. Strict increase is not implied by convexity alone: an affine payoff has constant option delta.

For the random constant Gaussian interest-rate mixture, represent the rate as $\rho ={\rho }_0+\tau Y$, with $Y,Z$ independent standard normal variables. This independence simply represents averaging the constant-rate price function over its Gaussian parameter. Completing the square in $Y$ yields the identityApply it inside the time-zero price formula:The combined normal term has variance ${\tau }^2{T}^2+{\sigma }^{2}T$. Define ${\rho }_{\ast }={\rho }_0-{\tau }^{2}T/2$ and ${\sigma }_{\ast }^2={\sigma }^2+{\tau }^{2}T$. Then the discount factor is $e^{-{\rho }_{\ast }T}$, and the log-price mean is $({\rho }_{\ast }-{\sigma }_{\ast }^2/2)T$. Thus the expectation is exactly the constant-parameter Black-Scholes model price with those parameters:The argument applies to nonnegative payoffs by Tonelli theorem, and to signed payoffs whenever absolute integrability justifies the expectations. It averages constant-rate prices; it does not assume a stochastic short-rate process.

Use the Black-Scholes equation with the claim's terminal condition $V(T,s)=g(s)$:For $t<T$, the Itô formula under the physical measure givesChoose the delta hedge and its bank-account holding byTheir portfolio value is $V(t,S_t)$, andThus the portfolio is self-financing and, with initial capital $V(0,S_0)$, has terminal wealth $g(S_T)$.

Localizing to compact stock-price intervals and times below $T$ justifies these stochastic integrals from the classical smoothness of $V$. The extension to maturity also follows from boundedness. Under the Risk-neutral measure for the Black-Scholes model, discounted $V(t,S_t)$ is a bounded local martingale, hence a true square-integrable martingale. Its quadratic variation has finite expectation, so the stock-integral coefficient is square integrable through $T$. Equivalence of measures preserves its almost-sure finiteness, and the physical drift integral is finite by Cauchy-Schwarz. The terminal limit is $g(S_T)$.

Finally, $B_t=B_0{e}^{rt}$ is bounded away from zero on this finite horizon, and bounded $V$ makes $V(t,S_t)/B_t$ bounded below by a deterministic constant. The hedge is therefore an admissible trading strategy. The displayed holdings replicate the payoff admissibly, even if the bounded payoff can be negative.

Set $\theta =(\mu -r)/\sigma$. The positive densityhas mean one, and the Girsanov theorem makes $W_t^Q=W_t+\theta t$ a Brownian motion under the equivalent measure $Q$. HenceThe Black-Scholes equation makes $e^{-rt}V(t,S_t)$ a local martingale. Bounded $V$ on the finite horizon makes it a true martingale, so its terminal condition gives the risk-neutral valuationThe geometric Brownian motion solution under $Q$ isThe Brownian increment is independent of ${\mathcal{F}}_t$ and equals $\sqrt{T-t}Z$ in distribution, where $Z$ has the standard normal distribution. ThereforeThe expectation on the right is only an integral against the standard normal law; the physical drift $\mu$ has disappeared.

Define the risk-neutral claim valueThen $V(0,S_0)=x$, $V(T,s)=g(s)$, and the Black-Scholes equation holds. Differentiation under the integral gives the deltaA Gaussian shift $Z\mapsto Z+\sigma \sqrt{\tau }$ rewrites this aswhich is exactly the stated ${\theta }_t$ at $s=S_t$.

Apply Itô formula to $V(t,S_t)$. The PDE givesThis is the same wealth equation as part a, with the same initial value $x$. Uniqueness therefore gives $X_t^{x,\theta }=V(t,S_t)$ and hence

In the Black-Scholes model,Conditioning on ${\mathcal{F}}_t$ and using the moment-generating function of the independent Gaussian increment $W_T-W_t$ givesThe function $c$ satisfies the zero-rate Black-Scholes equation, so Itô formula leaves only its stochastic term:Consequently the required delta hedge is

For running maximum $M_t={\max }_{u\le t}{S}_u$, a smooth price $v(t,S_t,M_t)$ has an Itô formula term $v_{m}d{M}_t$. The maximum increases only when $S_t=M_t$. A self-financing portfolio in stock and bank account cannot supply an extra singular finite-variation term there, so the smooth price must satisfy the displayed boundary condition. Inside $0<s<m$, it obeys the Black-Scholes equation in the price variable. The condition need only hold before maturity, where terminal corners may destroy differentiability.