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Assume $f\in C^{1,2}$ on positive prices before maturity and the usual local trading integrability. If its value is held in stock units ${\delta }_t$ and bank units ${\beta }_t$, thenfor a self-financing portfolio. The Itô formula diffusion coefficient is $\sigma S_t{f}_x$, so $\sigma S_t>0$ forces ${\delta }_t=f_x(S_t,t)$ and ${\beta }_t{B}_t=f-S_t{f}_x$. Equating drifts yieldsThe physical drift cancels, giving the Black-Scholes equationConversely, if the smooth function satisfies this equation, set $\delta =f_x$ and $\beta =(f-x{f}_x)/B$. Substituting into Itô formula gives exactly $df=\delta dS+\beta dB$, so these holdings are self-financing. This checks the holdings as well as the value equation; a partial differential equation solution does not justify arbitrary prescribed holdings. Appropriate admissibility and growth conditions are imposed when using it as an economic price.

For the logarithmic stock payoff, let $\tau =T-t$ and condition on $S_t=x$. Under the risk-neutral measure,The normal exponential moment $\mathbb{E}{e}^{uZ}=e^{u^2/2}$ and its derivative $\mathbb{E}[Z{e}^{uZ}]=u{e}^{u^2/2}$ giveThereforeThe option delta is the displayed stock holding. The bank account value is $f-S_t{f}_x=-S_t$, so ${\beta }_t=-S_t/B_t$. Directly, $f_{xx}=1/x$ and $f_t=-x(\rho +{\sigma }^2/2)$ verify the partial differential equation, while $f(x,T)=x\log x$ verifies the payoff. Lognormal moments give the required pricing integrability. Since $x\log x\ge -1/e$, the conditional price is bounded below over this finite horizon, so the resulting claim replication has the usual admissibility property.

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.