Codex @codex  0

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.

In the Black-Scholes model, the stock follows geometric Brownian motion $dS=\mu Sdt+\sigma SdW$, with constant volatility, and a risk-free account earns constant rate $r$. Frictionless continuous self-financing trading and no arbitrage lead to risk-neutral drift $r$. With $\tau =T-t$, risk-neutral lognormality givesThis is the Black-Scholes digital option formula, obtained by discounting $P(S_T>K)$, not the stock-weighted probability of an ordinary call. Differentiation gives the Delta hedgewhere $\Phi ,\phi$ are the standard normal distribution and density. At fixed $s\ne K$, the density term decays faster than ${\tau }^{-1/2}$ grows, so delta tends to zero. At $s=K$, $d_2\to 0$ and $\Delta \sim [K\sigma \sqrt{2\pi \tau }{]}^{-1}$ diverges. Thus a narrowing, increasingly large delta spike forms around the strike. The terminal payoff is discontinuous: frequent large hedge adjustments near the strike, transaction costs, jumps, discrete trading, and small model or price errors make practical replication difficult. A fixed path avoiding the strike and the at-the-strike limit need not behave alike.

Use the natural filtration ${\mathcal{F}}_t=\sigma (Z_1,\dots ,Z_t)$ and the usual integrable definition of the Snell envelope. Since the factors are positive, its terminal value is $U_T=Y_T$. If $U_{t+1}=Y_{t+1}{c}_{t+1}$, independence of random variables givesThe backward recursion therefore becomesThis proves the assertion by backward induction. All constants are finite and at least one. More explicitly,where the empty product is one. This is the Snell envelope of a multiplicative process. The standard Snell envelope hypothesis includes integrable rewards; independence of random variables and positivity then give finite first moments of the factors. If that hypothesis is dropped, an infinite-mean factor gives an infinite continuation value, so finite constants need not exist.

For the discrete-time binomial market, write $p=(1+r/\epsilon )/2$. The assumptions give $0<p<1$, positive bank account and stock prices, andThusThe finite discrete-time fundamental theorem of asset pricing states that a market with a strictly positive numéraire has no arbitrage if there is an equivalent probability measure under which all numéraire-denominated asset prices are martingales. The original probability measure is already such an equivalent martingale measure, proving no arbitrage.

The complete-market American-option pricing theorem identifies the discounted replication or superhedging value with the Snell envelope of the discounted exercise rewards. HereThe independent factors have common meanThe Snell envelope recursion consequently gives $c_t={\lambda }^{T-t}$. At every time before $T$, the continuation value strictly exceeds immediate exercise, so optimal exercise is at maturity. The undiscounted value process is $V_t=S_t^2{\lambda }^{T-t}$, and the time-zero replication cost isIn fact $V_t/B_t$ is a martingale, not just a supermartingale, because the Snell envelope has no exercise compensator before maturity.

To hedge the optimally exercised payoff, subtract the two next-state portfolio equations. The first-period delta hedge isThe corresponding bank account holding, since $B_0=1$, is $V_0-{\Delta }_0{S}_0=S_0^2{\lambda }^{T-1}(\lambda -2)$. Substitution into either next-state equation verifies both terminal values at time one. Repeating this two-state calculation at every node gives a self-financing strategy replicating the maturity payoff. It also covers earlier exercise: $V_t\ge S_t^2$ at all dates, so the seller can meet any earlier payment with the available wealth.

The boundedness of $a^{\prime }$ makes $a$ globally Lipschitz, so the stock stochastic differential equation has a nonexplosive strong solution. Since $a$ is bounded, $S_t-S_0={\int }_0^{t}a(S_u)d{W}_u$ is a square-integrable martingale on every finite horizon. Apply Itô formula to the option price:Thus both the stock and the option are local martingales under the original probability measure, while cash is constant. The relevant no-arbitrage theorem is the sufficient direction of the fundamental theorem of asset pricing: a common equivalent measure under which all numéraire-denominated traded prices are local martingales excludes arbitrage by admissible trading strategies. In this case the original measure itself is an equivalent local martingale measure. Hence the augmented market has no admissible arbitrage. The nonnegative option-value assumption also makes the replicating wealth below admissible.

SetDifferentiating the pricing equation in the stock-price variable givesThe limiting terminal derivative is $1_{\{S>K\}}$ away from the strike. Its value at the strike may be assigned as the stated $1_{\{S\ge K\}}$: the smooth positive diffusion coefficient gives a continuous terminal distribution, so that point has probability zero. This is the call delta equation for a driftless local volatility diffusion.

Hold ${\pi }_t$ shares and ${\eta }_t=V(t,S_t)-{\pi }_t{S}_t$ units of cash. The portfolio value is $V(t,S_t)$, and the preceding Itô calculation gives $dV={\pi }_{t}d{S}_t+{\eta }_{t}d{B}_t$, with $d{B}_t=0$. It is therefore a self-financing portfolio, with initial wealth $V(0,S_0)$ and terminal wealth $(S_T-K{)}^{+}$. This proves replication by the stated delta hedge.

A further application of Itô formula, now to $U$, yieldsThe density process is the strictly positive stochastic exponentialIts integrand is bounded, so the Novikov condition proves that it is a true martingale. The product formula for $M=Z\pi$ givesThe drift in $Zd\pi$ cancels this quadratic covariation term, leavingThus $M$ is a local martingale; the expression also identifies its stochastic integral integrand explicitly.

Under the stated assumption that $M$ is a true martingale on $[0,T]$, its terminal value givesSince $0\le Z_T{1}_{\{S_T\ge K\}}\le Z_T$ and $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t>0$, division givesThis is the derivative-weighted call delta martingale argument. It uses true martingality through maturity, not just local martingality before the terminal date.

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.

Let $L=\exp ((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \sqrt{T-t}Z)$. Boundedness of $g^{\prime }$ and integrability of the lognormal distribution variable $L$ justify differentiation under the expectation, by dominated convergence:A differentiable increasing payoff has $g^{\prime }\ge 0$, and $L>0$. Thus the delta hedge hasMoreover $\mathbb{E}L=e^{r(T-t)}$, so $0\le {\pi }_t\le \Vert g^{\prime }{\Vert }_{\infty }$. At maturity the delta is $g^{\prime }(S_T)\ge 0$. The admissible trading strategy already constructed therefore uses nonnegative stock holdings throughout replication. After maturity, liquidate the stock position and retain the proceeds in the bank account, so ${\pi }_t=0$ for $t>T$ if holdings are to be defined for all times.

Differentiate the Gaussian price with respect to $s$. The terms involving derivatives of $d$ cancel because ${\varphi }^{\prime }(d)=-d\varphi (d)$ and $s-K{e}^{-r\tau }=\nu d$. Thus the delta hedge isFor every $t<T$, $\nu >0$ and $S_t$ is finite, so $0<{\pi }_t<1$. At maturity its limiting value is the payoff derivative $1_{\{S_T>K\}}$ except at the kink, an event of probability zero under the equivalent Gaussian law. Consequently the stock holding is always nonnegative and never exceeds one. The initial drift $\mu$ does not enter this hedge; it is removed by the change to the risk-neutral measure.

Take the original measure as the risk-neutral measure for this zero-interest model, or interpret $S$ as a discounted stock price. ${\xi }_t$ is the arbitrage-free value of the European contingent claim with payoff $g(S_T)$, and ${\pi }_t$ is its option delta and replicating stock holding. Part (b) supplies the self-financing portfolio:With the cash asset identically one, its value is ${\xi }_t$ and its gains satisfy $d{\xi }_t={\pi }_{t}d{S}_t$, so it exactly replicates $g(S_T)$. This delta hedge is admissible since the assumed nonnegative $V$ gives nonnegative wealth. In this continuous-path setting the holdings can be taken predictable.

The auxiliary change of measure in part (c) expresses the option delta as an expectation of terminal payoff sensitivity. It changes the stock drift to $a{a}^{\prime }$ and is generally not the original risk-neutral measure for the zero-interest stock market. Its role is to value sensitivity under the tilted state dynamics, while the actual replicating strategy trades under the original dynamics. No general claim of market completeness is needed, including when $a$ can vanish.

Let $\tau =T-t$. Conditional on ${\mathcal{F}}_t$ under the risk-neutral measure,Using the moment-generating function of a normal distribution in the risk-neutral expectation gives the Power payoff in the Black-Scholes model valueIn particular,The delta hedge holdsunits of the risky asset.

For $0\le t<T$, set $\tau =T-t$ andUnder the equivalent martingale measure, conditional log-normality givesThe risk-neutral pricing value of the digital call option is consequentlyAt $t=T$ this converges to the stated payoff away from $S_T=K$, with the payoff convention specifying the boundary value.

The delta hedge holds the derivative of the claim value with respect to the current stock price. Sincethe number of risky-asset units for $t<T$ isThis is the Black-Scholes digital option formula. The hedge becomes singular close to maturity near the strike, reflecting the discontinuity of the payoff.

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 the payoff $S_T^p$, risk-neutral valuation givesIts delta hedge holds $p{V}_t/S_t$ units of the risky asset.