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The marked-to-market value of the current holdings is $X_t=H_t\cdot P_t$. In a self-financing portfolio, purchases are paid for by sales within the portfolio, so rebalancing itself changes no net wealth. Asset price changes contribute $H_t\cdot d{P}_t$, and nonnegative consumption removes $c_{t}dt$ from wealth. HenceThe holdings must be predictable and stochastically integrable. The second identity is the self-financing definition with withdrawals; it is not obtained by incorrectly setting the changes in the holdings themselves to zero.

Construct the strictly positive rolling one-period bond account using successive one-period zero-coupon bonds:At time $t-1$, invest the entire account value in the bond maturing at $t$. This gives a positive self-financing portfolio usable as a numéraire.

We use the finite-discrete-time fundamental theorem of asset pricing: in a frictionless market with finitely many adapted assets and trading dates, no arbitrage is equivalent to existence of an equivalent martingale measure for asset prices, including dividends, expressed in a positive traded numéraire. Maturing bond payoffs are reinvested in that numéraire. Let $\mathbb{Q}$ be such a measure and $L_t=\mathbb{E}[d\mathbb{Q}/d\mathbb{P}\mid {\mathcal{F}}_t]$ its positive density process. The martingale pricing relation for a unit zero-coupon bond isDefine $Z_t=L_t/{\beta }_t$. The Bayes formula for conditional expectation then givesThe positive expectations are finite because these are the traded finite bond prices; in particular $\mathbb{E}{Z}_T=Z_0{P}_0(T)$ when initial information is trivial. Normalize $Z_0=1$ without changing any ratio. With nontrivial initial information the same identities are conditional on that information. The state-price density need not be unique when the bond market is incomplete; existence is sufficient.

Let $h_t$ and $k_t$ be predictable holdings in the stock and account, with the stochastic integrability needed for a self-financing portfolio. Then $V=hS+kB$ and $dV=hdS+kdB$. The Itô product rule and self-financing identity giveSince $V=hS+kB$ and $\sigma \lambda =\mu -r$, the finite-variation term is zero. Thereforeand the deflated wealth is a local martingale.

For zero-capital nonnegative wealth under a local deflator, a nonnegative local martingale is a supermartingale, by localization and the Conditional Fatou lemma. Under the required nonnegative-wealth condition, $ZV\ge 0$ and starts at zero, so $0\le \mathbb{E}(Z_t{V}_t)\le 0$. Thus $Z_t{V}_t=0$ almost surely at every fixed $t$. Strict positivity of $Z$ gives $V_t=0$ almost surely. Taking a countable intersection over rational times and then using continuous wealth paths strengthens this toThis argument only needs the local deflator, so it remains valid without promoting $ZB$ to a true martingale.

Choose the market price of risk ${\lambda }_t=(\mu -r{S}_t)/\sigma$. The Girsanov theorem, with the hypotheses allowed in the question, gives an equivalent martingale measure $\mathbb{Q}$ under whichThus the stock under the risk-neutral measure is a linear Gaussian diffusion. In particular, for $\tau =T-t$ its conditional mean and standard deviation arePut $\varphi (x)=(2\pi {)}^{-1/2}{e}^{-x^2/2}$ and let $\Phi$ be the standard normal distribution function. For $N\sim N(0,1)$,since ${\int }_{-d}^{\infty }x\varphi (x)dx=\varphi (d)$. Discounting this expectation gives the call price in an arithmetic stock model with interest. A particularly convenient expression isAt maturity define $C(T,s)=(s-K{)}^{+}$. This value is nonnegative because it is a discounted expectation of a nonnegative payoff.

For $t<T$, hold ${\pi }_t=C_s(t,S_t)$ shares and hold ${\beta }_t=[C(t,S_t)-{\pi }_t{S}_t]/B_t$ units of the continuous-time bank account. The pricing function solvesThe Itô formula under the physical measure therefore givesThis proves self-financing and terminal replication, with wealth always $C(t,S_t)\ge 0$. The coefficients are locally smooth before maturity, and the strategy extends to maturity through its continuous wealth limit and the square-integrable discounted payoff representation.

To see minimality, any other nonnegative self-financing portfolio replicating the payoff has discounted wealth a nonnegative local martingale under $\mathbb{Q}$, hence a supermartingale. Its initial capital $x$ must satisfy $x\ge {\mathbb{E}}^{\mathbb{Q}}[e^{-rT}(S_T-K{)}^{+}]=C(0,S_0)$. The strategy constructed above attains equality. ThusThe additive physical diffusion may take negative stock values; the formula and nonnegative replicating wealth remain valid. Replacing it by a multiplicative Black–Scholes diffusion would give the wrong price and hedge.

A complete market permits claim replication for every finite maturity and every bounded claim measurable at that maturity: there are an initial capital and a predictable self-financing portfolio whose terminal wealth equals the claim almost surely. In a finite-state market this is equivalent to replicating every terminal payoff, since all such payoffs are bounded. Trading is allowed dynamically in the existing $n$ assets; adding a new security is not part of the definition.

The pathwise identity $(S_t-S_{t-1}{)}^2=S_t^2-S_{t-1}^2-2S_{t-1}(S_t-S_{t-1})$ telescopes toThis is the discrete realized-variance replication identity. Replicate $S_T^2$ using part (b). Add a self-financing portfolio with initial wealth $-S_0^2$ and stock holdings $-2S_{t-1}$ during $(t-1,t]$.

To give the cash positions explicitly, letThe added portfolio holds $G_{t-1}+2S_{t-1}^2$ units of the bond over that interval. Its starting value is $G_{t-1}$ and its change is exactly $-2S_{t-1}(S_t-S_{t-1})$. The combined holdings are thereforeThe terminal value equals the squared-increment sum by the telescoping identity. Its initial cost isThe dynamic stock hedge uses only prices known before each interval, while the calls remain static.

Normalize $B_0=1$ and put ${\tilde{S}}_t=e^{-rt}{S}_t$. Its dynamics are $d{\tilde{S}}_t={\tilde{S}}_t\sigma (t,S_t)d{W}_t$. Bounded volatility makes this stochastic exponential a true martingale on each finite horizon, by the Novikov condition. It also gives a finite second moment: stopping the Itô formula for $S^2$ and applying the Gronwall inequality yields $\mathbb{E}{S}_T^2\le S_0^2{e}^{(2r+L^2)T}$ when $\sigma \le L$.

The discounted payoff $\xi =e^{-rT}(S_T-K{)}^{+}$ is thus square-integrable. Define the nonnegative martingaleThe Brownian martingale representation theorem says that every square-integrable martingale in the Brownian filtration has a representation $M_t=M_0+{\int }_0^t{h}_{u}d{W}_u$ with predictable $h$ and $\mathbb{E}{\int }_0^T{h}_u^{2}du<\infty$. Since $\tilde{S}>0$ and $\sigma >0$, chooseThen discounted gains satisfy $d{M}_t={\pi }_{t}d{\tilde{S}}_t$. Consequently $X_t=B_t{M}_t={\varphi }_t{B}_t+{\pi }_t{S}_t$ is self-financing, nonnegative, and hence an admissible trading strategy. At maturity $X_T=(S_T-K{)}^{+}$, proving claim replication at cost $C(T,K)$.

For minimality, discounted wealth of any admissible self-financing portfolio is a local martingale bounded below, hence a supermartingale by localization and the conditional Fatou lemma. Thus any such replication with initial wealth $x$ obeysTogether with the constructed portfolio, this provesThis is Brownian representation replication in a local volatility market. The given drift $r$ means that the original probability measure already serves as the risk-neutral measure.

Assume $\sigma \ne 0$ and the usual Brownian filtration, so the single-risky-asset market is a complete market. With market price of risk $\kappa =(\mu -r)/\sigma$, the normalized state-price density isThe Itô formula shows that ${\zeta }_t{w}_t$ is a local martingale for a self-financing portfolio. For nonnegative admissible wealth it is a supermartingale, yielding the state-price budget constraintEvery integrable nonnegative terminal claim with equality is attainable in the complete market, by the Brownian martingale representation theorem. Its wealth process is $w_t={\zeta }_t^{-1}\mathbb{E}[{\zeta }_{T}X\mid {\mathcal{F}}_t]\ge 0$.

The Inada conditions $u^{\prime }(0+)=\infty$ and $u^{\prime }(\infty )=0$, together with strict concavity, make $I=(u^{\prime }{)}^{-1}$ a decreasing map from $(0,\infty )$ onto $(0,\infty )$. Pointwise optimization of $u(x)-y{\zeta }_{T}x$ gives the optimal terminal wealthwith scalar multiplier $y>0$. The question assumes a multiplier giving the required budget; utility expectations must also be well defined.

For any admissible terminal wealth $X$, the supporting-line inequality for a concave function givesTaking expectations and using the state-price budget constraint proves optimality:Strict concavity makes the optimal terminal wealth unique up to almost-sure equality.

Use dollar holdings ${\theta }_0,\dots ,{\theta }_n$, including the traded index, and let $m=\log M$. The Itô formula givesDefine the state-dependent correlation $q(m)=\rho (e^m)$, $d(m)=\sqrt{1-q(m{)}^2}>0$, and constants $p_i=({\mu }_i-r)/{\sigma }_i$. The self-financing portfolio with consumption has wealth equationSince all these assets are traded and $d(m)>0$, the volatility map is invertible. It is useful to optimize over Brownian portfolio exposureswhose market price of risk vector isIndeed ${\sum }_i({\mu }_i-r){\theta }_i=\ell \cdot y$. Put $L^2(m)=p_0^2+{\sum }_{i=1}^n{\ell }_i(m{)}^2$.

For $R\ne 1$, normalize constant relative risk aversion utility to $U(c)=c^{1-R}/(1-R)$. Scaling wealth, holdings, and consumption by $a>0$ leaves the index state unchanged and multiplies reward by $a^{1-R}$. ThusFor a finite smooth value with $V_{ww}<0$, the Hamilton-Jacobi-Bellman equation, including the shared-noise cross derivative, isThe last term is essential: index changes and the $W^0$ component of wealth are correlated. The first-order conditions yieldBecause $V_w=w^{-R}f$, $V_{ww}=-R{w}^{-R-1}f$, and $V_{wm}=w^{-R}{f}^{\prime }$, the optimized portfolio contribution isSubstitution gives the requested second-order nonlinear equationRecovering dollar holdings from the exposures gives the optimal controlsOnly the index holding carries the extra intertemporal hedging demand, because the index is the source of state variation. The stock positions hedge their common exposure through the index.

The power transformation of a complete-market investment equation gives a useful further simplification. Let $g=f^{1/R}$, and defineExpanding the squared term and substituting $f=g^R$ cancels the two $(g^{\prime }{)}^2$ terms. The power-transformed investment equation is linear:Then $c^{\ast }=w/g$ and $f^{\prime }/f=R{g}^{\prime }/g$. A practical finite difference method solves this linear differential equation on an expanding truncated interval in $m$, enforcing the economically relevant positive solution and checking domain and mesh convergence. If correlation approaches limits strictly inside $(-1,1)$ and the corresponding ${\delta }_{\pm }>0$, the constant-coefficient Merton consumption-investment problem gives endpoint approximations $g\to 1/{\delta }_{\pm }$. Without such asymptotics one must determine appropriate growth/transversality conditions; arbitrary fixed endpoint values are not justified. The statement's smooth decreasing correlation alone does not ensure globally bounded market prices of risk or a finite value. Any numerical candidate must also satisfy admissibility and the investment value transversality condition.

The printed $R>0$ includes $R=1$. In that case choose $U(c)=\log c$, whose scaling is additive:There is no shared-noise cross derivative because $V_{wm}=0$. The logarithmic case isThus the dollar holdings are the preceding formulas with $R=1$ and the $f^{\prime }/f$ hedge term omitted. This linear differential equation can be solved by the same truncation and convergence strategy.

A complete market can replicate every contingent claim in its specified payoff class: for every terminal ${\mathcal{F}}_T$-measurable payoff $X$, there is an admissible self-financing portfolio whose terminal wealth equals $X$ almost surely. Equivalently, in an arbitrage-free finite-state discrete-time model the equivalent martingale measure is unique. In notation,Market completeness is a replication property. The following price comparisons also use the usual absence of arbitrage, which the source leaves implicit: without it the initial cost of claim replication need not be unique. For example, in a deterministic one-period model with cash worth one at both dates and a stock worth one initially but zero finally, all terminal claims are replicable using cash. Adding any stock holding changes the initial cost without changing the terminal payoff. This is a complete model with arbitrage, so it has no well-defined unique replication price.

The positive-part function is convex, so pathwise Jensen inequality givesTo compare the Asian option price with European call option prices at different dates, carry each earlier payoff forward using the numéraire. Purchase $1/T$ of each replicating strategy for maturity $t$, and, when its payoff is received, reinvest it in $N$ until $T$. This is a self-financing portfolio, with terminal wealthThe first inequality uses $N_T\ge N_t$ and nonnegative payoffs. Since the Asian option is replicable in the complete market, absence of arbitrage makes its replication cost no larger than this superhedging cost. ThereforeEquivalently, divide the first pathwise bound by $N_T$, use $(S_t-K{)}^{+}/N_T\le (S_t-K{)}^{+}/N_t$, and take expectations under the numéraire equivalent martingale measure. Merely averaging earlier payoffs without reinvestment would miss the difference in payment dates.

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.