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Expected utility of terminal wealth. Let initial wealth be $x_0>0$, maturity $T=t_0$, and stock-dollar investment ${\pi }_t$. A self-financing strategy in the Black-Scholes model has wealthChoose admissible strategies with nonnegative wealth, and an increasing strictly concave differentiable utility function $U$ satisfying the Inada conditions. Assume the required expectations are finite and the optimization is well posed; the resulting budget equation below must have a solution. Write $\lambda =(\alpha -\rho )/\sigma$ andThis state-price density is the discounted density of the unique equivalent martingale measure. Nonnegative discounted wealth is a supermartingale under that measure, so every admissible terminal wealth $Y$ obeys the state-price budget constraint $\mathbb{E}[H_{T}Y]\le x_0$.

Conversely a nonnegative terminal claim with finite such cost is financed by its conditional martingale price, which remains nonnegative; Brownian Martingale representation theorem constructs its holdings. Thus market completeness turns dynamic optimization into a terminal-payoff problem. Let $I=(U^{\prime }{)}^{-1}$ be the inverse marginal utility, and choose $y>0$ from $\mathbb{E}[H_{T}I(y{H}_T)]=x_0$. Then the complete-market terminal utility optimizer isFor any feasible $Y$, concavity gives the pointwise tangent inequalityTaking expectations and using the budget proves optimality; strict concavity gives uniqueness up to null events. The optimal wealth process isIf its discounted martingale has representation $d(e^{-\rho t}{X}_t^{\ast })={\zeta }_{t}d{W}_t^Q$, hold ${\delta }_t=e^{\rho t}{\zeta }_t/(\sigma S_t)$ stock units and invest the remainder in the bank account. This derives the strategy, not only the terminal first-order condition.

The equivalent dynamic approach is the Bellman equation for terminal-wealth utility. For a smooth concave value $v(t,x)$, dynamic programming and Itô formula giveFor $v_{xx}<0$, maximizing the concave quadratic gives ${\pi }^{\ast }=-(\alpha -\rho )v_x/({\sigma }^2{v}_{xx})$. For an arbitrary admissible control the Itô drift of $v(t,X_t)$ is nonpositive; localization and integrability therefore bound its expected terminal utility by $v(0,x_0)$. The maximizing control makes the drift zero, giving equality when the verification expectations are valid.

For CRRA utility $U(x)=x^{1-\gamma }/(1-\gamma )$, $\gamma >0$, $\gamma \ne 1$, substitution givesThe optimal stock fraction is constant. Its geometric wealth dynamics stay positive, so it is feasible. For logarithmic utility the fraction is $(\alpha -\rho )/{\sigma }^2$ and $v(t,x)=\log x+(\rho +{\lambda }^2/2)(T-t)$. Risk aversion controls the risky exposure, while the market price of risk controls its reward.

Pricing claims depending on the path. For a path-dependent contingent claim $C=F((S_u{)}_{0\le u\le T})$, the risk-neutral pricing process isFor square-integrable discounted $C$, the Brownian Martingale representation theorem gives $d(e^{-\rho t}{V}_t)={\phi }_{t}d{W}_t^Q$. Since $d(e^{-\rho t}{S}_t)=\sigma e^{-\rho t}{S}_{t}d{W}_t^Q$, the replicating holding is ${\delta }_t={\phi }_t/(\sigma e^{-\rho t}{S}_t)$, with bank units $(V_t-{\delta }_t{S}_t)/B_t$. Thus path dependence does not destroy market completeness; it changes the information needed to determine the price and hedge.

For an arithmetic Asian option, introduce $A_t={\int }_0^t{S}_{u}du$ and seek $V_t=v(t,S_t,A_t)$. Because $d{A}_t=S_{t}dt$, Itô formula givesThe stock holding is $v_s$. The state $a$ records the already observed average; current stock price alone cannot recover it.

A Geometric Asian option admits an explicit conditional calculation. Let $I_t={\int }_0^t\log S_{u}du$ and $G_T=e^{I_T/T}$. Conditional on time $t$, the normal law of $\log G_T$ has mean and varianceIndeed its random part is $\sigma T^{-1}{\int }_t^T(T-u)d{W}_u^Q$, by stochastic Fubini; the Itô isometry gives $q_t$. Completing the square in a normal exponential integral proves the conditional geometric-average Asian option formulafor $K>0,t<T$. This explicitly includes the known past log-average. Holding that past integral fixed when differentiating the price gives the stock holdingThe normal-density derivative terms cancel because $e^{m_t+q_t/2}\phi (d_1)=K\phi (d_2)$.

For a lookback option use $M_t={\max }_{u\le t}{S}_u$. A floating-strike put pays $M_T-S_T$. Its price $v(t,s,m)$ obeys the ordinary Black-Scholes equation in $0<s<m$, with terminal value $m-s$. The extra Itô term is $v_{m}d{M}_t$; since $d{M}_t$ is carried by $S_t=M_t$, self-financing requires the running-maximum boundary for a lookback option $v_m(t,m,m)=0$ before maturity. Equivalently its price follows from the upper-hitting probabilities in Question 3:This uses $\mathbb{E}\max (m,Z)=m+{\int }_m^{\infty }P(Z\ge z)dz$ and ${\mathbb{E}}_Q{S}_T=s{e}^{\rho (T-t)}$. For a barrier option, the state must additionally record whether the barrier has already been hit; a no-rebate knockout price has an absorbing zero boundary. These augmentations give tractable PDEs or conditional-expectation computations while retaining the same martingale pricing and claim replication principle.

Use ${\theta }_t$ for the dollar amount invested in the stock; the number of shares is ${\theta }_t/S_t$. The bank account contains $w_t-{\theta }_t$ dollars. With no intermediate consumption, the self-financing portfolio equation isIf instead $\theta$ denotes share holdings, replace each dollar ${\theta }_t$ here by $S_t{\theta }_t$. The equation says that trading only reallocates existing portfolio wealth; gains come from cash interest and stock returns.

For the standard unconstrained complete-diffusion formulation, assume the completed natural Brownian filtration, initial wealth $w_0>0$, and nonnegative admissible wealth. Define the market price of risk ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. The boundedness assumptions make $\kappa$ bounded. The state-price density normalized at zero isIts money-market-adjusted process $Z_t=e^{{\int }_0^t{r}_{s}ds}{\zeta }_t$ is a true martingale by the Novikov condition. Under $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem gives $W_t^{\mathbb{Q}}=W_t+{\int }_0^t{\kappa }_{s}ds$ as a Brownian motion, and the discounted stock has zero drift.

The Itô product rule applied to the wealth equation givesFor nonnegative admissible wealth this is a nonnegative local martingale, hence a supermartingale. Therefore every affordable terminal payoff $X=w_T$ satisfies the state-price budget constraint $\mathbb{E}[{\zeta }_{T}X]\le w_0$. Nonzero volatility and the Brownian martingale representation theorem allow replication of nonnegative claims with finite state-price cost.

For a differentiable strictly concave utility function with the usual conditions ensuring an interior integrable optimizer, put $I=(U^{\prime }{)}^{-1}$, the inverse marginal utility. Pointwise maximization of $U(x)-y{\zeta }_{T}x$ givesThe second equation determines the positive constant $y$. To prove optimality, use the concave supporting-tangent inequality:Taking expectations and using the budget bound proves that the candidate dominates every admissible affordable payoff, whenever the utility expectations are well defined. This is the complete-market terminal utility optimizer.

The actual wealth and dollar portfolio are as explicit as general adapted coefficients permit. LetThe Brownian martingale representation theorem, with localization if only $L^1$ integrability is initially known, gives $h$. The dollar portfolio from a deflated wealth martingale isThe bank holds $w_t^{\ast }-{\theta }_t^{\ast }$ dollars. Equivalently, with $B_t=e^{{\int }_0^t{r}_{s}ds}$, $w_t^{\ast }=B_t{\mathbb{E}}^{\mathbb{Q}}[w_T^{\ast }/B_T\mid {\mathcal{F}}_t]$. These formulas need no unjustified differentiability of a value function with random coefficients.

The printed assumption that $U$ is only increasing and concave does not guarantee that the interior formula exists, or even that the optimal value is finite. In general replace the inverse formula by an affordable selectionwhere $\mathcal{D}$ is the wealth domain; boundary choices or supporting slopes handle nondifferentiability. Existence, finite cost and replication still have to hold. A concrete obstruction is the unbounded linear terminal-wealth utility: take $U(x)=x$, $r=0$, $\mu =\sigma =1$, and the Brownian filtration. Then ${\zeta }_T=e^{-W_T-T/2}$. For $A_n=\{{\zeta }_T<1/n\}$, the bounded nonnegative claimis replicable, costs $w_0$, and has $\mathbb{E}{X}_n>n{w}_0$. Thus there is no finite optimal expected utility under the literal utility hypotheses alone. Also, bounded coefficients do not ensure market completeness in an arbitrarily enlarged filtration containing untraded randomness; the full replication formula uses the Brownian market information assumption stated above.

In a complete market, terminal-wealth optimization can first be solved over terminal payoffs and then implemented by replication. Let initial wealth be $x_0>0$, and take an increasing differentiable strictly concave utility function $U$ on $(0,\infty )$ satisfying the Inada conditions. The state-price budget constraint for a terminal wealth $X$ is ${\mathbb{E}}_P[{\zeta }_{n}X]\le x_0$. Monotonicity spends the whole budget at the optimum.

Let $I=(U^{\prime }{)}^{-1}$ be the inverse marginal utility. Choose $y>0$ to satisfy ${\mathbb{E}}_P[{\zeta }_{n}I(y{\zeta }_n)]=x_0$. In a finite tree with positive state probabilities, this left side is continuous and strictly decreasing from infinity to zero, so $y$ exists uniquely. The complete-market terminal utility optimizer isTo prove optimality, concavity gives $U(X)\le U(X^{\ast })+U^{\prime }(X^{\ast })(X-X^{\ast })=U(X^{\ast })+y{\zeta }_n(X-X^{\ast })$. Taking expectations and using the budget proves $\mathbb{E}U(X)\le \mathbb{E}U(X^{\ast })$. Strict concavity gives uniqueness of the optimal terminal wealth. The finite-market setting avoids additional infinite-state existence and integrability issues.

Replicate $X^{\ast }$ using the backward recursion of part (i). Its wealth process is $V_k=B_k{\mathbb{E}}_Q[X^{\ast }/B_n\mid {\mathcal{F}}_k]$, and the two-successor difference quotient gives its stock holdings. This self-financing strategy starts with $x_0$ by the budget equality and remains positive at every node, since it is the discounted conditional expectation of a strictly positive terminal wealth.

For logarithmic utility, $U(x)=\log x$, $I(z)=1/z$, the multiplier is $y=1/x_0$ and $X^{\ast }=x_0/{\zeta }_n=x_0{B}_n/L_n$. For CRRA utility with risk aversion $\gamma >0$, $\gamma \ne 1$, the normalized payoff isConstraints such as prohibiting short sales change the attainable terminal-payoff set and require a constrained optimization; the unconstrained complete-market formula uses the stated assumptions.