= Solution
\b[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 wealth
$$
dX_t=[\rho X_t+(\alpha-\rho)\pi_t]dt+\sigma\pi_t\,dW_t.
$$
Choose 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 $\lambda=(\alpha-\rho)/\sigma$ and
$$
H_T=e^{-\rho T}\exp(-\lambda W_T-\lambda^2T/2).
$$
This <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> $\mathbb E[H_TY]\le x_0$.
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 $I=(U')^{-1}$ be the <inverse marginal utility>, and choose $y>0$ from $\mathbb E[H_T I(yH_T)]=x_0$. Then the <complete-market terminal utility optimizer> is
$$
\boxed{Y^*=I(yH_T).}
$$
For any feasible $Y$, concavity gives the pointwise tangent inequality
$$
U(Y)\le U(Y^*)+U'(Y^*)(Y-Y^*)
=U(Y^*)+yH_T(Y-Y^*).
$$
Taking <expectations> and using the budget proves optimality; strict concavity gives uniqueness up to null events. The optimal wealth process is
$$
X_t^*=e^{-\rho(T-t)}\mathbb E_Q[Y^*\mid\mathcal F_t]
=H_t^{-1}\mathbb E_P[H_TY^*\mid\mathcal F_t].
$$
If its discounted <martingale> has representation $d(e^{-\rho t}X_t^*)=\zeta_t dW_t^Q$, 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> give
$$
v_t+\rho xv_x+\sup_\pi\left[(\alpha-\rho)\pi v_x+\frac12\sigma^2\pi^2v_{xx}\right]=0,
\qquad v(T,x)=U(x).
$$
For $v_{xx}<0$, maximizing the concave quadratic gives $\pi^*=-(\alpha-\rho)v_x/(\sigma^2v_{xx})$. 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> $U(x)=x^{1-\gamma}/(1-\gamma)$, $\gamma>0$, $\gamma\ne1$, substitution gives
$$
v(t,x)=\frac{x^{1-\gamma}}{1-\gamma}
\exp\!\left((1-\gamma)\left[\rho+\frac{\lambda^2}{2\gamma}\right](T-t)\right),
\qquad\boxed{\frac{\pi_t^*}{X_t^*}=\frac{\alpha-\rho}{\gamma\sigma^2}.}
$$
The 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.
\b[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 is
$$
V_t=e^{-\rho(T-t)}\mathbb E_Q[C\mid\mathcal F_t].
$$
For square-integrable discounted $C$, the Brownian <Martingale representation theorem> gives $d(e^{-\rho t}V_t)=\varphi_t dW_t^Q$. Since $d(e^{-\rho t}S_t)=\sigma e^{-\rho t}S_t dW_t^Q$, the replicating holding is $\delta_t=\varphi_t/(\sigma e^{-\rho t}S_t)$, with bank units $(V_t-\delta_tS_t)/B_t$. 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 $A_t=\int_0^tS_u du$ and seek $V_t=v(t,S_t,A_t)$. Because $dA_t=S_tdt$, <Itô formula> gives
$$
v_t+\rho s v_s+\frac12\sigma^2s^2v_{ss}+s v_a-\rho v=0,
\qquad v(T,s,a)=(a/T-K)^+.
$$
The <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 <variance>
$$
m_t=\frac{I_t+(T-t)\log S_t+\frac12(\rho-\sigma^2/2)(T-t)^2}{T},
\qquad q_t=\frac{\sigma^2(T-t)^3}{3T^2}.
$$
Indeed its random part is $\sigma T^{-1}\int_t^T(T-u)dW_u^Q$, by stochastic Fubini; the <Itô isometry> gives $q_t$. Completing the square in a normal exponential integral proves the <conditional geometric-average Asian option formula>
$$
V_t=e^{-\rho(T-t)}\left[e^{m_t+q_t/2}\Phi(d_1)-K\Phi(d_2)\right],
\quad d_2=\frac{m_t-\log K}{\sqrt{q_t}},\quad d_1=d_2+\sqrt{q_t},
$$
for $K>0,t<T$. This explicitly includes the known past log-average. Holding that past integral fixed when differentiating the price gives the <stock> holding
$$
\delta_t=\frac{T-t}{TS_t}e^{-\rho(T-t)}e^{m_t+q_t/2}\Phi(d_1).
$$
The normal-density derivative terms cancel because $e^{m_t+q_t/2}\varphi(d_1)=K\varphi(d_2)$.
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 $v_m dM_t$; since $dM_t$ 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:
$$
v(t,s,m)=e^{-\rho(T-t)}\left[m+\int_m^\infty P_Q\left(\max_{t\le u\le T}S_u\ge z\,\middle|\,S_t=s\right)dz\right]-s.
$$
This uses $\mathbb E\max(m,Z)=m+\int_m^\infty P(Z\ge z)dz$ and $\mathbb E_QS_T=se^{\rho(T-t)}$. 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.
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