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A utility function on monetary amounts need not be increasing merely because preferences obey the expected utility hypothesis. The outcome labels do not themselves supply a preference order. For example, values $u(1)=2$, $u(2)=0$, $u(4)=3$ define consistent expected utility comparisons even though the amount $2$ is least preferred. Preferring more money is an additional assumption. It is distinct from risk aversion, which concerns the effect of spreading uncertainty for a specified increasing utility function.

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.

Risk aversion means that a sure mean is preferred to the corresponding risky portfolio wealth. An increasing concave utility function, with $u^{\prime }>0$ and $u^{\prime \prime }\le 0$, has this property by the Jensen inequality; strictly concave utility gives strict preference for nondegenerate risks. Differentiating the expected utility from part (a) givesAn interior optimum therefore satisfiesThis balances the expected loss of marginal utility in the good state against its expected gain in the bad state. The borrowing constraint also requires checking the endpoints: $x_{\ast }=0$ if $U^{\prime }(0)\le 0$, while $x_{\ast }=T$ would require $U^{\prime }(T)\ge 0$. At $x=T$ both state wealths equal $T(1+r)$, soHence an optimum never puts all wealth into deposits. If $U^{\prime }(0)>0$, continuity and the displayed endpoint sign give an interior root, and concavity makes every such root globally optimal. Under strict concavity it is unique.

Constant relative risk aversion coefficient means that $\alpha =-W{U}^{\prime \prime }(W)/U^{\prime }(W)$ does not depend on wealth, for $W>0$ and $U^{\prime }(W)>0$. The defining differential equation integrates asA second integration therefore givesThe additive constant and positive scale do not affect expected utility preferences. Since $U^{\prime \prime }(W)=-\alpha c{W}^{-\alpha -1}$, positive $\alpha$ gives risk aversion, zero gives risk neutrality, and negative $\alpha$ gives risk seeking. These are respectively concave, affine and convex utility functions. The usual CRRA utility class is the positive-$\alpha$ case.

The power-wealth investment with running utility objective is homogeneous of degree $\gamma$ in wealth: scaling both wealth and positions by $a>0$ scales the reward by $a^{\gamma }$. Accordingly set $J(w,t)=A(t)w^{\gamma }$. ThenFor completeness the remaining equation is $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, whereThis confirms $J_{ww}<0$. The constant optimal risky fraction also makes wealth a positive geometric Brownian motion when started positive.

Financially the relative risk aversion coefficient is $1-\gamma$. Increasing $\gamma$ reduces risk aversion, raises the risky fraction, and decreases the bond fraction: its derivative is $-(\mu -r)/[{\sigma }^2(1-\gamma {)}^2]<0$. If the risky fraction is below one, some wealth is lent; at one, none is invested in bonds; above one, $B^{\ast }<0$ represents borrowing to leverage the risky position. As $\gamma \uparrow 1$ the risky fraction becomes unbounded, consistent with the absence of a finite unconstrained optimum for linear utility and positive excess drift. As $\gamma \downarrow 0$ the limiting fraction is $(\mu -r)/{\sigma }^2$, the logarithmic-utility allocation obtained by the appropriately rescaled power-utility limit; the literal objective at $\gamma =0$ would be constant and would not select a unique strategy.

For positive wealth and increasing twice differentiable utility function, this dimensionless coefficient measures sensitivity to proportional wealth risk. If it is constant, integration gives $U^{\prime }(W)=c{W}^{-\alpha }$ with $c>0$, and hence $U(W)=c{W}^{1-\alpha }/(1-\alpha )+d$ for $\alpha \ne 1$ or $c\log W+d$ for $\alpha =1$. Positive, zero and negative coefficients correspond respectively to risk aversion, risk neutrality and risk seeking. The usual CRRA utility subclass takes $\alpha >0$.

A risk-neutral measure $Q$ is a probability measure equivalent to the physical probability measure $P$, meaning they agree on which events have zero probability. For assets without intermediate payments, prices expressed in the chosen numéraire must be martingales under $Q$. For a positive bank account $B$ and a stock without dividends, this requires $S/B$ to be a martingale; with dividends, use discounted dividend gains. In an unrestricted finite-state one-period financial market, existence of such a measure is equivalent to absence of arbitrage, by the fundamental theorem of asset pricing.

Physical probability $p=P(\text{up})$ describes the modeled frequency or belief about an up move. The risk-neutral probability $q=Q(\text{up})$ is constrained by traded prices. In the worked binomial market, $q=5/8$ even if $p=1/2$ or $p=4/5$. Calling $Q$ risk-neutral does not assume that investors have no risk aversion; it changes the weights used for risk-neutral pricing.