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An arbitrage is an allowed self-financing portfolio with zero initial wealth, nonnegative terminal portfolio wealth almost surely under the physical probability measure $P$, and strictly positive terminal wealth with positive probability. In a finite model where every state has positive probability, nonnegativity means no loss in any state. Positive expected return alone does not establish arbitrage: losses may still occur.

For the binomial market example, suppose the European call option sells for 13 while its replicating portfolio costs $250/21$. Sell the call, buy its replicating portfolio, and lend the surplus $13-250/21=23/21$ in the bank account. The initial net cost is zero. At maturity the call obligation and portfolio cancel in either state, leaving $(21/20)(23/21)=23/20$. A price below $250/21$ permits the reversed construction, using short selling and borrowing. This is the law of one price made operational; the assumed trading permissions are essential.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

If $V$ satisfies the pricing equation for a local volatility model, hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.

At each node a positive stock price moves to $u{S}_t$ or $d{S}_t$, with $0<d<u$, while a bank account grows by a fixed gross factor $R>0$. Assume both successors have positive physical probability, no dividends, a frictionless market with no trading costs, divisible holdings, unrestricted short selling, and borrowing and lending at the same fixed $R$. A self-financing portfolio receives no outside cash after its initial funding; subsequent rebalancing uses its own proceeds.

This binomial market excludes arbitrage exactly when $d<R<u$. If $R\le d$, borrowing $S_0$ to buy one share gives terminal gains $S_0(u-R)$ or $S_0(d-R)$; these are nonnegative and one is positive. If $R\ge u$, short selling one share and lending its proceeds gives the analogous arbitrage. Strict inequalities allow a positive risk-neutral probability for each state. The two traded payoffs also span both successor states, giving market completeness relative to the information generated by this tree.

A traded claim or instrument carrying a current price and future payments or resale value. In a one-period financial market, its terminal payoff is a random variable and a portfolio combines payoffs linearly through its asset holdings. A stock, a zero-coupon bond and a bank account are basic examples. Dividends require treating the full gain process rather than only resale prices.

Mathematical finance uses probability theory, linear algebra and mathematical optimization to study financial markets, portfolio decisions and contingent claim prices. One question asks which portfolio best serves a specified objective; another asks what price a contingent claim must have to exclude arbitrage. The latter can have an exact answer even when future stock prices are uncertain.

The central idea is claim replication: construct a self-financing portfolio delivering the same financial payoff as a contract in every possible state. The law of one price then identifies the contract's price with the portfolio's initial cost. The binomial market makes this beautiful connection between uncertainty and linear algebra visible using just two equations. Risk-neutral pricing expresses the same cost as a discounted expectation.

The one-step example below fixes today's stock price at 100, terminal prices at 120 or 80, and the bank account's gross growth factor at 1.05. A European call option with strike price 100 pays 20 or zero. Its replicating portfolio in a binomial market costs $250/21$, independently of the physical up probability, provided both states remain possible. These are conclusions within the specified financial market model.

If the bank account $B$ is positive and nondecreasing and $S/B$ is a martingale under an equivalent martingale measure $Q$, then for $K\ge 0$ the European call price $C(T,K)$ is nondecreasing in $T$. Put $M_t=S_t/B_t$. For $u\ge t$, $K/B_u\le K/B_t$ and hence $(M_u-K/B_u{)}^{+}\ge (M_u-K/B_t{)}^{+}$. Conditional convexity gives ${\mathbb{E}}_Q[(M_u-K/B_t{)}^{+}\mid {\mathcal{F}}_t]\ge (M_t-K/B_t{)}^{+}$. Taking expectations and multiplying by $B_0$ proves the claim. The result is not generally true for negative strikes: with $S_t=B_t=e^{rt}$ and $r>0$, a negative-strike call costs $1-K{e}^{-rT}$, decreasing in maturity.

Fix a finite horizon $N$, a filtration $({\mathcal{F}}_n)$, a positive bank account $B_n$ with predictable one-step growth, ex-dividend risky prices $S_n$, and asset dividends $D_n$ paid at date $n$. Put $s_n=S_n/B_n$ and $d_n=D_n/B_n$. An equivalent martingale measure $Q$ makes the discounted cum-dividend gainsmartingales; equivalently ${\mathbb{E}}_Q[s_n-s_{n-1}+d_n\mid {\mathcal{F}}_{n-1}]=0$. It is these gains, not necessarily the discounted ex-dividend prices alone, which enter the definition.

Let ${\theta }_n$ be risky holdings chosen at date $n-1$ and carried to date $n$. Let $C_n$ be the portfolio's net cash distribution to its owner at date $n$, and $V_n$ its remaining value after that distribution and rebalancing. A self-financing portfolio with these distributions has the budget identityTo see this, value the previous stock and bank holdings at the new prices, add the asset dividends, subtract the owner distribution, and observe that the discounted bank holding is unchanged. Rebalancing then alters holdings but not value. An asset dividend reinvested in the portfolio is already in the gain term; it is not also an owner distribution.

For a predictable strategy whose gains are integrable under $Q$ (bounded holdings in a finite-state market suffice), conditioning the gain increment gives zero. Therefore discounted remaining wealth plus cumulative discounted owner payments is a martingale. Conditioning its terminal value proves the multi-period dividend pricing identityThe relation holds under every equivalent martingale measure for which these integrability conditions hold. Each payment is discounted from its own payment date, not from one common terminal date. The expression inside the expectation is in bank account units; multiplication by $B_n$ returns current currency units.

When the portfolio is liquidated and all its final value is paid out, $V_N=0$ after liquidation and that value is included in $C_N$. The displayed identity then involves only future discounted distributions. Without liquidation, the terminal value must remain. A bank account holding with no coupons before redemption has positive value despite having no intervening dividends; omitting its redemption or terminal holding value would plainly give the wrong result. In an infinite horizon, deleting the terminal term instead requires a dividend-price transversality condition, namely convergence of its conditional discounted expectation to zero.

For a traded asset the same proof gives its discounted expected dividends plus terminal sale value. For a replicated claim paid only at $N$, it gives $V_n=B_n{\mathbb{E}}_Q[H/B_N\mid {\mathcal{F}}_n]$. In a complete market this prices every admissible claim uniquely. In an incomplete market only the portfolio's attainable payout stream has this common value under all measures; an arbitrary unattainable payoff can have different expectations under different measures. For a merely dominated measure the conditional identity holds $Q$-almost surely; it makes no assertion on physical states assigned zero mass by $Q$. External capital injections must also be counted, as negative owner distributions. Thus the valuation principle follows from the gain martingale and the trading budget, together with the terminal convention and integrability, rather than from unconditional averaging of arbitrary future cash flows.

Assume $f\in C^{1,2}$ on positive prices before maturity and the usual local trading integrability. If its value is held in stock units ${\delta }_t$ and bank units ${\beta }_t$, thenfor a self-financing portfolio. The Itô formula diffusion coefficient is $\sigma S_t{f}_x$, so $\sigma S_t>0$ forces ${\delta }_t=f_x(S_t,t)$ and ${\beta }_t{B}_t=f-S_t{f}_x$. Equating drifts yieldsThe physical drift cancels, giving the Black-Scholes equationConversely, if the smooth function satisfies this equation, set $\delta =f_x$ and $\beta =(f-x{f}_x)/B$. Substituting into Itô formula gives exactly $df=\delta dS+\beta dB$, so these holdings are self-financing. This checks the holdings as well as the value equation; a partial differential equation solution does not justify arbitrary prescribed holdings. Appropriate admissibility and growth conditions are imposed when using it as an economic price.

For the logarithmic stock payoff, let $\tau =T-t$ and condition on $S_t=x$. Under the risk-neutral measure,The normal exponential moment $\mathbb{E}{e}^{uZ}=e^{u^2/2}$ and its derivative $\mathbb{E}[Z{e}^{uZ}]=u{e}^{u^2/2}$ giveThereforeThe option delta is the displayed stock holding. The bank account value is $f-S_t{f}_x=-S_t$, so ${\beta }_t=-S_t/B_t$. Directly, $f_{xx}=1/x$ and $f_t=-x(\rho +{\sigma }^2/2)$ verify the partial differential equation, while $f(x,T)=x\log x$ verifies the payoff. Lognormal moments give the required pricing integrability. Since $x\log x\ge -1/e$, the conditional price is bounded below over this finite horizon, so the resulting claim replication has the usual admissibility property.

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.

The entire term structure is naturally indexed by observation time and maturity. Work under a risk-neutral measure $Q$ and let $f(t,T)$ be the instantaneous forward rate, with $0\le t\le T$. Define the short rate $r_t=f(t,t)$, the bank account $B_t=\exp ({\int }_0^t{r}_{s}ds)$ and the zero-coupon bond priceAn arbitrary Gaussian mean and covariance need not make these bond prices consistent. Absence of arbitrage requires the discounted traded bonds to be martingales under a suitable equivalent measure. We derive the constraint explicitly.

Consider the Gaussian forward-rate field $f(t,T)=m(t,T)+X(t,T)$ with deterministic mean and centered Gaussian random field satisfyingThe maturity kernel is symmetric, and its increments in the first parameter must be positive semidefinite kernels. Assume continuity, separability and sufficient integrability for the following mean-square integrals and maturity derivatives. This covariance gives independent observation-time increments, while permitting correlated fluctuations across all maturities. The filtration contains the history of the whole observed forward curve.

For a fixed maturity $T$, collect the past short rate and the present curve into the integrated Gaussian forward-rate processThen $P(t,T)/B_t=e^{-A(t,T)-Y(t,T)}$. Its centered Gaussian variance isFor $s<t$ and an observed field coordinate $X(z,U)$ with $z\le s$, the covariance of $Y(t,T)-Y(s,T)$ with it is zero: the integrand isThe uncorrelated jointly Gaussian variables are independent principle therefore makes the increment independent of the earlier field filtration. Its variance is $v(t,T)-v(s,T)$, since the same covariance calculation gives $\mathrm{Cov}(Y(t,T),Y(s,T))=v(s,T)$. Conditional Gaussian exponential expectation now givesHence the exact bond-martingale condition isDifferentiating in maturity, using symmetry of the covariance, gives the Gaussian forward-rate covariance drift restrictionThis is necessary and sufficient. For sufficiency, the derivatives of the half-variance identity agree by the displayed restriction. Its integration constant also agrees at $T=t$, becauseThus the full identity follows for every $T\ge t$, and the conditional expectation calculation proves a true martingale, not merely zero formal drift. The initial mean $f(0,T)=-{\partial }_T\log P(0,T)$ fits the observed initial curve exactly; thereafter the mean adjustment is fixed by the covariance.

If $c(t;T,U)={\int }_0^t{k}_s(T,U)ds$ is time-differentiable, the forward drift is $a(t,T)={\int }_t^T{k}_t(T,u)du$. Writingwith deterministic finite- or Hilbert-space factor loadings gives the Heath-Jarrow-Morton modelWith $\Sigma (t,T)={\int }_t^T\sigma (t,u)du$, Itô formula independently verifies bond drift $r_t-{\int }_t^{T}a(t,u)du+\Vert \Sigma (t,T){\Vert }^2/2=r_t$. Finite-factor models give low-rank maturity correlations; a genuine random field or an infinite factor space permits a much richer covariance structure. Gaussian forward rates and short rates may be negative, although the exponential bond prices stay positive. Under a physical measure, a Girsanov theorem market-risk-premium change adds $⟨\sigma ,\lambda ⟩$ to the forward drift; deterministic premia preserve Gaussianity, whereas arbitrary adapted premia need not.

For a concrete two-parameter example, let $X$ be a standard Brownian sheet, so $c(t;T,U)=t\min (T,U)$. The restriction givesIndeed ${\int }_0^T\min (t,u)udu=t^3/3+t(T^2-t^2)/2$. This demonstrates directly how a Gaussian random field determines the compensating forward drift.

For the one-factor constant-volatility case, $\sigma (t,T)=\eta$, the Constant-coefficient Ho-Lee model hasandThese follow by integrating the explicit curve, not by assigning the short-rate drift independently. Exponentially decaying loadings $\eta e^{-\kappa (T-t)}$ instead give a Gaussian mean-reverting Hull-White model, with bond-rate loading $(1-e^{-\kappa (T-t)})/\kappa$.

A practical advantage is an explicit Gaussian bond-option formula. For a call of strike $K>0$ exercised at $T$ on a bond maturing at $U>T$, use the T-forward measure with numéraire $P(t,T)$. The forward bond ratio $F_t=P(t,U)/P(t,T)$ has a lognormal distribution under that measure, with remaining integrated varianceIts martingale property makes the conditional log mean $\log F_t-q/2$. Evaluating the positive-part lognormal integral givesFor $q=0$, the limit is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure accounts for the stochastic short rate; discounting by one deterministic interest rate would generally be wrong. Calibration selects a positive-semidefinite covariance structure fitting observed rate co-movements and option prices, with the no-arbitrage mean restriction imposed afterward. With more independent factors than the available traded bond exposures span, the market is incomplete and additional claims need a specified pricing measure or risk-premium model. The Gaussian field is therefore a flexible curve model with explicit consistency and pricing equations, rather than a claim that all possible rate risks are hedgeable.

One alternative is a stochastic volatility model, such as the Heston model:The parameters satisfy $\kappa ,\theta ,\xi >0$ and $|\rho |\le 1$, and $v_0\ge 0$. The square-root Itô diffusion has a nonnegative solution; $2\kappa \theta \ge {\xi }^2$ is a standard sufficient condition for the positive initial variance not to hit zero. Mean reversion of $v_t$ allows persistent high- and low-volatility episodes, and negative $\rho$ can produce the empirical association between falling prices and rising volatility. Mixtures of conditional return distributions produce richer tails and option implied volatility shapes than constant-volatility geometric Brownian motion.

For data showing volatility clustering and option smiles, this is often a more useful fit. It is not universally superior: more parameters create estimation and calibration uncertainty, the model is more expensive to use, and its continuous price paths still exclude price jumps. With two independent noise directions and only a stock and a bank account, the market is generally incomplete, so a volatility risk premium or an extra traded instrument is needed for unique contingent claim pricing. The alternative improves flexibility at the cost of calibration and hedging complexity.

Assume a frictionless market with no taxes or transaction costs, unrestricted short selling and borrowing/lending at the same constant rate $r$, continuous trading, and absence of arbitrage. The non-dividend-paying stock has geometric Brownian motion dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma >0$, and the bank account satisfies $dB=rBdt$. A contingent claim with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal financial payoff kinks can be handled by the smooth value for earlier times. Trading strategies are adapted, self-financing portfolios, and admissible trading strategies so that doubling strategies are excluded.

The Itô formula givesChoose $\Delta =f_S$ shares and put the remaining value $f-S{f}_S$ in the bank account. Its self-financing gain isMatching the contingent claim's gain to this replicating gain cancels the random increment and equates the drifts:This is the Black-Scholes equation. The argument uses the self-financing gain $\Delta dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the bank account. Conversely, a smooth solution with suitable growth and boundary conditions defines a delta hedge whose value replicates the contingent claim; law of one price identifies this value with the contingent claim price. Here $S$ is the current stock price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the contingent claim price. The physical expected return $\mu$ cancels because exposure to price risk has been hedged.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

Take bank-account values $B_k=R^k$ with $R>0$, and let the stock multiply independently by $u$ or $d$ at each step, where $0<d<R<u$. Under the physical probability measure $P$, the up probability is $p\in (0,1)$; the filtration records the successive moves. These assumptions specify the standard binomial market. If $R$ lies outside the strict interval $(d,u)$, a long or short stock position financed through the bank account gives an arbitrage.

The risk-neutral probability in a binomial market is fixed by $qu+(1-q)d=R$, givingFor successor claim values $V_u,V_d$ at a node with current stock price $S$, solve the two replication equations. The current stock holding and current cash value areThe cost of this replicating portfolio in a binomial market isApply this backward option pricing recursion from any terminal contingent claim $H$, including claims depending on the whole sequence of moves. It constructs a self-financing portfolio and proves market completeness by induction on the remaining number of periods. Taking successive conditional expectations under the measure $Q$ with independent up probability $q$ gives the unique risk-neutral pricing valueAny other price would give an arbitrage by buying the cheaper of the claim and its replica and selling the dearer.

If $K$ is the number of up moves, a particular path has probabilities $p^K(1-p{)}^{n-K}$ under $P$ and $q^K(1-q{)}^{n-K}$ under $Q$. Therefore the binomial-market probability density, namely the Radon-Nikodym derivative, isThis last expression makes $Z$ a function of the terminal stock price. Summing over paths gives ${\mathbb{E}}_{P}Z=1$, and strict positivity proves equivalence of the two probability measures.

For unrestricted terminal portfolio wealth $H$, its initial replication cost is $R^{-n}{\mathbb{E}}_{Q}H$. Hence feasible wealth satisfies the state-price budget constraint ${\mathbb{E}}_{Q}H=R^n{w}_0$. For the stated exponential utility, $v^{\prime }(x)=e^{-ax}$ and $v^{\prime \prime }(x)=-a{e}^{-ax}<0$. The binomial exponential-utility terminal wealth iswhereIt satisfies the budget, and its marginal utility is $v^{\prime }(H^{\ast })=\lambda Z$ with $\lambda =\exp (-a{R}^n{w}_0-{\mathbb{E}}_Q\log Z)>0$. The concave supporting-tangent inequality givesTaking expectations under $P$ makes the last term zero by the budget. Strict concavity makes equality possible only when $H=H^{\ast }$ in every positive-probability state. The already constructed complete market supplies its replicating strategy, so this proves global optimality and uniqueness. The formula allows negative final wealth, as permitted by the unrestricted trading problem.

The positive state-price density alternative characterizes one-period arbitrage. Interpret $A_0$ as the vector of current asset prices and $A_1$ as their terminal payments. A fixed vector of holdings produces initial cost $x^{\top }{A}_0$ and terminal payment $x^{\top }{A}_1$. The first alternative is a nonpositive-cost investment with a nonnegative payment and strict improvement either initially or on an event of positive probability. A strictly positive state-price density prices every such payment and excludes both forms of arbitrage, as proved below.

If there is a traded bank account costing one and paying a deterministic $R>0$, the density satisfies $1=R\mathbb{E}\nu$. Hence $dQ/dP=R\nu$ defines an equivalent martingale measure andConversely this pricing relation under an equivalent martingale measure gives $\nu =R^{-1}dQ/dP$. Without such an account, $\nu$ is a pricing density, not necessarily a probability density or even an integrable random variable by itself; the stated requirement is integrability of $\nu A_1$. This distinction keeps the abstract alternative valid without an unstated asset paying one.

Specify the trading conventions first. Let the bank account have value $R_r>0$, with $R_0=1$ and its next gross return known at time $r$. Write $B_r=1/R_r$ for the discount factor. Let $S_r$ be the ex-dividend asset-price vector, and $d_r$ the vector of cash dividends paid per unit at time $r$; set $d_r=0$ for non-dividend-paying assets. Under an equivalent martingale measure $Q$, each discounted cumulative asset gainis a martingale. Equivalently, ${\mathbb{E}}_Q[B_{r+1}(S_{r+1}+d_{r+1})\mid {\mathcal{F}}_r]=B_r{S}_r$. This is the gain-process form of risk-neutral pricing, rather than a martingale assumption on raw ex-dividend prices.

Let $X_r$ be the ${\mathcal{F}}_r$-measurable holdings after trading and distributions at time $r$. The previous holdings $X_{r-1}$ earn the next price change and dividends. Define the strategy's net cash payment and its residual portfolio wealth byA negative payment is an external infusion; a self-financing portfolio with no consumption has $D_r^X=0$. The discounted budget equation isAssume these gains and payments are integrable under $Q$, for example by bounded predictable holdings and integrable discounted asset gains. Thenis a true martingale, by conditioning the displayed one-step gains. Merely assuming a local martingale without appropriate integrability would not justify the conditional equality.

Conditioning $M_n$ on ${\mathcal{F}}_r$ proves the general multi-period dividend pricing identityFor a strategy liquidated at time $n$, $X_n=0$ and all liquidation proceeds are included in $D_n^X$. The terminal value disappears, giving exactly the requested dividend-only relation. That liquidation convention is necessary: holding one unit of the bank account with no payments would otherwise have a positive left side and zero right side.

An attainable claim $H$ paid at $n$ can be replicated by a strategy with no earlier payments and final liquidation payment $D_n^X=H$. Its time-$r$ value is thereforeTwo admissible replicating strategies have the same value by this identity, and the value is the same under every equivalent martingale measure for which the replication gains are true martingales. For unattainable claims this expectation is a possible model price, but the replication argument does not establish a unique price. With density process $Z_r={\mathbb{E}}_P[dQ/dP\mid {\mathcal{F}}_r]$, the equivalent state-price density form is $V_r=(B_r{Z}_r{)}^{-1}{\mathbb{E}}_P[B_n{Z}_{n}H\mid {\mathcal{F}}_r]$.

Assume $\sigma >0$, $x>0$ and sufficient $C^{2,1}$ regularity to use the Itô formula. Under the physical measure write $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ and let the bank account be $R_t=e^{\rho t}$. If $V_t=f(S_t,t)$ is a self-financing portfolio with ${\Delta }_t$ stock units and ${\beta }_t$ bank-account units, thenThe Itô formula gives $df=(f_t+\mu S_t{f}_x+\frac{1}{2}{\sigma }^2{S}_t^2{f}_{xx})dt+\sigma S_t{f}_{x}d{W}_t$. Equating diffusion terms forces ${\Delta }_t=f_x(S_t,t)$; equating drifts and eliminating ${\beta }_t{R}_t=f-S_t{f}_x$ gives the Black-Scholes equationConversely, if this equation holds, take $\Delta =f_x$ and $\beta =(f-x{f}_x)/R$. The same comparison shows $df=\Delta dS+\beta dR$, so the constructed portfolio is self-financing. This is a local replication equivalence for smooth functions on the state space; global admissibility and a terminal condition specify a unique economically admissible claim price.