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Let ${\theta }_t$ and ${\eta }_t$ be the predictable holdings of the stock and the bank account. With no consumption, the self-financing portfolio has $X=\theta S+\eta B$ and $dX=\theta dS+\eta dB$. The Itô product rule, together with the dynamics from part (a), givesThe drift vanishes because $X=\theta S+\eta B$ and $\sigma \lambda =\mu -r$. Thus $XY$ is a local martingale. It is nonnegative by the assumed nonnegative wealth and strict positivity of $Y$, and henceFor example, this last fact follows directly from Conditional Fatou lemma applied to a localizing sequence; the initial capital is assumed finite. As usual, holdings must be integrable against the asset semimartingales for the self-financing portfolio equation to be defined.

Fix the maturity horizon. Since $YB$ is a nonnegative local martingale and $r$ is bounded, $\mathbb{E}{Y}_T<\infty$: $B_T\ge B_0{e}^{-\Vert r{\Vert }_{\infty }T}$ and $\mathbb{E}(Y_T{B}_T)\le B_0$. The bounded nonnegative payoff thus makes $Y_T{\xi }_T$ integrable. SetThe Brownian martingale representation theorem yields $N_t=N_0+{\int }_0^t{\zeta }_{s}d{W}_s$, using its locally square-integrable version for an integrable terminal random variable. The Itô formula for $N/Y$ givesChoose the replicating strategyIts self-financing portfolio equation has exactly the displayed drift and diffusion, because $\mu -r=\sigma \lambda$. All coefficients are locally integrable after stopping; the holdings are predictable in the augmented natural Brownian filtration. This construction has $X_t\ge 0$ and $X_T={\xi }_T$, so it is an admissible replicating strategy under the question's nonnegative-wealth convention.

For any other nonnegative self-financing portfolio $\hat{X}$ replicating the same payoff, part (b) implies ${\hat{X}}_0\ge \mathbb{E}[Y_T{\xi }_T]$. The constructed strategy attains equality, since $Y_0=1$ and $N_0=\mathbb{E}[Y_T{\xi }_T]$ in the augmented natural Brownian filtration. HenceThis is deflator-based claim replication; it does not require upgrading the local martingale deflator to a true martingale density.

With constant coefficients, $\lambda =(\mu -r)/\sigma$ is constant, so the stochastic exponential $Z_t=e^{rt}{Y}_t$ is a true martingale. Under its equivalent martingale measure $\mathbb{Q}$, the stock follows the Black-Scholes model $d{S}_t=S_t(rdt+\sigma d{W}_t^{\mathbb{Q}})$. The Gaussian exponential moment gives the value of the power optionThe option delta is ${\partial }_{s}V=V/(2s)$, so the replicating strategy holdsHalf the wealth value is in the stock and half in the bank account, with continuous rebalancing. To check the self-financing portfolio property under the original measure, the Itô formula gives $d{V}_t=\frac{1}{2}(r+\mu )V_{t}dt+\frac{1}{2}\sigma V_{t}d{W}_t={\theta }_{t}d{S}_t+{\eta }_{t}d{B}_t$. Also $V_T=\sqrt{S_T}$ and $V_t>0$.

The payoff is unbounded, so the bounded-payoff restriction of part (c) is not invoked automatically; the Black-Scholes model has the necessary finite Gaussian moments, and the explicit strategy attains $V_0=\mathbb{E}[Y_T\sqrt{S_T}]$. The supermartingale bound from part (b) therefore proves its minimality. The cost is independent of the physical drift $\mu$.

Let $V_t$ denote wealth measured in units of the strictly positive numéraire. A predictable self-financing portfolio is an arbitrage ifEquivalently, one may permit $V_0\le 0$ provided one of the two gains is strict.

The model is a complete market when every contingent claim in the specified class, usually every ${\mathcal{F}}_T$-measurable payoff satisfying the required integrability condition, is the terminal value of some admissible self-financing portfolio.

Let $\varphi =({\varphi }_n^0,{\varphi }_n^1,\dots ,{\varphi }_n^d)$ be a predictable self-financing portfolio, and let $V_n^{\varphi }={\varphi }_n\cdot S_n$ be its value. An arbitrage is such a strategy with

Suppose the claim $H=(S_1{)}^2$, payable at time $1$, were replicable. Holdings chosen at time zero are constants, so the time-$1$ value of any self-financing portfolio has the affine formBut $S_1=S_0{Z}_1$ has a continuous positive density on $(0,\infty )$. No affine function of $S_1$ can equal the quadratic function $S_1^2$ almost surely. Therefore $H$ is not attainable and the market is not complete.

The same argument applies if $H$ is viewed as a time-$T$ claim: once $H$ is known at time $1$, its required discounted value at time $1$ is fixed, and the first-period portfolio value would still have to reproduce a nonaffine function of $S_1$.

An $x$-admissible strategy is a predictable vector of holdings $\varphi =({\varphi }^0,\dots ,{\varphi }^d)$ that is integrable against the asset prices, is self-financing, has initial wealthand whose wealth process $V_t={\sum }_i{\varphi }_t^i{S}_t^i$ remains nonnegative.

If ${\theta }_n$ shares are held during period $n$, the remaining wealth $X_{n-1}-{\theta }_n{S}_{n-1}$ is in the bank account. Self-financing therefore giveswhich rearranges to the required equation.

A predictable strategy $H$ is self-financing when its wealth $X=H\cdot P$ satisfiesIt is admissible when its wealth obeys the stipulated lower bound, here taken to be nonnegative. A strictly positive Itô process $Y$ is a martingale deflator when every deflated asset price $Y{P}^i$ is a local martingale.

Between times $n-1$ and $n$, the investor holds ${\theta }_n$ shares worth ${\theta }_n{S}_{n-1}$ and places the remaining wealth $X_{n-1}-{\theta }_n{S}_{n-1}$ in the risk-free asset. The self-financing condition therefore gives

Let $B_t=e^{rt}$ and let ${\psi }_t$ be the bank-account holding. Then $X_t={\psi }_t{B}_t+{\theta }_t{S}_t$. The self-financing portfolio condition gives

The stock tree has $S_0=5$, first-period values $6,4$, and terminal successors $7,5$ from $6$ and $5,3$ from $4$. When $r=1/6$, the risk-free growth factor is $R=7/6$. At the upper time-one node, investing $6$ in the risk-free asset produces $7$ at time two, while one share bought for $6$ produces only $7$ or $5$.

Use the following predictable self-financing portfolio. Hold nothing initially. If $S_1=6$, short one share and invest the proceeds $6$ in the risk-free asset; if $S_1=4$, continue to hold nothing. Its terminal payoff ison the upper branch, and zero on the lower branch. It costs zero, is never negative, and is positive with positive probability, so it is an arbitrage.

The strategy $\eta$ is a self-financing portfolio because ${\xi }_t^{\eta }=0$, and its price is known one period in advance. Suppose its price first became nonpositive. On the event, known immediately before that date, that the next price is nonpositive while the current price is positive, an investor can short or buy the self-financing portfolio with the sign that gives no downside, finance the position at the current date, and close it at the known next price. This gives a nonnegative gain and a strictly positive gain whenever the price changes sign or reaches zero from a positive value.

More formally, stopping and scaling $\eta$ on the first such predictable event constructs an arbitrage. Since the market has no arbitrage and ${\pi }_0^{\eta }>0$, induction over dates gives

The value of an investment portfolio, obtained by summing each holding times its current asset price. Consumption removes money from portfolio wealth; a self-financing portfolio changes its value only through asset gains. Nonnegative portfolio wealth is a common admissibility restriction in an investment-consumption problem.

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.

Reinvest each maturity payment in the next unit-principal zero-coupon bond. Positive bond prices make the resulting self-financing portfolio strictly positive, so it can be used as a numéraire. Its one-step growth is the reciprocal of the current one-period bond price.

A state-price density converts terminal wealth and consumption into initial cost. For a nonnegative admissible self-financing portfolio with consumption, the deflated wealth plus cumulative deflated consumption is a supermartingale, giving $\mathbb{E}[{\zeta }_{T}X]+\mathbb{E}{\int }_0^T{\zeta }_t{c}_{t}dt\le w$. In a complete market, an integrable nonnegative terminal claim with full budget equality is replicable.