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Construct the strictly positive rolling one-period bond account using successive one-period zero-coupon bonds:At time $t-1$, invest the entire account value in the bond maturing at $t$. This gives a positive self-financing portfolio usable as a numéraire.

We use the finite-discrete-time fundamental theorem of asset pricing: in a frictionless market with finitely many adapted assets and trading dates, no arbitrage is equivalent to existence of an equivalent martingale measure for asset prices, including dividends, expressed in a positive traded numéraire. Maturing bond payoffs are reinvested in that numéraire. Let $\mathbb{Q}$ be such a measure and $L_t=\mathbb{E}[d\mathbb{Q}/d\mathbb{P}\mid {\mathcal{F}}_t]$ its positive density process. The martingale pricing relation for a unit zero-coupon bond isDefine $Z_t=L_t/{\beta }_t$. The Bayes formula for conditional expectation then givesThe positive expectations are finite because these are the traded finite bond prices; in particular $\mathbb{E}{Z}_T=Z_0{P}_0(T)$ when initial information is trivial. Normalize $Z_0=1$ without changing any ratio. With nontrivial initial information the same identities are conditional on that information. The state-price density need not be unique when the bond market is incomplete; existence is sufficient.

The terminal density is strictly positive and has expectation one under the usual deterministic initial bond-price convention. Its density process isFor $0\le s\le t\le T$, the Bayes formula for conditional expectation under a change of measure givesThe same calculation at $s=0$ gives the finite expectation $1/P(0,T)$, so this is a true martingale, not just a formal conditional identity. Thus the continuous-time bank account measured in units of the maturity-$T$ bond is a ${\mathbb{Q}}_T$-martingale. This is the forward measure change of numéraire. If the initial bond price were random rather than given, integrability of its reciprocal would need to be included for this true-martingale assertion.

Use zero-interest cash as the one-period numéraire, consistent with the stated expectation-price formula. For $p>1$, differentiate the proposed call-price curve twice. The first derivative and the candidate density areThis is the power call-curve pricing density. It is strictly positive. Its integral is $C^{\prime }(\infty )-C^{\prime }(0+)=1$, and its survival function is $-C^{\prime }(u)$. Since $C(0+)=1$ and $C(\infty )=0$,Likewise, integrating the survival function from $K$ onwards givesThus a market whose terminal stock has this law under an equivalent martingale measure prices the stock at one, every proposed call at $C(K)$, and any integrable claim $g(S_1)$ at $\int g(u)f_p(u)du$. The finite-market fundamental theorem of asset pricing says that an equivalent measure pricing every traded discounted payoff by expectation excludes arbitrage. This proves the intended conclusion when such an equivalent pricing law is part of the model. For example, take the canonical terminal state space $(0,\infty )$ with stock equal to its coordinate and physical law equivalent to the positive density $f_p$.

There is, however, a genuine insufficiency in the literal finite-strike formulation: a finite list of call prices and no-arbitrage alone do not force this pricing law, nor even a continuous terminal distribution. Here is an explicit counterexample. Take $p=2$, one strike $K=1$, and two terminal stock valuesGive the lower stock value the remaining strictly positive probability and take this as the physical measure too. Direct calculation gives $\mathbb{E}{S}_1=1$ andso the stock/cash/call market is arbitrage-free. Now letThis bounded nonnegative function is zero at both actual stock values, so $g(S_1)=0$ almost surely. Yet $\int g(u)f_2(u)du>0$. Charging that positive amount for the identically zero payoff creates an arbitrage by selling it. In fact no Lebesgue probability density can price every claim correctly on this two-state market.

Therefore the displayed $f_p$ is the intended continuous pricing density, but the promised no-arbitrage extension requires an equivalent pricing measure with this terminal law; a full call curve identifies that law if such a measure exists, but it does not follow from the printed finite-strike hypotheses alone. This is the finite-strike nonidentification of a pricing density. The counterexample and the corrected sufficient hypothesis account for the literal and intended readings separately.

Under the usual no-arbitrage interpretation from part (a), let $Q^N$ be an equivalent martingale measure for the strictly positive numéraire $N$. Then $M_t=S_t/N_t$ is a martingale. Define the discounted European call option payoff $Y_t=(S_t-K{)}^{+}/N_t=(M_t-K/N_t{)}^{+}$. Because $N_{t+1}\ge N_t$ and $K>0$, we have $K/N_{t+1}\le K/N_t$. The conditional Jensen inequality therefore givesFor the middle step, the strike $K/N_t$ is fixed conditionally on ${\mathcal{F}}_t$; alternatively use $\mathbb{E}[X^{+}\mid {\mathcal{F}}_t]\ge \max \{\mathbb{E}[X\mid {\mathcal{F}}_t],0\}$. Thus $Y$ is a submartingale. Risk-neutral valuation in units of $N$ gives $C(T,K)=N_0{\mathbb{E}}^{Q^N}{Y}_T$, with the usual fixed initial prices. ConsequentlyThe source's word “increasing” means nondecreasing: a European call option can have zero payoff at successive maturities, so strict increase is not guaranteed. Integrability is understood in the claim class for which the stated replication prices exist.

Let $b$ and $h$ be the numbers of units of the numéraire and the stock in a replicating strategy. The two terminal payoffs of the European call option are $2$ and $0$, soSolving these simultaneous equations gives $b=-6/35$ and $h=8/35$. The initial replication cost isAs an independent pricing check, let $q$ be the risk-neutral probability of the state $(15,20)$ under the numéraire measure. The discounted stock must have initial value one and terminal mean one:Risk-neutral valuation then gives $10(3/7)(2/15)=4/7$. The physical probabilities $1/2,1/2$ are not the numéraire-measure probabilities.

The positive-part function is convex, so pathwise Jensen inequality givesTo compare the Asian option price with European call option prices at different dates, carry each earlier payoff forward using the numéraire. Purchase $1/T$ of each replicating strategy for maturity $t$, and, when its payoff is received, reinvest it in $N$ until $T$. This is a self-financing portfolio, with terminal wealthThe first inequality uses $N_T\ge N_t$ and nonnegative payoffs. Since the Asian option is replicable in the complete market, absence of arbitrage makes its replication cost no larger than this superhedging cost. ThereforeEquivalently, divide the first pathwise bound by $N_T$, use $(S_t-K{)}^{+}/N_T\le (S_t-K{)}^{+}/N_t$, and take expectations under the numéraire equivalent martingale measure. Merely averaging earlier payoffs without reinvestment would miss the difference in payment dates.

For a finite-state, finite-horizon discrete-time frictionless market with a strictly positive numéraire, the fundamental theorem of asset pricing states no arbitrage if and only if there exists an equivalent martingale measure under which all discounted asset prices are martingales. Equivalence means the same null events as the physical measure. In this setting market completeness is equivalent to uniqueness of the equivalent martingale measure, and a replicable payoff $H$ has price $S_t^0{\mathbb{E}}_Q[H/S_T^0\mid {\mathcal{F}}_t]$. Without finite-state hypotheses, stronger integrability and closure conditions are needed; this is the standard multi-period course version.

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.

An equivalent martingale measure is a probability measure $\mathbb{Q}$ on the terminal sigma-algebra such that $\mathbb{Q}$ and the physical measure $\mathbb{P}$ are mutually absolutely continuous and every asset price discounted by the chosen numéraire is a $\mathbb{Q}$-martingale with respect to the market filtration.

A T-forward measure is a probability measure $Q^T$, equivalent to the physical measure, under which prices expressed in units of the positive maturity-$T$ bond are martingales. Equivalently, every attainable payoff $X_T$ has time-$t$ priceThe zero-coupon bond is the numéraire.

In the Black-Scholes model, take $B_t=e^{-\rho (T-t)}$ as a zero-coupon bond numéraire. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The self-financing conditions for smooth stock and bond holdings giveunits of the bond. Since $V_t=g_t{S}_t+h_t{B}_t$, this yields the displayed wealth. Under the risk-neutral measure, $V_t/B_t=w_0/B_0+\theta \sigma W_t^Q$: the discounted gain is a constant multiple of Brownian motion.

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.

In a finite-state discrete-time financial market, a replicable terminal financial payoff $X$ has the displayed value under any equivalent martingale measure $Q$ for the bank account numéraire. Here ${\mathcal{F}}_t$ represents current information and conditional expectation averages over future states. The self-financing portfolio's discounted portfolio wealth is a martingale, so its current value equals the discounted expectation of its terminal value. In an incomplete market, nonreplicable claims need not have a unique price; continuous-time extensions require further conditions on admissible trading strategies and integrable discounted wealth.

For the European call option in the example,Discounting the physical expected value instead gives $200/21\approx 9.52381$ when $p=1/2$, or $320/21\approx 15.23810$ when $p=4/5$. Those amounts describe discounted physical expectations, not the replicating portfolio's cost. Holding $S_0,u,d,R$ fixed is crucial: changing beliefs can influence traded prices in a richer financial market model.

The probability of a market event under an equivalent martingale measure for a specified numéraire. It can differ from the physical probability. For a random numéraire, impose the martingale condition on asset-price ratios rather than treating the numéraire return as deterministic.