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An admissible trading strategy is a permitted stochastically integrable trading rule satisfying the model's wealth restriction. A common finite-horizon convention, relative to a fixed positive numéraire $B$, requires a deterministic $a\ge 0$ such thatThe bound may depend on the strategy. Nonnegative wealth is sufficient. The choice of numéraire and whether the restriction is imposed on nominal or discounted wealth must be stated; the corresponding conditions need not agree for an arbitrary random account.

Under an equivalent local martingale measure, discounted self-financing gains are local martingales. The deterministic lower bound makes the discounted wealth a supermartingale, supporting the usual no-arbitrage and minimal-replication-cost arguments. Restrictions only on terminal wealth do not exclude all doubling strategies.

Fischer Black developed the Black-Scholes model with Myron Scholes. Their 1973 analysis shows how continuous adjustment of a replicating portfolio can eliminate the local exposure to the underlying random price change, leading to a no-arbitrage equation for the value of a contingent claim.^[ref]

This approach makes the hedge part of the mathematical explanation of the price. It also makes the assumptions visible: ideal continuous adjustment and the specified price dynamics are ingredients of the derivation, so changing the trading rules or dynamics can change the valuation problem. The Black-Scholes equation is therefore a useful starting point for studying both a tractable model and the effects of relaxing its assumptions.^[ref]

Josef Teichmann authored lectures on the fundamental theorem of asset pricing. The notes connect no-arbitrage, equivalent local martingale measures and replication, including a finite-state convex-geometric argument. This explains why a collection of quoted prices can be consistent precisely when an appropriate positive pricing measure exists under the theorem's hypotheses.^[ref]

Mark Rubinstein worked on option pricing, including the Cox--Ross--Rubinstein model developed with John C. Cox and Stephen A. Ross. Their 1979 paper constructs an accessible discrete-time model in which successive up and down moves form a binomial price tree. Replicating a contingent payment with traded assets determines its value under the model's no-arbitrage assumptions.^[ref]

The construction makes the relation between hedging and valuation explicit. Its risk-neutral probabilities are weights consistent with the model's prices, rather than predictions of the actual frequency of an upward market move. Refining the time steps also connects the discrete model to the Black-Scholes model.^[ref]

Myron Scholes, with Fischer Black, developed the Black-Scholes model and its option pricing equation. Their 1973 paper connects a contingent claim's value to the cost of a dynamically adjusted replicating portfolio, under explicit assumptions about trading and the underlying price process.^[ref]

The mathematical insight is that a carefully chosen hedge can cancel the instantaneous random component of the claim's value. No-arbitrage then constrains the remaining locally riskless return, producing the Black-Scholes equation together with a final payoff condition. The result is a conditional conclusion within a model: it does not require an ability to predict the actual stock-price path.^[ref]

Institutional profile: www.gsb.stanford.edu/faculty-research/faculty/myron-s-scholes

A financial market satisfies no-arbitrage when there is no admissible trading strategy that is self-financing, starts at zero cost, and has nonnegative terminal wealth almost surely with a strictly positive gain on an event of positive probability:The allowed trading class matters; in continuous time its lower-bound restriction excludes doubling strategies. The law of one price follows for identical cash flows when the offsetting long-short position is itself admissible, as in an unconstrained finite one-period model. Under merely local martingale prices, a required short position may have unbounded possible interim losses and be inadmissible; the unrestricted price-equality conclusion must then be justified separately.

Write $R=e^r$, $u=4/3$ and $d=2/3$. Since both moves have positive physical probability, the binomial options pricing model is free of arbitrage exactly whenIf $R\le d$, borrowing the share price and buying one share gives terminal gain $S_t(u-R)$ or $S_t(d-R)$, both nonnegative and at least one positive. If $R\ge u$, shorting one share and investing its proceeds gives the opposite comparison, with gain $S_t(R-u)$ or $S_t(R-d)$. Equality at either endpoint still permits arbitrage because the other outcome has positive probability.

Conversely, strict inequalities give a risk-neutral probability $q=(R-d)/(u-d)\in (0,1)$, for which the discounted share is a martingale. Any zero-cost self-financing portfolio with nonnegative terminal value has zero discounted expectation under this equivalent positive-probability law and hence cannot have a strictly positive payoff in either state. Applying this argument conditionally at each node proves no-arbitrage over the tree.

A replicating strategy holds $\Delta$ shares and a current cash amount $b$ in the bank account. Matching the two next values givesSolving this linear system,By no-arbitrage, the claim costs $V_t=\Delta S_t+b$. Collecting the coefficients of its two future values yieldsThe physical probability $p$ does not appear: replication fixes the price of this contingent claim from traded prices.

A European call option has nonnegative financial payoff, so its price is nonnegative under no-arbitrage. A second arbitrage comparison buys one call and a zero-coupon bond paying $K$, and shorts one share. Its terminal value is $(S_T-K{)}^{+}+K-S_T=(K-S_T{)}^{+}\ge 0$, so its initial cost cannot be negative. ConsequentlyThe holder of an American call option can choose to wait until expiry, so $C_t^a\ge C_t^e$. Since $r>0$ and $t<T$, an in-the-money call has $C_t^a\ge S_t-K{e}^{-r(T-t)}>S_t-K$: immediate exercise sacrifices value. An out-of-the-money exercise pays zero and cannot improve the holder's position. This proves no early exercise of a call without dividends: waiting is always at least as good, and is strictly better for positive immediate exercise payoff. In a degenerate market with zero continuation value an out-of-the-money holder can be indifferent; the Black-Scholes model with positive volatility gives strictly positive continuation value before expiry.

The portfolio value is ${\Pi }_t={\varphi }_t{S}_t+{\psi }_t{B}_t$. The self-financing condition means that changes in holdings are financed by trading within the portfolio, with no external infusion or withdrawal. ThusThe holdings are predictable and satisfy the usual stochastic-integrability and admissibility conditions; without admissibility, doubling strategies would spoil the intended no-arbitrage interpretation.

A self-financing replicating portfolio for a terminal payoff $F(S_T)$ has the same terminal value on every relevant path, with all earlier rebalancing funded internally. In a complete market its initial cost is unique under no-arbitrage: otherwise buy the cheaper payoff and sell the more expensive identical payoff.

In the Black-Scholes model, the portfolio holds $V_s$ shares and bond value $V-S{V}_s$, where $V$ solves the Black-Scholes equation with terminal data $F$. Itô's formula verifies the self-financing condition. Equivalently,This relates claim replication, the Black-Scholes equation, and risk-neutral valuation.

For a long position in the claim, hold $-V_S$ shares and bank value $S{V}_S-V$ so that claim plus hedge has zero initial value. The Black-Scholes equation and Itô's formula show that the total position is self-financing and its Brownian and drift terms cancel. Equivalently, the positive replicating portfolio holds $V_S$ shares and bond value $V-S{V}_S$.

Under the usual growth, integrability and admissibility conditions, no-arbitrage forces the claim price to be $V$: a different price could be traded against the replicating portfolio. The qualification matters because an arbitrary formal PDE solution without a suitable financial class need not be a valid price.

In an arbitrage-free finite market, a risk-neutral price expresses a replicable terminal financial payoff $H$ as a discounted expectation under an equivalent martingale measure $Q$. For a positive bank account $B$, its time-$t$ price isThis is a statement about claim replication and no-arbitrage, rather than the physical probability of future outcomes. For a one-period binomial market with multipliers $d<R<u$, the risk-neutral up probability is $q=(R-d)/(u-d)$ and $V_0=(q{H}_u+(1-q)H_d)/R$. A nonreplicable payoff in an incomplete market can give different values under different martingale measures; uniqueness then needs additional assumptions.

Stephen A. Ross developed arbitrage pricing theory, which relates expected asset returns to exposures to common sources of risk under assumptions limiting arbitrage. His mathematical approach asks how restrictions on feasible trading strategies constrain prices, rather than starting solely from an individual investor's preferences.^[ref]

With John C. Cox and Mark Rubinstein, he also developed the Cox--Ross--Rubinstein model for option pricing. The binomial price tree separates the possible stock-price paths from the valuation rule: a replicating portfolio and no-arbitrage produce risk-neutral pricing even when the actual up-move probability is not specified.^[ref]