# Risk-Neutral Pricing: Replication, Martingale Probabilities, and Limits

Understand why replicable derivative payoffs can be priced as discounted risk-neutral expectations, how to derive one-period probabilities, and why they are not real-world forecasts.

Canonical: https://wiki.fcontext.com/options/risk-neutral-pricing/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Risk-neutral pricing** values a replicable future payoff as its expected value under a specially constructed pricing measure, discounted at the appropriate financing rate. Under that measure, discounted tradable asset prices behave as martingales after carry and distributions are treated consistently.

The method does not assume real investors are risk-neutral. Risk aversion and risk premiums are absorbed into the change from real-world probabilities to risk-neutral pricing weights. The decisive economic argument is no arbitrage and replication: two portfolios with identical state-by-state cash flows should have the same current value, subject to the model's market and trading assumptions.

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## From no arbitrage to a pricing measure

In a one-period binomial model, a stock is `S₀` today and becomes either `Sᵤ` or `S_d`. Let the risk-free gross return be `R`. The risk-neutral up probability is:

`q = (R S₀ − S_d) / (Sᵤ − S_d)`

and the down probability is `1 − q`. For a derivative paying `Vᵤ` or `V_d`:

`V₀ = [q Vᵤ + (1 − q) V_d] / R`

No arbitrage requires `S_d < R S₀ < Sᵤ`, which makes `0 < q < 1`. The same price comes from a replicating position with stock Delta:

`Δ = (Vᵤ − V_d) / (Sᵤ − S_d)`

and a borrowing or lending position chosen so the portfolio matches both terminal payoffs. The “probabilities” are algebraic state-price weights generated by that replication, not estimates of how often the stock will rise.

In a complete arbitrage-free model, the equivalent martingale measure is unique and every modeled contingent payoff can be replicated. In incomplete markets, multiple pricing measures may be consistent with traded prices, so additional model assumptions, bounds, preferences, or hedging criteria are needed. Jumps, transaction costs, discrete trading, funding differences, borrow constraints, and liquidity break the frictionless replication ideal.

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## One-period Call example

Assume:

- stock price today `S₀ = $100`;
- in one period, `Sᵤ = $120` or `S_d = $90`;
- risk-free gross return `R = 1.02`;
- a Call with strike `$105`, paying `$15` in the up state and `$0` in the down state.

The pricing probability is:

`q = (1.02 × 100 − 90) / (120 − 90) = 12 / 30 = 0.40`

Therefore:

`Call value = (0.40 × $15 + 0.60 × $0) / 1.02 = $5.88`

Replication gives the same result. The hedge ratio is:

`Δ = ($15 − $0) / ($120 − $90) = 0.50`

Holding `0.50` share and borrowing `$44.12` today produces approximately `$15` in the up state and `$0` in the down state:

`0.50 × $100 − $44.12 = $5.88`

Suppose an analyst believes the real-world up probability is `70%`. Discounting the physical expected payoff gives `(0.70 × $15) / 1.02 = $10.29`, not the no-arbitrage price. That calculation ignores the stock's required risk premium and conflicts with the cheaper replicating portfolio. The physical probability can be useful for expected return or risk analysis, but not substituted directly into this replication price.

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## Model and implementation checklist

- Define the numeraire, rate curve, carry, dividends, borrow, and compounding consistently.
- Verify model states span the payoff being priced; unmatched risks prevent exact replication.
- Check no-arbitrage inequalities and that inferred probabilities are nonnegative and sum to one.
- Distinguish risk-neutral valuation from physical forecasting, expected return, and scenario probability.
- Use term-specific rates and dividend forecasts rather than one policy rate for all dates.
- Reprice American options with an early-exercise method; a single terminal expectation is insufficient.
- Include transaction costs, discrete rebalancing, gaps, trading halts, and liquidity stresses outside the base model.
- Treat counterparty, collateral, margin, funding, and settlement terms as valuation inputs where relevant.
- Validate convergence when using a tree, finite differences, or Monte Carlo under the risk-neutral process.
- Compare model value with executable Bid/Ask, not only Mid or a theoretical mark.
- Record measure, dynamics, calibration data, timestamp, and limitations so “risk-neutral” is not an undocumented label.

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## Common misconceptions

- “Risk-neutral pricing assumes investors do not dislike risk.” It changes the measure; it does not erase real-world preferences.
- “Risk-neutral probabilities are market forecasts.” They are pricing weights that include risk premiums.
- “Use the most likely real-world outcome and discount it.” Pricing requires all states and their replicating state prices.
- “Every payoff has one model-free risk-neutral price.” Uniqueness depends on market completeness and assumptions.
- “Discount every derivative at one Treasury rate.” Funding, collateral, dividends, borrow, and tenor can matter.
- “A risk-neutral expected payoff guarantees the market quote.” Model inputs, frictions, and Bid/Ask still matter.
- “No-arbitrage means riskless trading profits are easy to execute.” Apparent violations can disappear after costs and constraints.
- “Risk-neutral valuation measures investment attractiveness.” A fair replication price does not provide a physical expected return or decision rule.

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## Related topics

- [Risk-neutral density](/options/risk-neutral-density/)
- [Binomial option pricing](/options/binomial-option-pricing/)
- [Black-Scholes model](/options/black-scholes-model/)
- [Put-call parity](/options/put-call-parity/)

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## Sources

- [Cox, Ross, and Rubinstein: Option Pricing—A Simplified Approach](https://doi.org/10.1016/0304-405X(79)90015-1)
- [Harrison and Kreps: Martingales and Arbitrage in Multiperiod Securities Markets](https://doi.org/10.1016/0022-0531(79)90043-7)
- [Black and Scholes: The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062)
- [Options Industry Council: Put/Call Parity](https://www.optionseducation.org/advancedconcepts/put-call-parity)