Risk-neutral pricing is not a risk-free probability

What you will be able to do

After this lesson, you should be able to:

distinguish a real-world probability from a risk-neutral pricing weight;
explain why changing measure does not remove uncertainty or volatility;
calculate an expected discounted payoff in a finite-state model;
state why a simple CDS model can omit a separate volatility input without claiming that volatility never matters.

Two measures for two different uses

Valuation is at valuation-time $0$ . Let the local risk-neutral-pricing-horizon $\explain{risk-neutral-pricing-horizon}{T}$ be a future time in exact model-years, and let the terminal-random-payoff $\explain{terminal-random-payoff}{X_{\explain{risk-neutral-pricing-horizon}{T}}}$ be the signed USD amount that a claim holder receives or pays at $\explain{risk-neutral-pricing-horizon}{T}$ .

A probability measure assigns mutually consistent weights to events. The real-world-probability-measure $\explain{real-world-probability-measure}{\mathbb{P}}$ describes the modeled probabilities of events in the real world. It is the measure for a forecast. The risk-neutral-probability-measure $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ gives the weights of an arbitrage-free pricing model. So the same event can have different probabilities under $\explain{real-world-probability-measure}{\mathbb{P}}$ and $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ . Shreve makes this distinction between the actual and the risk-neutral measure in the introduction of the change of measure [1]. Hull makes the same distinction between forecasting and valuation [2].

There is no “risk-free measure” in this model. The measure is risk-neutral. “Risk-free” describes the cash account and discounting convention used as the pricing benchmark. It does not describe the event probabilities, and it does not make the payoff safe.

The name “risk-neutral” also does not assume that every investor is indifferent to risk. Risk preferences and risk premia are among the reasons why $\explain{real-world-probability-measure}{\mathbb{P}}$ and $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ differ. The pricing calculation uses $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ after $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ has been selected or calibrated.

From a partition to a price

In a no-arbitrage model with the cash account as numeraire, $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ is chosen so that discounted traded prices are martingales. In words: after discounting with the cash account, today’s traded price equals the conditional $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ -expectation of its discounted future value. Shreve states the martingale condition and its discounted-payoff consequence [3]. Tuckman and Serrat give the same expected-discounted-value construction in a finite-state fixed-income setting [4].

Write the expectation under the risk-neutral-probability-measure as $\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}$ . It is the same averaging operator, taken with the pricing weights of $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ . Use the local risk-neutral-scenario-index $\explain{risk-neutral-scenario-index}{j}$ and risk-neutral-scenario-count $\explain{risk-neutral-scenario-count}{m}$ . Let $\explain{risk-neutral-scenario-payoff}{x_{\explain{risk-neutral-scenario-index}{j}}}$ be the local risk-neutral-scenario-payoff, in USD at $\explain{risk-neutral-pricing-horizon}{T}$ , and let $\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}$ be its dimensionless risk-neutral-scenario-probability. For disjoint, exhaustive scenarios on which the payoff is constant, the risk-neutral expected payoff is:

\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{terminal-random-payoff}{X_{\explain{risk-neutral-pricing-horizon}{T}}}] =\sum_{\explain{risk-neutral-scenario-index}{j}=1}^{\explain{risk-neutral-scenario-count}{m}}\explain{risk-neutral-scenario-payoff}{x_{\explain{risk-neutral-scenario-index}{j}}}\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}, \qquad \explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}\ge 0, \qquad \sum_{\explain{risk-neutral-scenario-index}{j}=1}^{\explain{risk-neutral-scenario-count}{m}}\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}=1

(1.7.1)

This expected payoff is in USD at $\explain{risk-neutral-pricing-horizon}{T}$ . It is not a present value. Let the input discount-factor be $\explain{discount-factor}{D}(0,\explain{risk-neutral-pricing-horizon}{T})$ , in USD at time zero per USD at $\explain{risk-neutral-pricing-horizon}{T}$ . The claim’s present-value is the discounted risk-neutral expected payoff:

\explain{present-value}{PV_0} =\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}\!\left[\explain{discount-factor}{D}\left(0,\explain{risk-neutral-pricing-horizon}{T}\right)\explain{terminal-random-payoff}{X_{\explain{risk-neutral-pricing-horizon}{T}}}\right] =\explain{discount-factor}{D}(0,\explain{risk-neutral-pricing-horizon}{T})\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{terminal-random-payoff}{X_{\explain{risk-neutral-pricing-horizon}{T}}}] =\explain{discount-factor}{D}(0,\explain{risk-neutral-pricing-horizon}{T})\sum_{\explain{risk-neutral-scenario-index}{j}=1}^{\explain{risk-neutral-scenario-count}{m}}\explain{risk-neutral-scenario-payoff}{x_{\explain{risk-neutral-scenario-index}{j}}}\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}

(1.7.2)

The second equality uses this lesson’s assumption that $\explain{discount-factor}{D}(0,\explain{risk-neutral-pricing-horizon}{T})$ is deterministic. If the discount factor is random, it must in general stay inside the expectation. The expectation of a product equals the product of the expectations only under additional assumptions.

Worked example

Suppose $\explain{discount-factor}{D}(0,\explain{risk-neutral-pricing-horizon}{T})=0.95$ and the claim has two disjoint, exhaustive outcomes:

Table 1.7.1 Scenarios of the worked example. Each scenario's risk-neutral probability, payoff at the horizon, and contribution to the risk-neutral expected payoff, in USD at the horizon.

Scenario $\explain{risk-neutral-scenario-index}{j}$	Risk-neutral probability $\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}$	Payoff $\explain{risk-neutral-scenario-payoff}{x_{\explain{risk-neutral-scenario-index}{j}}}$ , USD at $\explain{risk-neutral-pricing-horizon}{T}$	Contribution $\explain{risk-neutral-scenario-probability}{q_{\explain{risk-neutral-scenario-index}{j}}}\explain{risk-neutral-scenario-payoff}{x_{\explain{risk-neutral-scenario-index}{j}}}$ , USD at $\explain{risk-neutral-pricing-horizon}{T}$
1	0.25	120	30.00
2	0.75	40	30.00

The risk-neutral expected payoff (1.7.1) and time-zero value (1.7.2) are

\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{terminal-random-payoff}{X_{\explain{risk-neutral-pricing-horizon}{T}}}] =0.25(120)+0.75(40) =\text{USD }60\text{ at }\explain{risk-neutral-pricing-horizon}{T}

(1.7.3)

\explain{present-value}{PV_0} =0.95(60) =\text{USD }57\text{ at time }0

(1.7.4)

No $\explain{real-world-probability-measure}{\mathbb{P}}$ probabilities enter this price, because the example states the $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ pricing weights as inputs. An estimate of the real-world expected payoff would be a different calculation.

Uncertainty remains

A change from $\explain{real-world-probability-measure}{\mathbb{P}}$ to $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ changes the weights of the modeled outcomes. It does not remove the randomness. In the standard change-of-measure result for a diffusion, the drift changes and the diffusion coefficient stays the same [5]. So volatility remains a model input for nonlinear claims such as options. A restricted linear model can omit a separate volatility input for reasons specific to its product. The lesson on that product states those reasons.

Advanced boundary: uniqueness

The pricing measure need not be unique. Under the usual technical conditions, the absence of arbitrage is tied to the existence of a suitable martingale measure. Uniqueness of the measure requires a stronger condition: the modeled market must be complete. This section is a boundary note, and the assessment of this lesson does not test it. Shreve develops the existence and uniqueness distinction [6].

Check your understanding

These checks separate interpretation from arithmetic and include unfamiliar scenario weights and a signed negative payoff.

Knowledge check 1.7.1 Risk-neutral pricing

Model boundary and review note

This lesson uses one terminal payoff, a finite scenario partition, deterministic risk-free discounting, and input $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ weights. It does not estimate $\explain{real-world-probability-measure}{\mathbb{P}}$ , calibrate $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ , generate a yield curve, treat a stochastic numeraire, establish market completeness, or value counterparty, liquidity, funding, or option effects.

All source records, notation, formulas, examples, and answer keys remain draft pending independent human source, quantitative, and editorial review.

References

Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.6, printed pp. 32-35. draft ↩
Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 12 §12.2, printed pp. 257-260; Ch. 14 §14.7, printed pp. 311-313. draft ↩
Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 5 §5.2.2, printed pp. 214-217, eqs. 5.2.22-5.2.24; §5.2.4, printed pp. 218-219, eqs. 5.2.29-5.2.31. draft ↩
Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). Ch. 7 §7.3, printed pp. 182-184, eqs. 7.7-7.8. draft ↩
Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 5 §5.2.2, printed pp. 214-217, especially eqs. 5.2.22-5.2.24. draft ↩
Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 5 §§5.4.3-5.4.4, printed pp. 228-233. draft ↩

Notation used on this page (18)

T

Risk-neutral pricing horizondraft

Future model time at which the one terminal payoff is delivered. It is measured in exact model-years from valuation time zero and is not a calendar date.

\explain{risk-neutral-pricing-horizon}{T}>0

Units: years from valuation time

m

Risk-neutral scenario countdraft

Counts the disjoint and exhaustive scenarios in the finite pricing partition. It is a positive integer and does not measure time or money.

\explain{risk-neutral-scenario-count}{m} \in \mathbb{N}

j

Risk-neutral scenario indexdraft

Selects one scenario in the finite pricing partition. It is a bookkeeping label rather than a probability or time.

\explain{risk-neutral-scenario-index}{j} \in \{1,\ldots,m\}

x_j

Risk-neutral scenario payoffdraft

Realized signed terminal amount in one scenario. Positive means received and negative means paid by the claim holder at the horizon.

Units: stated currency at the horizon

q_j

Risk-neutral scenario probabilitydraft

Pricing weight assigned by the risk-neutral measure to one scenario. The weights are dimensionless, non-negative, and sum to one over the finite partition.

\explain{risk-neutral-scenario-probability}{q_j} \ge 0,\quad \sum_{j=1}^{m}\explain{risk-neutral-scenario-probability}{q_j}=1

Units: dimensionless probability

A(0,t)

Accumulation factordraft

Grows one current unit over a stated future horizon under the selected rate model.

Units: dimensionless currency-units per current currency-unit

m

Compounding frequencydraft

Number of equal compounding periods per year under the stated rate convention, a positive integer fixed by the model convention.

Units: compounding periods per year

D(0,t)

Discount factordraft

Converts one deterministic future unit into value at valuation time.

Units: current currency-units per future currency-unit

\mathbb{E}

Expectationdraft

Operator returning the average of a random quantity, each outcome weighted by its probability under a stated measure; a superscript names that measure when more than one is in play.

\explain{expectation}{\mathbb{E}}[\explain{expectation.random-variable}{X}]=\int_{\explain{expectation.sample-space}{\Omega}} \explain{expectation.random-variable}{X}(\explain{expectation.outcome}{\omega})\,d\explain{expectation.probability-measure}{\mathbb{P}}(\explain{expectation.outcome}{\omega})

Symbols: $X$ random variable; $\Omega$ sample space; $\omega$ outcome; $\mathbb{P}$ probability measure(dimensionless probability weights between zero and one)

Units: value units

j^{(m)}

Nominal annual ratedraft

Annualized rate quote that must be paired with its compounding frequency.

Units: decimal rate per year

t_k

Payment timedraft

Time from the valuation date to one scheduled cash-flow event; a time coordinate, not a calendar date or payment index.

Units: years from the valuation date

r_m

Periodic ratedraft

Rate applied once in each compounding period under the stated convention, derived from the stated nominal annual quote in this model.

Units: decimal per compounding period

PV_0

Present valuedraft

Combines dated signed cash flows into one value at valuation time, using the same holder perspective as the signed cash flows.

Units: stated currency at the valuation time

\mathbb{P}

Real-world probability measuredraft

Assigns modeled probabilities intended to describe actual-world event likelihoods; used for forecasting and statistical statements under the stated real-world model.

Units: dimensionless probability weights between zero and one

\mathbb{Q}

Risk-neutral probability measuredraft

Gives model pricing weights under which discounted traded prices satisfy the martingale condition; it prices payoffs relative to a stated numeraire and is not a forecast of actual event frequencies.

Units: dimensionless probability weights between zero and one

CF_k

Signed cash flowdraft

Amount received or paid at one event from the stated holder perspective; positive means received and negative means paid by that holder.

Units: stated currency units

X_T

Terminal random payoffdraft

Signed amount delivered by a claim at the stated future horizon, before its outcome is known; positive means received and negative means paid by the claim holder.

Units: stated currency at the future horizon

0

Valuation timedraft

Common origin from which later model times and present values are measured.