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Asset pricing

Asset pricing connects uncertain future payments to prices and required returns. Follow a two-state valuation through state prices, discount factors and replication.

Asset pricing is the study of how uncertain future payments translate into prices today and the returns those prices imply. A quant uses it both to value a new claim consistently with traded assets and to ask whether an asset’s expected return compensates for the risks it adds to an investor’s portfolio.

Those are related tasks, but they start from different information. A hedge can fix the price of a replicable payoff without forecasting its average return. An economic model can instead try to explain the return investors require, and therefore the price they will pay. Naming a model does not supply the missing assumptions or inputs.

The mean payoff does not set the price

Consider a stipulated, frictionless one-year USD market. An asset costs USD 100 today and makes a single terminal payment of USD 80 or USD 140. Each outcome has physical probability one-half. There are no interim distributions, fees or taxes. Cash can be borrowed or lent at a 5% one-year simple rate, so USD 100 in cash becomes USD 105. Positions are divisible and short selling is permitted; both future states are possible.

The physical expected payment is

EP[X]=12(80)+12(140)=110.\mathbb{E}^{\mathbb{P}}[X]=\tfrac12(80)+\tfrac12(140)=110.
Here P\mathbb{P} describes the assumed real-world outcome probabilities. Dividing USD 110 by 1.05 gives about USD 104.7619, which fails to reproduce the traded price. Discounting the average uncertain payment as though it were a certain payment has omitted the risk adjustment.
The given price and forecast imply a 10% expected return: 110/100−1=0.10110/100-1=0.10. Equivalently, 110/1.10=100110/1.10=100. But this calculation has inferred the return from the price. It has not independently established that 10% is the appropriate required return. The risk premium is the resulting five percentage points above cash; that article follows the expected-versus-realised return consequences of this same market.

Price the dollars where they arrive

A useful way to value uncertainty is to imagine two elementary claims: one pays USD 1 only in the lower-payoff state; the other pays USD 1 only in the higher-payoff state. Denote their prices today by πd\pi_d and πu\pi_u. These state prices measure today’s dollars per state-specific future dollar.

Buying both elementary claims guarantees USD 1 in a year. Buying 80 lower-state claims and 140 higher-state claims reproduces the asset. The law of one price therefore requires

πd+πu=11.05,80πd+140πu=100.\begin{aligned} \pi_d+\pi_u&=\frac{1}{1.05},\\ 80\pi_d+140\pi_u&=100. \end{aligned}
Solving gives πd=5/9\pi_d=5/9 and πu=25/63\pi_u=25/63. They are positive and sum to 20/2120/21, approximately USD 0.9524: the one-year discount factor. State prices do not sum to one and are not probabilities.
A synthetic one-year USD asset pays 80 in the lower state and 140 in the higher state. Prices per state-specific future dollar are 5/9 and 25/63; the corresponding present-value contributions are approximately USD 44.4444 and USD 55.5556, totaling USD 100. The state prices sum to 20/21.
A synthetic one-year USD asset pays 80 in the lower state and 140 in the higher state. Prices per state-specific future dollar are 5/9 and 25/63; the corresponding present-value contributions are approximately USD 44.4444 and USD 55.5556, totaling USD 100. The state prices sum to 20/21.
Original synthetic one-year USD ledger. Each state has physical probability one-half. State prices incorporate both timing and the relative value of payments across states; calculations use exact fractions before rounding.

This provides a reusable valuation rule. For any claim paying YdY_d in the lower state and YuY_u in the higher state,

V0=πdYd+πuYu.V_0=\pi_dY_d+\pi_uY_u.

For example, a claim paying USD 30 or USD 90 costs

V0=59(30)+2563(90)=110021≈USD 52.3810.V_0=\frac59(30)+\frac{25}{63}(90) =\frac{1100}{21}\approx\text{USD }52.3810.
There is an independent check: buy one unit of the original asset and borrow enough today to repay USD 50 in a year. The net terminal payments are 80−50=3080-50=30 and 140−50=90140-50=90. The portfolio costs 100−50/1.05=1100/21100-50/1.05=1100/21 today. A different price for the new claim would allow buying the cheaper side and selling the dearer side, with matching future payments cancelling under the stated trading assumptions. That is arbitrage, not merely a disagreement about average returns.
The public sample The Binomial Model develops this replication argument and extends it to repeated steps. It uses continuous compounding; the example here uses a single one-year simple rate.

Two equivalent ways to write the risk adjustment

A stochastic discount factor mm attaches a different discount multiplier to each future state. With physical probabilities psp_s, state prices satisfy πs=psms\pi_s=p_sm_s. For this market,
md=109,mu=5063.m_d=\frac{10}{9},\qquad m_u=\frac{50}{63}.

The pricing equation becomes

P0=EP[mX]=12(109)80+12(5063)140=100.\begin{aligned} P_0&=\mathbb{E}^{\mathbb{P}}[mX]\\ &=\tfrac12\left(\frac{10}{9}\right)80 +\tfrac12\left(\frac{50}{63}\right)140 =100. \end{aligned}

Also, EP[m]=20/21=1/1.05\mathbb{E}^{\mathbb{P}}[m]=20/21=1/1.05, as required to price a certain payment. The lower-state multiplier exceeds one. That is compatible with a positive risk-free rate: a state-dependent multiplier values a contingent dollar, whereas its physical average prices the certain dollar.

Alternatively, normalise the state prices by their sum. This produces risk-neutral probabilities qd=7/12q_d=7/12 and qu=5/12q_u=5/12, which sum to one:
P0=EQ[X]1.05=(7/12)80+(5/12)1401.05=1051.05=100.P_0=\frac{\mathbb{E}^{\mathbb{Q}}[X]}{1.05} =\frac{(7/12)80+(5/12)140}{1.05} =\frac{105}{1.05}=100.
The same risk adjustment appears either in the state-dependent multiplier under P\mathbb{P} or in the probability weights under Q\mathbb{Q}. It is not applied twice. Neither representation changes the assumed physical probability of the higher payment from one-half to 5/125/12. MIT’s pricing notes explain this distinction between probabilities inferred from prices and ordinary outcome probabilities.

What no-arbitrage can explain—and what needs economics

In this two-state market, the original asset and cash can reproduce any pair of state payments. Once their prices are supplied, no-arbitrage fixes every other claim’s price. This is a complete market. In an incomplete market, some payments cannot be replicated; no-arbitrage can leave a range of admissible valuations rather than choose one price.

Crucially, the construction has taken the original asset’s USD 100 price as an input. The physical payoff distribution and 5% cash rate alone do not uniquely select that price. No-arbitrage restricts the relative prices of available trading opportunities; it does not by itself explain all expected risk premia.

An equilibrium model adds economic structure: investors’ preferences and beliefs, their income or endowments, available investments, and market clearing. In a consumption-based model, a dollar is especially valuable when consumption is scarce and its marginal utility is high. A payoff arriving in those states can command a higher price. A payoff concentrated in prosperous states provides less insurance. Cochrane’s author-hosted sample chapters derive the discount multiplier from this marginal trade-off between consumption today and tomorrow.
The higher value of the lower-state dollar in our ledger is consistent with that interpretation, but the numbers were inferred from stipulated prices. The calculation has not estimated consumption preferences or proved a particular equilibrium model. Likewise, the capital asset pricing model makes an additional restriction linking expected excess returns to market exposure; it does not turn any forecast into a guaranteed return.
For an empirical pricing model, the practical test is whether the same proposed risk adjustment explains prices or expected returns across assets. A forecast return above a model’s required return may signal attractive value if the forecast and model are sound. It can also reflect estimation error, omitted risks or trading frictions. That discrepancy alone does not construct an arbitrage.

Sources

  • Edwin J. Elton, Martin J. Gruber, Stephen J. Brown and William N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition (Wiley, 2014), Chapter 10 appendix, pp. 216–217, and Chapter 13 appendix, pp. 304–305: physical probabilities, state-dependent discounting and the economic interpretation of pricing kernels.
  • John H. Cochrane, Asset Pricing (Princeton University Press, 2001), author-hosted sample chapters, §§1.2–1.4 and §§2.2–2.4: pricing multipliers, consumption and alternative pricing approaches.
  • Scott Sheffield, MIT 18.175, Lecture 15, PDF pp. 9–17 and 24–29: state-contingent prices, pricing probabilities and discounted expectations.
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