State price deflators explained

CM2 · Financial Engineering & Loss Reserving · 10 min read

Two ways to say the same thing

Risk-neutral valuation prices a payoff by changing the probabilities and discounting at the risk-free rate:

price = e^−rT E_Q[payoff]

A state price deflator prices the same payoff by keeping the real probabilities and changing the discounting, state by state:

price = E_P[A × payoff]

In words: attach a number A to each state of the world, multiply each possible payoff by the A for its state, and take the ordinary expectation using the probabilities you actually believe.

These give identical prices. The deflator is not a different theory and it is not more powerful; it is the same no-arbitrage content packaged so that the real-world probabilities stay in view. Knowing both, and knowing they agree, is what the syllabus is after.

Where the deflator comes from

Write the two expressions for the price of the same payoff and match them state by state. In a one-period model with an up state of real probability p and risk-neutral probability q:

e^−rT[q X_u + (1−q)X_d] = p A_uX_u + (1−p)A_dX_d

For this to hold for every payoff, the coefficients of X_u and of X_d must match separately. That gives the deflators immediately:

A_u = e^−rT q / p A_d = e^−rT(1−q)/(1−p)

In words: the deflator for a state is the discount factor multiplied by the ratio of the risk-neutral probability of that state to its real probability.

That ratio has a name, the Radon-Nikodym derivative, and it is worth knowing only because it says what the deflator is doing: it is carrying the change of measure and the discounting in a single number. Everything that distinguishes the risk-neutral world from the real one has been folded into A.

Three properties that are all checks

Its expectation is the discount factor. Price a payoff of 1 in every state — a risk-free bond:

E_P[A] = p A_u + (1−p)A_d = e^−rT

So the deflator is a discount factor that varies by state, and averages out to the ordinary one. This is the first thing to verify in any calculation.

It is lower in the good states. A rupee delivered when markets are strong is worth less today than a rupee delivered when they are weak, because it arrives when you need it least. In a one-period market that means A_u < A_d, and if your arithmetic gives the reverse, check the sign of the risk premium in the data.

It is positive in every state. A negative deflator would be an arbitrage: you could buy a payoff that is positive in one state and zero elsewhere for a negative price. Positivity of the deflators is exactly the no-arbitrage condition, which is the same statement as 0 < q < 1 in the risk-neutral language.

Why bother, if risk-neutral valuation already works

Four reasons, and a question asking "why use deflators" wants two or three of them.

Real-world probabilities stay visible. This is the practical one. An actuary modelling a guarantee usually needs both the value and the real-world distribution of outcomes — what it is worth, and how often it will bite. A deflator approach runs one set of real-world scenarios and produces both, from the same simulation, instead of needing a separate risk-neutral run.

It extends to several periods by multiplication. The deflator over two periods is the product of the one-period deflators along the path. So a deflator can be carried along a simulated path like an accumulating discount factor, which is exactly how a real asset-liability model is built.

It separates discounting from the risk adjustment. Writing price = E[A]E[X] + Cov(A, X) splits any price into a plain discounted expectation plus a correction for how the payoff covaries with the state of the world. The entire risk premium on any asset is that covariance, and the worked example below computes it. That is a genuinely useful way to see where a risk premium comes from, and it is invisible in the risk-neutral formulation.

It still exists when a unique risk-neutral measure does not. In an incomplete market there are many measures that price the traded assets, which is the same thing as saying there are many deflators. The deflator language makes that plural situation natural to describe, whereas "the" risk-neutral measure invites the mistake of assuming uniqueness.

The same object under other names

A question may use any of these and mean exactly this:

State price deflator, the actuarial term;
Stochastic discount factor, the term in economics;
Pricing kernel, the term in asset pricing;
Deflator, on its own, in insurance reporting.

A related but distinct term is a state price, which is the price today of one rupee in one state and nothing elsewhere. The state price is p A_u rather than A_u — the deflator with its probability already in it. Mixing the two up is an error of exactly a factor of p, and the clue is that state prices sum to the discount factor while deflators average to it.

Where marks are lost

Multiplying by the risk-neutral probabilities as well as by the deflator. The deflator already contains the change of measure. Use the real probabilities with it, or you have applied the adjustment twice.

Discounting again after using a deflator. The discount factor is inside A. A deflator price is a price, not a future value.

Confusing a deflator with a state price. Covered above: the factor is p.

Assuming the deflator is the same for every asset. It is a property of the market, not of the payoff. One set of deflators prices everything, and that is the whole point — which is also why it is a strong assumption, and why the deflators are not unique when the market is incomplete.

Getting the real probability from the model rather than the question. p is a real-world belief and has to be given. If a question provides only u, d and r, it is a risk-neutral question and no deflator can be constructed from it.

Questions students ask

Is the deflator observable?

Not directly, any more than the risk-neutral measure is. What is observable is prices, and any set of deflators consistent with them will do. In a complete market the prices pin the deflators down exactly; in an incomplete one they do not, and a modeller has to choose — which is an assumption that should be stated rather than buried.

How does this connect to utility theory?

Closely, and it is the deepest point on the page. In an equilibrium model the deflator turns out to be proportional to the marginal utility of the representative investor in each state: A ∝ U′(wealth in that state). Because utility is concave, marginal utility is low when wealth is high, which is exactly why the deflator is low in the good states. The property stated above as a check is a consequence of risk aversion, and the two topics at the start and end of the asset-valuation syllabus turn out to be the same topic.

Can a deflator be used with the real-world drift in a simulation?

Yes, and that is its main practical use. Simulate the fund under the real-world measure, carry the accumulated deflator along each path, and the average of deflator times payoff is the price. The answer must agree with a separate risk-neutral run, and comparing the two is the standard way to check that a deflator model has been built correctly.

Why does the deflator appear in the binomial topic rather than on its own?

Because the one-period binomial market is the smallest place it can be constructed completely, and because its agreement with risk-neutral valuation can be checked there by hand. Everything on this page is three numbers and two probabilities; the generalisation to continuous time adds machinery and no new idea.

A worked example

A share trades at ₹500. In one year it will be worth either ₹600, which the investor believes has probability 0.6, or ₹420 with probability 0.4. The continuously compounded risk-free rate is 5%.

The share and all three figures are invented for this example.

Step 1: the risk-neutral probability

Under the risk-neutral measure the share must earn the risk-free rate.

500 × e^0.05 = 500 × 1.0512711 = 525.635548
q = (525.635548 − 420) / (600 − 420) = 105.635548 / 180 = 0.5868642

Note that q = 0.5869 is below the real probability of 0.6. That is the risk adjustment: the risk-neutral measure is pessimistic relative to what the investor believes, and the gap is what produces the share's risk premium.

Step 2: the deflators

e^−0.05 = 0.9512294
A_u = 0.9512294 × 0.5868642 / 0.6 = 0.9304041
A_d = 0.9512294 × 0.4131358 / 0.4 = 0.9824674

A_d / A_u = 1.0560, so a rupee in the bad state is worth 5.6% more today than a rupee in the good state. The ordering is the right way round.

The first check: the deflator must average to the discount factor

0.6 × 0.9304041 + 0.4 × 0.9824674 = 0.5582425 + 0.3929870 = 0.9512294

which is e^−0.05 exactly. Pricing a certain payoff of 1 gives the ordinary discount factor, as it must.

The second check: the deflators must reprice the share

0.6 × 0.9304041 × 600 + 0.4 × 0.9824674 × 420
= 334.945476 + 165.054524 = ₹500.000000

The share comes back to ₹500 to six decimal places, from the real probabilities and no discounting step at all.

Step 3: price a call and a put with them

Take a strike of ₹500. The call pays 100 in the up state and nothing in the down state; the put pays nothing up and 80 down.

	Through the deflators	By risk-neutral valuation
Call	0.6 × 0.9304041 × 100 = 55.824245	0.9512294 × 0.5868642 × 100 = 55.824245
Put	0.4 × 0.9824674 × 80 = 31.438958	0.9512294 × 0.4131358 × 80 = 31.438958

Identical to six decimal places, by two routes that share no intermediate figure except q. And put-call parity holds on the deflator prices:

c − p = 55.824245 − 31.438958 = 24.385288
S − Ke^−rT = 500 − 500 × 0.9512294 = 24.385288

Step 4: where the risk premium comes from

This is what the deflator shows and the risk-neutral formulation hides. Under the real probabilities:

E[S₁] = 0.6 × 600 + 0.4 × 420 = ₹528.00
expected return = ln(528/500) = 0.0544882, so 5.4488% a year
risk premium = 5.4488% − 5.0000% = 0.4488 percentage points

Now split the price:

E[A] × E[S₁] = 0.9512294 × 528.00 = 502.249136
price = 500.000000
so Cov(A, S₁) = 500.000000 − 502.249136 = −2.249136

Read that as a sentence. If the deflator and the payoff were unrelated, the share would be worth ₹502.25 — its expected value, discounted at the risk-free rate. It is worth ₹500 instead, and the entire ₹2.25 difference is the negative covariance between the deflator and the payoff: the share pays most when a rupee is worth least.

That ₹2.25 is the risk premium, expressed as a price rather than as a rate. There is no other source of it. An asset whose payoff were uncorrelated with the deflator would earn exactly the risk-free rate however volatile it was — which is the same statement as the Capital Asset Pricing Model's claim that specific risk is not rewarded, reached from a different direction.

Step 5: two periods, by multiplying

Over two years with the same one-period structure, the deflator along a path is the product of the one-period deflators:

A_uu = 0.9304041² = 0.8656518
A_ud = 0.9304041 × 0.9824674 = 0.9140917
A_dd = 0.9824674² = 0.9652422

And the check, which is the same check as before one period further out:

0.36 × 0.8656518 + 2 × 0.24 × 0.9140917 + 0.16 × 0.9652422 = 0.9048374
e^{−2 × 0.05} = 0.9048374

The two-period deflator averages to the two-period discount factor. That multiplicative property is what makes a deflator usable in a simulation: carry a running product along each path, exactly as you would carry an accumulating discount factor, and the price of anything is the average of deflator times payoff at the end.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

State price deflators: pricing with real-world probabilities