No-Arbitrage Condition

#termstructure #economics #measure #finance

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

From Consumption-CAPM Consumption-CAPM, we show that the price of asset can be represented using the pricing formula, specifically as As pioneered by Ross (1976), in Arbitrage Pricing Theory, we starts at the pricing formula and investigates the sufficient conditions for strictly positive SDF to exist.

This condition is called as no-arbitrage condition, which has much weaker assumptions required in a general equilibrium pricing model, such as Consumption-CAPM > ^187b53 Consumption-CAPM > Assumption 6 (assumptions for C-CAPM).

For simplicity of the notation, we assume a two-period model, and ignore the time-subscription. Now we find a sufficient conditions for there to exists a strictly positive SDF such that

No-Arbitrage Assumption

Before beginning, remark the following theorem

Theorem 8 (Riesz Representation).

Let be a Hilbert space over complex field. For any linear functional that is continuous at , there exists a unique element s.t. and if , then there exists at least one s.t. .

which implies that any continuous linear functional

can be represented as a unique inner product.

Existence of Non-zero SDF

Assumption (assumptions for existence of Non-zero SDF).

Complete and linear payoff space: is a Hilbert space (complete inner product space) over the real field, where is a linear subspace of and is a norm of .
Law of one price (LOP): is a linear functional, i.e. implying LOP(if the expected payoff is the same, then it has the same price).
Continuity at : is continuous at , i.e.
Risk-free asset: there exists the risk-free payoff , whose price is . additionally, we have as we have driven in Consumption-CAPM Consumption-CAPM.

Theorem (existence of non-zero SDF).

Under ^937520 Assumption 1 (assumptions for existence of Non-zero SDF), there exists a SDF such that and

Proof.Under ^937520 Assumption 1 (assumptions for existence of Non-zero SDF), using the Hilbert and Lp spaces > ^5180f9 Hilbert and Lp spaces > Theorem 8 (Riesz Representation), there exists a unique non-zero element such that for any . □

Existence of Positive SDF

Assumption (assumptions for existence of strictly positive SDF).

Complete and linear payoff space: is a Hilbert space over the real space.
Law of one price (LOP): is a linear functional.
Continuity at : is continuous at .
Risk-free asset: there exists the risk-free payoff , such that . additionally, we have and .
Payoff space includes all derivatives: if , then for any measurable function such that is square integrable.
No-arbitrage opportunities: if an investor incurs no losses from an asset and there is a non-negligible chance for that asset to deliver a positive payoff, then the price of that payoff should be positive. i.e.

Theorem (existence of strictly positive SDF).

Under ^c75a75 Assumption 3 (assumptions for existence of strictly positive SDF), there exists a such that and

Proof.From ^094f43 Theorem 2 (existence of non-zero SDF), there exists an such that and Now, it remains to show that when we additionally assume 5 and 6 from ^c75a75 Assumption 3 (assumptions for existence of strictly positive SDF).

RTA: assume there we have , i.e. . Now define then since

by 5, contains all derivatives with fundamentals in
by construction,
by construction, with a positive probability of . i.e. As , by the no-arbitrage assumption, we have Since is the Riesz representation of , we have Then we have This implies which is a contradiction to our initial assumption that . Therefore, we have . i.e. is strictly positive almost surely. □

Risk-Neutral Measure

Physical and Risk-Neutral Measure

Definition (physical measure).

The physical measure, or the -measure, denoted as , is the probability measure probability measure that, for a given event, yields the actual probability of the event occuring.

Given the probability space , the no-arbitrage equation can be expressed as

Definition (risk-neutral measure).

The risk-neutral measure, or -measure, denoted as , is defined as where is the indicator function that equals if an outcome is included in and otherwise.

Proposition (Q-measure is a probability measure.).

Proof.Remark probability measure probability measure, we must prove the following four conditions.

non-negativity: , we have where the last inequality holds since and .
assigns to the empty set: .
assigns to the whole set:
countable additivity: for any disjoint countable collection , we have Therefore, is a probability measure. □

The below theorem shows the reason why is called as a 'risk-neutral' measure.

Theorem (first fundamental theorem of asset pricing).

For any payoff such that is -integrable, we have

Proof.From we have Then, we have which equals to This completes the proof. □

Under ^c75a75 Assumption 3 (assumptions for existence of strictly positive SDF), the above ^51420f Theorem 8 (first fundamental theorem of asset pricing) shows that there exists a measure under which the price of any asset is equal to the present value of its expected payoff.

measures incorporates the SDF and adjust it with respect to the risk-free rate . If SDF is a strictly positive measurable function, then the can be understood as a weighted measure of with respect to the value of discounted with .

If the investors are risk-neutral, then under the no-arbitrage assumption, the expected return from selling an asset () equals the expected discounted return from holding the asset and selling it next period ().

Simplified Dimension of Problem

intuitive meaning of risk-neutral measure

In -measure,

if an investor is risk-neutral, she only considers the 1st moment of the events. i.e. .
if an investor is risk-averse, she also considers the 2nd moment (risk) of the events. i.e. .

In -measure,

it allows for a risk-averse investors to only consider 1st moment of the events. i.e. .

Assume that the asset market is complete, so that any random variable is contained in the payoff space . Then, by ^3191fa Definition 6 (risk-neutral measure), the probability of any event under risk-neutral measure is This implies that the risk-neutral measure is proportional to the price of an asset that yields a payoff of if and only if the event occurs.

Now, suppose there are two events , of equal physical probability, but, is riskier than . Then, for the risk-neutral investors, two assets are indifferent since implying that the two assets have the same expected payoffs.

On the other hand, if the investors are risk-averse, than they would prefer the asset with payoff over the asset with payoff . While the two assets would yield the same expected payoff in terms of -measure, in -measure, we have as the risk-averse investors would require for the compensation for the risky asset, and thus the price of () would be more expensive than the asset .

This shows that the risk-neutral measure implements the information about the risk of the asset into its expected payoff, making it possible for risk-averse investors to rely only on an asset's expected payoff when making investment decisions.

Radon-Nikodym Derivative

Definition (equivalent measure).

Let be a measurable space. Then two probability measures and on are equivalent if for all .

Proposition (P-measure and Q-measure are equivalent).

The physical measure physical measure and risk-neutral measure risk-neutral measure are equivalent measure equivalent measure.

Proof.Note that by ^3191fa Definition 6 (risk-neutral measure), -measure is defined as () Assume , for some . Then, we have Since by ^ae2295 Theorem 4 (existence of strictly positive SDF), we must have () Assume . Then we have implying that is the null set null set. Then, by the definition, we have thus . □

Theorem (Radon-Nikodym derivative).

Let and be equivalent measures defined on . Then there exists an almost surely positive random variable such that and Moreover, we say is the Radon-Nikodym derivative of with respect to , which defined as

See Monotone Convergence Theorem > ^bd29e6 Monotone Convergence Theorem > Corollary 5 (Radon-Nikodym derivative) for the proof.

By ^bb6978 Proposition 10 (P-measure and Q-measure are equivalent), we can calculate Radon-Nikodym derivative, which is derived as

Empirical SDF

Using Taylor's expansion, we have for every event . Here, we choose a general form for the SDF under the Consumption-CAPM > Case of Log-Normal Returns.

Girsanov's Theorem

Definition (empirical SDF).

The empirical SDF is defined as where is some -dimensional vector, and is a random vector follows -dimensional standard normal distribution under physical measure.

Note that from Consumption-CAPM Consumption-CAPM can be shown for the empirical SDF by

The following theorem shows that for under the physical measure, we can make it also follows the standard normal distribution under the risk-neutral measure by shifting it.

Theorem (one-time Girsanov's theorem).

For the empirical SDF defined as , where follows the standard normal distribution under , then also follows the standard normal distribution under .

Define . Since follows under , we have . Then for any , we have which is the MGF of the standard normal distribution. Thus we can conclude that under . □

Application to C-CAPM

Note that under Consumption-CAPM > Case of CRRA Utility Consumption-CAPM > Case of CRRA Utility, where the CRRA utility function is defined as , the SDF was given as where the subjective discount rate is given as , and the last equality holds since Then, by taking logs on both sides, we have where is consumption growth.

Suppose under measure. Then since , we have then Thus, we have which is exactly the same form of the ^f3c6a1 Definition 12 (empirical SDF).

Note that denotes the Consumption-CAPM > ^38ea47 Consumption-CAPM > Definition 5 (market price of risk), representing how much compensation investors demand for an additional unit of risk. Meanwhile, (normalized) consumption growth represents the systemic risk factor, which presented in SDF proportionally by the correlation of an asset's rate of return with ().

This suggest from

Definition 12 (empirical SDF).

The empirical SDF is defined as where is some -dimensional vector, and is a random vector follows -dimensional standard normal distribution under physical measure.

the

represents the market price of risk, while

can be interpreted as a risk factor proportional to systematic risk.

Multi-Period SDF Process

Now we extend the model into the multi-period.

denotes the payoff the investor receives at from the investment at .
holds, where is the payoff space at .
is the price formula, where , meaning that the current price is determined by the future payoff .

Definition (SDF process).

Let ^c75a75 Assumption 3 (assumptions for existence of strictly positive SDF) adjusted for the multi-period settings holds for every . Then, there exists an SDF such that for every . By letting , the sequence is called SDF process.

Note that under the additional assumption that is uniformly integrable, a risk-neutral measure exists as for any .

Definition (empirical SDF process).

The SDF process is said to be an empirical SDF process if for any , where is an dimensional standard normally distributed random vector under , and is some dimensional random vector known at .

Theorem (Girsanov's theorem).

Define under the above empirical SDF process. Then follows an dimensional standard normal distribution.