1Arbitrage theory

In this chapter, we study the mathematical structure of a simple one-period model of a financial market. We consider a finite number of assets. Their initial prices at time t = 0 are known, their future prices at time t = 1 are described as random variables on some probability space. Trading takes place at time t = 0. Already in this simple model, some basic principles of mathematical finance appear very clearly. In Section 1.2, we single out those models which satisfy a condition of market efficiency: There are no trading opportunities which yield a profit without any downside risk. The absence of such arbitrage opportunities is characterized by the existence of an equivalent martingale measure. Under such a measure, discounted prices have the martingale property, that is, trading in the assets is a way of playing a fair game. As explained in Section 1.3, any equivalent martingale measure can be identified with a pricing rule: It extends the given prices of the primary assets to a larger space of contingent claims, or financial derivatives, without creating new arbitrage opportunities. In general, there will be several such extensions. Agiven contingent claim has a unique price if and only if it admits a perfect hedge. In our one-period model, this will be the exception rather than the rule. Thus, we are facing market incompleteness, unless our model satisfies the very restrictive conditions discussed in Section 1.4. The geometric structure of an arbitrage-free model is described in Section 1.5.

The one-period market model will be used throughout the first part of this book. On the one hand, its structure is rich enough to illustrate some of the key ideas of the field. On the other hand, it will provide an introduction to some of the mathematical methods which will be used in the dynamic hedging theory of the second part. In fact, the multi-period situation considered in Chapter 5 can be regarded as a sequence of one-period models whose initial conditions are contingent on the outcomes of previous periods. The techniques for dealing with such contingent initial data are introduced in Section 1.6.

1.1Assets, portfolios, and arbitrage opportunities

Consider a financial market with d + 1 assets. The assets can consist, for instance, of equities, bonds, commodities, or currencies. In a simple one-period model, these assets are priced at the initial time t = 0 and at the final time t = 1. We assume that the ith asset is available at time 0 for a price πi ≥ 0. The collection

is called a price system. Prices at time 1 are usually not known beforehand at time 0. In order to model this uncertainty, we fix a measurable space (Ω,F) and describe the asset prices at time 1 as nonnegative measurable functions

on (Ω,F) with values in [0,∞). Every ω ∈ Ω corresponds to a particular scenario of market evolution, and Si(ω) is the price of the ith asset at time 1 if the scenario ω occurs.

However, not all asset prices in a market are necessarily uncertain. Often there is a riskless bond which will pay a sure amount at time 1. In our simple model for one period, such a riskless investment opportunity will be included by assuming that

for a constant r, the return of a unit investment into the riskless bond. In many situations it would be natural to assume r ≥ 0, but for our purposes it is enough to require that S0 > 0, or equivalently that

thus accounting for the possibility of negative interest rates. In order to distinguish S0 from the risky assets S1, . . . , Sd, it will be convenient to use the notation

and in the same way we will write = (1, π).

At time t = 0, an investor will choose a portfolio

where ξi represents the number of shares of the ith asset. The price for buying the portfolio equals

At time t = 1, the portfolio will have the value

depending on the scenario ω ∈ Ω. Here we assume implicitly that buying and selling assets does not create extra costs, an assumption which may not be valid for a small investor but which becomes more realistic for a large financial institution. Note our convention of writing x · y for the inner product of two vectors x and y in Euclidean space.

Our definition of a portfolio allows the components ξi to be negative. If ξ0 < 0, this corresponds to taking out a loan such that we receive the amount |ξ0| at t = 0 and pay back the amount (1+ r)|ξ0| at time t = 1. If ξi < 0 for i ≥ 1, a quantity of |ξi| shares of the ith asset is sold without actually owning them. This corresponds to a short sale of the asset. In particular, an investor is allowed to take a short position ξi < 0, and to use up the received amount πi|ξi| for buying quantities ξj ≥ 0, j ≠ i, of the other assets. In this case, the price of the portfolio is given by

Remark 1.1. So far we have not assumed that anything is known about probabilities that might govern the realization of the various scenarios ω ∈ Ω. Such a situation is often referred to as Knightian uncertainty, in honor of F. Knight [182], who introduced the distinction between “risk” which refers to an economic situation in which the probabilistic structure is assumed to be known, and “uncertainty” where no such assumption is made.

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Let us now assume that a probability measure P is given on (Ω,F). The asset prices S1, . . . , Sd and the portfolio values can thus be regarded as random variables on (Ω,F, P).

Definition 1.2. A portfolio is called an arbitrage opportunity if but P-a.s. and

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Intuitively, an arbitrage opportunity is an investment strategy that yields with strictly positive probability a strictly positive profit and is not exposed to any downside risk. The existence of such an arbitrage opportunity may be regarded as a market inefficiency in the sense that certain assets are not priced in a reasonable way. In real-world markets, arbitrage opportunities are rather hard to find. If such an opportunity showed up, it would generate a large demand, prices would adjust, and the opportunity would disappear. Later on, the absence of such arbitrage opportunities will be our key assumption. Absence of arbitrage implies that Si vanishes P-a.s. once πi = 0. Hence, there is no loss of generality if we assume from now on that

Remark 1.3. Note that the probability measure P enters the definition of an arbitrage opportunity only through the null sets of P. In particular, the definition can be formulated without any explicit use of probabilities if Ω is countable. In this case, we can simply apply Definition 1.2 with an arbitrary probability measure P suchthat P[{ω}] > 0 for every ω ∈ Ω. Then an arbitrage opportunity is a portfolio with with for all ω ∈ Ω, and such that for at least one ω0 ∈ Ω.

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The following lemma shows that absence of arbitrage is equivalent to the following property of the market: Any investment in risky assets which yields with positive probability a better result than investing the same amount in the risk-free asset must be exposed to some downside risk.

Lemma 1.4. The following statements are equivalent.

(a) The market model admits an arbitrage opportunity.

(b) There is a vector ξ ∈ ℝd such that

Proof. To see that (a) implies (b), let be an arbitrage opportunity. Then Hence,

Since is P-a.s. nonnegative and strictly positive with nonvanishing probability, the same must be true of ξ · S − (1 + r)ξ · π.

Next let ξ be as in (b). We claim that the portfolio (ξ0, ξ) with ξ0 := −ξ· π is an arbitrage opportunity. Indeed, by definition. Moreover, which is P-a.s. nonnegative and strictly positive with nonvanishing probability.

Exercise 1.1.1. On Ω = {ω1, ω2, ω3}we fix a probability measure P with P[ ωi ] > 0 for i = 1, 2, 3. Suppose that we have three assets with prices

at time 0 and

at time 1. Show that this market model admits arbitrage.

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Exercise 1.1.2. We consider a market model with a single risky asset defined on a probability space with a finite sample space Ω and a probability measure P that assigns strictly positive probability to each ω ∈ Ω. We let

Show that the model does not admit arbitrage if and only if a < π(1 + r) < b.

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Exercise 1.1.3. Show that the existence of an arbitrage opportunity implies the following seemingly stronger condition.

(a) There exists an arbitrage opportunity such that

Show furthermore that the following condition implies the existence of an arbitrage opportunity.

(b) There exists such that and P-a.s.

What can you say about the implication (a)⇒(b)?

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1.2Absence of arbitrage and martingale measures

In this section, we are going to characterize those market models which do not admit any arbitrage opportunities. Such models will be called arbitrage-free.

Definition 1.5. A probability measure P∗ on (Ω,F) is called a risk-neutral measure, or a martingale measure, if

Remark 1.6. In (1.1), the price of the ith asset is identified as the expectation of the discounted payoff under the measure P∗. Thus, the pricing formula (1.1) can be seen as a classical valuation formula which does not take into account any risk aversion, in contrast to valuations in terms of expected utility which will be discussed in Section 2.3. This is why a measure P∗ satisfying (1.1) is called risk-neutral. The connection to martingales will be made explicit in Section 1.6.

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The following key result is sometimes called the “fundamental theorem of asset pricing” or, in short, FTAP. It characterizes arbitrage-free market models in terms of the set

of risk-neutral measures which are equivalent to P. Recall that two probability measures P∗ and P are said to be equivalent (P∗ ≈ P) if, for A ∈ F, P∗[ A ] = 0 if and only if P[ A ] = 0. This holds if and only if P∗ has a strictly positive density dP∗/dP with respect to P; see Appendix A.2. An equivalent risk-neutral measure is also called a pricing measure or an equivalent martingale measure.

Theorem 1.7. A market model is arbitrage-free if and only if P ≠ ∅. In this case, there exists a P∗ ∈ P which has a bounded density dP∗/dP.

We show first that the existence of a risk-neutral measure implies the absence of arbitrage.

Proof of the implication “⇐” of Theorem 1.7. Suppose that there exists a risk-neutral measure P∗ ∈ P. Take a portfolio such that P-a.s. and Both properties remain valid if we replace P by the equivalent measure P∗. Hence,

Thus, cannot be an arbitrage opportunity.

For the proof of the implication “⇒” of Theorem 1.7, it will be convenient to introduce the random vector Y = (Y1, . . . , Yd) of discounted net gains:

With this notation, Lemma 1.4 implies that the absence of arbitrage is equivalent to the following condition:

Since Yi is bounded from below by −πi, the expectation E∗[ Yi ] of Yi under any measure P∗ is well-defined, and so P∗ is a risk-neutral measure if and only if

Here, E∗[ Y ] is a shorthand notation for the d-dimensional vector with components E∗[ Yi ], i = 1, . . . , d. The assertion of Theorem 1.7 can now be read as follows:

Condition (1.3) holds if and only if there exists some P∗ ≈ P such that E∗[ Y ] = 0, and in this case, P∗ can be chosen such that the density dP∗/dP is bounded.

Proof of the implication “⇒” of Theorem 1.7. We have to show that (1.3) implies the existence of some P∗ ≈ P such that (1.4) holds and such that the density dP∗/dP is bounded. We will do this first in the case in which

Let Q denote the convex set of all probability measures Q ≈ P with bounded densities dQ/dP, and denote by EQ[ Y ] the d-dimensional vector with components EQ[ Yi ], i = 1, . . . , d. Due to our assumption E[ |Y| ] < ∞, all these expectations are finite. Let

and note that C is a convex set in and then has the bounded density

and hence belongs to Q. Therefore,

lies in C .

Our aim is to show that C contains the origin. To this end, we suppose by way of contradiction that 0 /∈ C . Using the “separating hyperplane theorem” in the elementary form of Proposition A.5, we obtain a vector such that ξ · x ≥ 0 for all x ∈ C , and such that ξ · x0 > 0 for some x0 ∈ C . Thus, ξ satisfies EQ[ ξ · Y ] ≥ 0 for all Q ∈ Q and EQ0 [ ξ · Y ] > 0 for some Q0 ∈ Q. Clearly, the latter condition yields that P[ ξ·Y > 0 ] > 0. We claim that the first condition implies that ξ·Y is P-a.s. nonnegative. This fact will be a contradiction to our assumption (1.3) and thus will prove that 0 ∈ C.

To prove the claim that ξ · Y ≥ 0 P-a.s., let A := {ξ · Y < 0}, and define functions

Normalizing the φn yields densities for new probability measures Qn:

Since 0 < φn ≤ 1, it follows that Qn ∈ Q, and thus that

Hence, Lebesgue’s dominated convergence theorem yields that

This proves the claim that ξ · Y ≥ 0 P-a.s. and completes the proof of Theorem 1.7 in case E[ |Y|] < ∞.

If Y is not P-integrable, then we simply replace the probability measure P by a suitable equivalent measure whose density d/dP is bounded and for which [ |Y| ] < ∞. For instance, one can define by

Recall from Remark 1.3 that replacing P with an equivalent probability measure does not affect the absence of arbitrage opportunities in our market model. Thus, the first part of this proof yields a risk-neutral measure P∗ which is equivalent to and whose density dP∗/d Pis bounded. Then P∗ ∈ P, and

is bounded (compare Exercise A.2.1). Hence, P∗ is as desired, and the theorem is proved.

Remark 1.8. Note that neither the absence of arbitrage nor the definition of the class P involve the full structure of the probability measure P; they only depend on the class of nullsets of P. In particular, the preceding theorem can be formulated in a situation of Knightian uncertainty, i.e., without fixing any initial probability measure P, whenever the underlying set Ω is countable.

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Remark 1.9. Our assumption that asset prices Si are nonnegative implies that the components of Y are bounded from below. Note however that this assumption was not needed in our proof. Thus, Theorem 1.7 also holds if we only assume that S is ℝ-valued and In this case, the definition of a risk-neutral measure P∗ via (1.1) is meant to include the assumption that Si is integrable with respect to P∗ for i = 1, . . . , d.

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Example 1.10. Let P be any probability measure on the finite set Ω := {ω1, . . . , ωN} that assigns strictly positive probability pi to each singleton {ωi}. Suppose that there is a single risky asset defined by its price π = π1 at time 0 and by the random variable S = S1.We assume that N ≥ 2 and that the values si := S(ωi) are distinct and arranged in increasing order: s1 < ·· · < sN. According to Theorem 1.7, this model does not admit arbitrage opportunities if and only if

Moreover, P∗ is a risk-neutral measure if and only if the probabilities p∗i:= P∗[ {ωi} ] solve the linear equations

under the constraints p∗i ≥ 0, i = 1, . . . , N. By Theorem 1.7, a solution exists if and only if (1.5) holds. It will be unique if and only if N = 2, and there will be infinitely many solutions for N > 2.

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Exercise 1.2.1. On Ω = {ω1, ω2, ω3} we fix a probability measure P with P[ {ωi}] > 0 for i = 1, 2, 3. Suppose that we have three assets with prices

at time and

at time 1. Show that this market model does not admit arbitrage and find all risk-neutral measures. Note that this model differs from the one in Exercise 1.1.1 only in the value of S2(ω3).

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Exercise 1.2.2. Consider a market model with risk-free rate r > −1 and one risky asset that is such that π1 > 0. Assume that the distribution of S1 has a strictly positive density function f : (0,∞) → (0,∞), that is, Find an equivalent risk-neutral measure P∗.

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Exercise 1.2.3. Consider an arbitrage-free market model with risk-free rate r > −1 and one risky asset satisfying π1 > 0 and E[ S1 ] < ∞. Construct P∗ ∈ P whose density with respect to P is constant on each of the sets {S1 > c}, {S1 < c}, and {S1 = c} with c := π1(1 + r).

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Remark 1.11. The economic reason for working with the discounted asset prices

is that one should distinguish between one unit of a currency (e.g. €) at time t = 0 and one unit at time t = 1. Usually people tend to prefer a certain amount today over the same amount which is promised to be paid at a later time. Such a preference is reflected in an interest r > 0 paid by the riskless bond: Only the amount 1 /(1+r)€ must be invested at time 0 to obtain 1€ at time 1. This effect is sometimes referred to as the time value of money. Similarly, the price Si of the ith asset is quoted in terms of € at time 1,while πi corresponds to time-zero euros. Thus, in order to compare the two prices πi and Si, one should first convert them to a common standard. This is achieved by taking the riskless bond as a numéraire and by considering the discounted prices in (1.6).

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Remark 1.12. One can choose as numéraire any asset which is strictly positive. For instance, suppose that π1 > 0 and P[ S1 > 0 ] = 1. Then all asset prices can be expressed in units of the first asset by considering

Clearly, the definition of an arbitrage opportunity is independent of the choice of a particular numéraire. Thus, an arbitrage-free market model should admit a risk-neutral measure with respect to the new numéraire, i.e., a probability measure ∗ ≈ P such that

Let us denote by the set of all such measures ∗. Then

Indeed, if ∗ lies in the set on the right, then

and so ∗ ∈ . Reversing the roles of and P then yields the identity of the two sets. Note that

as soon as S1 is not P-a.s. constant, because Jensen’s inequality then implies that

and hence ∗[ S1 ] > E∗[ S1 ] for all ∗ ∈ and P∗ ∈ P.

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Let

denote the linear space of all payoffs which can be generated by some portfolio. An element of V will be called an attainable payoff. The portfolio that generates V ∈ V is in general not unique, but we have the following law of one price.

Lemma 1.13. Suppose that the market model is arbitrage-free and that V ∈ V can be written as P-a.s. for two different portfolios and Then

Proof. We have a.s. for any P∗ ∈ P. Hence,

due to (1.1).

By the preceding lemma, it makes sense to define the price of V ∈ V as

whenever the market model is arbitrage-free.

Exercise 1.2.4. Give an example of a market model that satisfies the law of one price but is not arbitrage-free.

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Remark 1.14. Via (1.7), the price system π can be regarded as a linear form on the finite-dimensional vector space V . For any P ∗ ∈ P we have

Thus, an equivalent risk-neutral measure P∗ defines a linear extension of π onto the larger space L1(P∗) of P∗-integrable random variables. Since this space is usually infinite-dimensional, one cannot expect that such a pricing measure is in general unique; see however Section 1.4.

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We have seen above that, in an arbitrage-free market model, the condition P-a.s. implies that In fact, one may assume without loss of generality that

for otherwise we can find i ∈ {0, . . . , d} such that ξ i ≠ 0 and the ith asset can be represented as a linear combination of the remaining ones:

In this sense, the ith asset is redundant and can be omitted.

Definition 1.15. The market model is called nonredundant if (1.8) holds.

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Exercise 1.2.5. Show that in a nonredundant market model the components of the vector Y of discounted net gains are linearly independent in the sense that

Show then that condition (1.9) implies nonredundance if the market model is arbitrage-free.

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Exercise 1.2.6. Show that in a nonredundant and arbitrage-free market model the set

is compact for any w > 0.

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Definition 1.16. Suppose that the market model is arbitrage-free and that V ∈ V is an attainable payoff such that π(V) ≠ 0. Then the return of V is defined by

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Note that we have already seen the special case of the risk-free return

If an attainable payoff V is a linear combination of nonzero attainable payoffs Vk, then

The coefficient βk can be interpreted as the proportion of the investment allocated to Vk. As a particular case of the formula above, we have that

for all nonzero attainable payoffs (recall that we have assumed that all πi are strictly positive).

Proposition 1.17. Suppose that the market model is arbitrage-free, and let V ∈ V be an attainable payoff such that π(V) ≠ 0.

(a) Under any risk-neutral measure P∗, the expected return of V is equal to the risk-free return r:

(b) Under any measure Q ≈ P such that the expected return of V is given by

where P∗ is an arbitrary risk-neutral measure in Pand covQ denotes the covariance with respect to Q.

Proof. (a): Since we have

(b): Let P∗ ∈ P and φ∗ := dP∗/dQ. Then EQ[ φ∗ ] = 1 and hence

Using part (a) yields the assertion.

Remark 1.18. Let us comment on the extension of the fundamental equivalence in Theorem 1.7 to market models with an infinity of tradable assets S0, S1, S2, . . . . We assume that S0 ≡ 1 + r for some r > −1 and that the random vector

takes values in the space ∞ of bounded real sequences. The space ∞ is a Banach space with respect to the norm

A portfolio is chosen in such a way that belongs to the Banach space 1 of all real sequences x = (x1, x2, . . . ) such that We assume that the corresponding price system satisfies π ∈ ∞ and π0 = 1. Clearly, this model class includes our model with d + 1 traded assets as a special case.

Our first observation is that the implication “⇐” of Theorem 1.7 remains valid, i.e., the existence of a measure P∗ ≈ P with the properties

implies the absence of arbitrage opportunities. To this end, suppose that is a portfolio strategy such that

Then we can replace P in (1.10) by the equivalent measure P∗. Hence, cannot be an arbitrage opportunity since

Note that interchanging summation and integration is justified by dominated convergence, because

The following example shows that the implication “⇒” of Theorem 1.7, namely that absence of arbitrage opportunities implies the existence of a risk-neutral measure, may no longer be true for an infinite number of assets.

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Example 1.19. Let Ω = {1, 2, . . . }, and choose any probability measure P which assigns strictly positive probability to all singletons {ω}. We take r = 0 and define a price system πi = 1, for i = 0,1, . . . . Prices at time 1 are defined by S0 ≡ 1 and, for i = 1,2, . . . , by

Let us show that this market model is arbitrage-free. To this end, suppose that is a portfolio such that ξ ∈ 1 and such that for each ω ∈ Ω, but such that Considering the case ω = 1 yields

Similarly, for ω = i > 1,

It follows that 0 ≥ ξ1 ≥ ξ2 ≥· · · . But this can only be true if all ξ i vanish, since we have assumed that ξ ∈ 1. Hence, there are no arbitrage opportunities.

However, there exists no probability measure P∗ ≈ P such that E∗[ Si ] = πi for all i. Such a measure P∗ would have to satisfy

for i > 1. This relation implies that P∗[ {i}] = P∗[ {i + 1} ] for all i > 1, contradicting the assumption that P∗ is a probability measure and equivalent to P.

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