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Our next aim is to characterize the structure of the set of arbitrage-free prices of a discounted claim H. It follows from Theorem 5.29 that every arbitrage-free price πH of H must lie between the two numbers

We also know that πinf(H) and πsup(H) are equal if H is attainable. The following theorem shows that also the converse implication holds, i.e., H is attainable if and only if πinf(H) = πsup(H).

Theorem 5.32. Let H be a discounted claim.

(a) If H is attainable, then the set Π(H) of arbitrage-free prices for H consists of the single element V0, where V is the value process of any replicating strategy for H.

(b) If H is not attainable, then πinf(H) < πsup(H) and

Proof. The first assertion follows from Remark 5.26 and Theorem 5.29.

To prove (b), note first that

is an interval because P is a convex set. We will show that Π(H) is open by constructing for any π ∈ Π(H) two arbitrage-free prices and for H such that < π < To this end, take P∗ ∈ P such that π = E∗[ H ]. We will first construct an equivalent martingale measure ∈ P such that [ H ] > E∗[ H ]. Let

so that

Since H is not attainable, there must be some t ∈ {1, . . . , T} such that Ut − Ut−1 / ∈ Kt ∩ L1(P∗), where

By Lemma 1.69, Kt∩L1(P∗) is a closed linear subspace of L1(Ω,Ft , P∗). Therefore, Theorem A.60 applied with B := {Ut − Ut−1} and C := Kt ∩ L1(P∗) yields some Z ∈ L∞(Ω,Ft , P∗) such that

From the linearity of Kt ∩ L1(P∗) we deduce that

and hence that

There is no loss of generality in assuming that |Z| ≤ 1/3, so that

can be taken as the density d/dP∗ = of a new probability measure ≈ P. Since Z is Ft-measurable, the expectation of H under satisfies

where we have used (5.22) in the last step. On the other hand,Thus, := [ H ] will yield the desired arbitrage-free price larger than π if we have ∈ P.

Let us prove that ∈ P. For k > t, the Ft-measurability of and Proposition A.16 yield that

For k = t, (5.21) yields E∗[ (Xt − Xt−1) Z | Ft−1 ] = 0. Thus, it follows from E∗[| Ft−1 ] = 1 that

Finally, if k < t then P∗ and coincide on Fk. Hence

and we may conclude that ∈ P.

It remains to construct another equivalent martingale measure such that

But this is simply achieved by letting

which defines a probability measure ≈ P, because the density d/dP∗ is bounded from above by 5/3 and below by 1/3. P ∈ P is then obvious as is (5.23).

Remark 5.33. So far, we have assumed that a contingent claim is settled at the terminal time T. A natural way of dealing with an FT0-measurable payoff C0 ≥ 0 maturing at some time T0 < T is to apply our results to the corresponding discounted claim

in the market model with the restricted time horizon T0. Clearly, this restricted model is arbitrage-free. An alternative approach is to invest the payoff C0 at time T0 into the numéraire asset S0. At time T, this yields the contingent claim

whose discounted claim

is formally identical to H0. Moreover, our results can be directly applied to H. It is intuitively clear that these two approaches for determining the arbitrage-free prices of C0 should be equivalent. A formal proof must show that the set Π(H) is equal to the set

of arbitrage-free prices of H0 in the market model whose time horizon is T0. Here, P0 denotes the set of measures P∗0on (Ω,FT0 ) which are equivalent to P on FT0 and which are martingale measures for the restricted price process (Xt)t=0,...,T0 . Clearly, each P∗ ∈ P defines an element of P0 by restricting P∗ to the σ-algebraFT0 . In fact, Proposition 5.34 below shows that every element in P0 arises in this way. Thus, the two sets of arbitrage-free prices for H and H0 coincide, i.e.,

It follows, in particular, that H 0 is attainable if and only if H is attainable.

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Proposition 5.34. Consider the situation described in Remark 5.33 and let be given. Then there exists some P∗ ∈ P whose restriction to FT0 is equal to

Proof. Let ∈ P be arbitrary, and denote by Z T0 the density of with respect to the restriction of to the σ-algebra FT0 . Then Z T0 is FT0-measurable, and

defines a probability measure on F. Clearly, P∗ is equivalent to and to P, and it coincides with P∗0on FT0 . To check that P∗ ∈ P, it suffices to show that Xt − Xt−1 is a martingale increment under P∗ for t > T0. For these t, the density ZT0 is Ft−1-measurable, so Proposition A.16 implies that

Example 5.35. Let us consider the situation of Example 5.30, where the numéraire S0 is a locally riskless bond. Remark 5.33 allows us to compare the arbitrage-free prices of two European call options and with the same strikes and underlyings but with different maturities T0 < T. As in Example 5.30, we get that for P∗ ∈ P

Hence, if P∗ is used to calculate arbitrage-free prices for C0 and C, the resulting price of C0 is lower than the price of C:

This argument suggests that the price of a European call option should be an increasing function of the maturity.

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Exercise 5.3.2. Let Yk (k = 1, . . . , T) be independent identically distributed random variables in L1(Ω,F, P), and suppose that the Yk are not P-a.s. constant and satisfy E [ Yk ] = 0. Let furthermore

Then X is a P-martingale when we consider the filtration given by F0 = {∅, Ω} and Ft = σ(Y1, . . . , Yt) for t = 1, . . . , T. We now enlarge the filtration by adding “insider information” of the terminal value X T. That is, we consider the enlarged filtration

(a) Show that X is no longer a P-martingale with respect to (Ft).

(b) Prove that the process

is a P-martingale with respect to the enlarged filtration (Ft).

(c) The insider information of the terminal value XT implies the existence of self-financing strategies with positive expected profit. Construct a strategy ∗ that maximizes the expected profit

within the class of all (Ft)-predictable strategies ξ with |ξt| ≤ 1 P-a.s. for all t.

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5.4Complete markets

We have seen in Theorem 5.32 that any attainable claim in an arbitrage-free market model has a unique arbitrage-free price. Thus, the situation becomes particularly transparent if all contingent claims are attainable.

Definition 5.36. An arbitrage-free market model is called complete if every contingent claim is attainable.

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Complete market models are precisely those models in which every contingent claim has a unique and unambiguous arbitrage-free price. However, in discrete time, only a very limited class of models enjoys this property. The following characterization of market completeness is sometimes called the “second fundamental theorem of asset pricing”.

Theorem 5.37. An arbitrage-free market model is complete if and only if there exists exactly one equivalent martingale measure. In this case, the probability space (Ω,FT , P) can be decomposed into finitely many atoms, whose number is bounded from above by (d + 1)T.

Proof. If the model is complete, then H := A for A ∈ FT is an attainable discounted claim. It follows from the results of Section 5.3 that the mapping is constant over the set P. Hence, there can be only one equivalent martingale measure.

Conversely, if |P| = 1, then the set Π(H) of arbitrage-free prices of every discounted claim H has exactly one element. Hence, Theorem 5.32 implies that H is attainable.

To prove the second assertion, note first that the asserted bound on the number of atoms in FT holds for T = 1 by Corollary 1.42. We proceed by induction on T. Suppose that the assertion holds for T − 1. By assumption, any bounded FT-measurable random variable H ≥ 0 can be written as

where both VT−1 and ξT are FT−1-measurable and hence constant on each atom A of (Ω,FT−1, P). It follows that the dimension of the linear space L∞(Ω,FT , P[ · |A]) is less than or equal to d + 1. Thus, Proposition 1.41 implies that (Ω,FT , P[ · |A]) has at most d + 1 atoms. Applying the induction hypothesis concludes the proof.

Below we state additional characterizations of market completeness. Denote by Q the set of all martingale measures in the sense of Definition 5.13. Then both P and Q are convex sets. Recall that an element of a convex set is called an extreme point of this set if it cannot be written as a nontrivial convex combination of members of this set.

Property (d) in the following theorem is usually called the predictable representation property, or the martingale representation property, of the P∗-martingale X.

Theorem 5.38. For P∗ ∈ P the following conditions are equivalent:

(a) P = {P∗}.

(b) P∗ is an extreme point of P

(c) P∗ is an extreme point of Q.

(d) Every P∗-martingale M can be represented as a “stochastic integral” of a d-dimensional predictable process ξ:

Proof. (a)⇒(c): If P∗ can be written as P∗ = αQ1 + (1 − α)Q2 for α ∈ (0, 1) and Q1, Q2 ∈ Q, then Q1 and Q2 are both absolutely continuous with respect to P∗. By defining

we thus obtain two martingale measures P1 and P2 which are equivalent to P∗. Hence, P1 = P2 = P∗ and, in turn, Q1 = Q2 = P∗.

(c)⇒(b): This is obvious since P ⊂ Q.

(b)⇒(a): Suppose that there exists a ∈ P which is different from P∗. We will show below that in this case can be chosen such that the density d/dP∗ is bounded by some constant c > 0. Then, if ε > 0 is less than 1/c,

defines another measure Pʹ ∈ P different from P∗. Moreover, P∗ can be represented as the convex combination

which contradicts condition (b). Hence, P∗ must be the unique equivalent martingale measure.

It remains to prove the existence of ∈ P with a bounded density d/dP∗ if there exists some ∈ P which is different from P∗. Since ≠ P∗, there exists a set A ∈ FT such that P∗[ A ] ≠ [ A ]. We enlarge our market model by introducing the additional asset

and we take P∗ instead of P as our reference measure. By definition, is an equivalent martingale measure for (X0, X1, . . . , Xd , Xd+1). Hence, the extended market model is arbitrage-free, and Theorem 5.16 guarantees the existence of an equivalent martingale measure such that the density d/dP∗ is bounded. Moreover, must be different from P∗, since P∗ is not a martingale measure for Xd+1:

(a)⇒(d): The terminal value MT of a P∗-martingaleM can be decomposed into the difference of its positive and negative parts:

and can be regarded as two discounted claims, which are attainable by Theorem 5.37. Hence, there exist two d-dimensional predictable process ξ+ and ξ− such that

for two nonnegative constants and Since the value processes

are P∗-martingales by Theorem 5.25, we get that

This proves that the desired representation of M holds in terms of the d-dimensional predictable process ξ := ξ+ − ξ−.

(d)⇒(a): Applying our assumption to the martingale Mt := P∗[ A | Ft ] shows that H = A is an attainable contingent claim. Hence, it follows from the results of Section 5.3 that the mapping is constant over the set P. Thus, there can be only one equivalent martingale measure.

Exercise 5.4.1. Consider the sample space

and denote by Yt(ω) := yt, for ω = (y1, . . . , yT), the projection on the tth coordinate. As filtration we take F0 := {∅, Ω} and Ft := σ(Y1, . . . , Yt) for t = 1, . . . , T. We consider a financial market model with two assets such that the discounted price process is of the form

for a constant X0 > 0 and two predictable processes (σt) and (mt). We suppose that 0 ≤ |mt| < σt for all t.

(a) Show that if P is a probability measure on Ω for which P[ {ω} ] > 0 for all ω, then there exists a unique equivalent martingale measure P∗.

(b) By using the binary structure of the model, and without using Theorem 5.37, prove the following martingale representation result. If P∗ is as in (a), every P∗-martingale M can be represented as

where the predictable process ξ is given by

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Exercise 5.4.2. Consider a complete market model with T = 2 and discounted price process (Xt)t=0,1,2, defined on some filtered probability space (Ω,F, (Ft)t=0,1,2, P). By P∗ we denote the unique equivalent martingale measure in this complete model. Now we enlarge the model by adding two external states ω+ and ω−, corresponding to additional information revealed at time t = 2. That is, we define := Ω×{ω+, ω−}, Fˆ := σ(A × {ω+}| A ∈ F), and

The fact that the additional information is revealed at time t = 2 means that we choose the price process

and the following filtration,

For 0 < p < 1 we furthermore define the probability measure on Fˆ via

(a) Show that each measureis an equivalent martingale measure for the extended model.

(b) Argue that the extended model is incomplete and find a nonattainable contingent claim.

(c) Does every equivalent martingale measure belong to the set If yes, give a proof, if not, find a counterexample.

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5.5The binomial model

A complete financial market model with only one risky asset must have a binary tree structure, as we have seen in Theorem 5.37. Under an additional homogeneity assumption, this reduces to the following particularly simple model, which was introduced by Cox, Ross, and Rubinstein in [61]. It involves the riskless bond

with r > −1 and one risky asset S1 = S, whose return

in the tth trading period can only take two possible values a, b ∈ ℝ such that

Thus, the stock price jumps from St−1 either to the higher value St = St−1(1+b) or to the lower value St = St−1(1 + a). In this context, we are going to derive explicit formulas for the arbitrage-free prices and replicating strategies of various contingent claims.

Let us construct the model on the sample space

Denote by

the projection on the tth coordinate, and let

The price process of the risky asset is modeled as

where the initial value S0 > 0 is a given constant. The discounted price process takes the form

As filtration we take

Note that F0 = {∅, Ω}, and

F := FT coincides with the power set of Ω. Let us fix any probability measure P on (Ω,F) such that

Such a model will be called a binomial model or a CRR model. The following theorem characterizes those parameter values a, b, r for which the model is arbitrage-free.

Theorem 5.39. The CRR model is arbitrage-free if and only if a < r < b. In this case, the CRR model is complete, and there is a unique martingale measure P∗. The martingale measure is characterized by the fact that the random variables R1, . . . , RT are independent under P∗ with common distribution

Proof. A measure Q on (Ω,F) is a martingale measure if and only if the discounted price process is a martingale under Q, i.e.,

for all t ≤ T − 1. This identity is equivalent to the equation

i.e., to the condition

But this holds if and only if the random variables R1, . . . , RT are independent under Q with common distribution Q[ Rt = b ] = p∗. In particular, there can be at most one martingale measure for X.

If the market model is arbitrage-free, then there exists an equivalent martingale measure P∗. The condition P∗ ≈ P implies

which holds if and only if a < r < b.

Conversely, if a < r < b then we can define a measure P∗ ≈ P on (Ω,F) by

where k denotes the number of occurrences of +1 in ω = (y1, . . . , yT). Under P∗, Y1, . . . , YT, and hence R1, . . . , RT, are independent random variables with common distribution P∗[ Yt = 1] = P∗[ Rt = b ] = p∗, and so P∗ is an equivalent martingale measure.

From now on, we consider only CRR models which are arbitrage-free, and we denote by P∗ the unique equivalent martingale measure.

Remark 5.40. Note that the unique martingale measure P∗, and hence the valuation of any contingent claim, is completely independent of the initial choice of the “objective” measure P within the class of measures satisfying (5.26).

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Let us now turn to the problem of pricing and hedging a given contingent claim C. The discounted claim can be written as

for a suitable function h.

Proposition 5.41. The value process

of a replicating strategy for H is of the form

where the function vt is given by

Proof. Clearly,

Each quotient St+s/St is independent of Ft and has under P∗ the same distribution as

Hence (5.28) follows from the standard properties of conditional expectations.

Since V is characterized by the recursion

we obtain a recursive formula for the functions vt defined in (5.28):

where

Example 5.42. If H = h(ST) depends only on the terminal value ST of the stock price, then Vt depends only on the value St of the current stock price:

Moreover, the formula (5.28) for vt reduces to an expectation with respect to the binomial distribution with parameter p∗:

In particular, the unique arbitrage-free price of H is given by

For h(x) = (x−K)+/(1+r)T or h(x) = (K−x)+/(1+r)T ,we obtain explicit formulas for the arbitrage-free prices of European call or put options with strike price K. For instance, the price of Hcall := (ST − K)+/(1 + r)T is given by

Example 5.43. Denote by

the running maximum of S, and consider a discounted claim H = h(ST , M T). For instance, H can be an up-and-in or up-and-out barrier option or a lookback put. Then the value process of H is of the form

where

This follows from (5.28) or directly from the fact that

where maxt≤u≤T Su/St is independent of Ft and has the same law as MT−t/S0 under P∗. The same argument works for options that depend on the minimum of the stock price such as lookback calls or down-and-in barrier options.

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Exercise 5.5.1. For an Asian option depending on the average price

during a predetermined set of periods ⊂ {0, . . . , T}, we introduce the process

Show that the value process Vt of the Asian option is a function of S t, At, and t.

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Let us now derive a formula for the hedging strategy = (ξ0, ξ) of our discounted claim H = h(S0, . . . , ST).

Proposition 5.44. The hedging strategy is given by

where

Proof. For each ω = (y1, . . . , yT), ξt must satisfy

In this equation, the random variables ξt, X t−1, and Vt−1 depend only on the first t −1 components of ω. For a fixed t, let us define ω+ and ω− by

Plugging ω+ and ω− into (5.30) shows

Solving for ξt(ω) and using our formula (5.27) for Vt, we obtain

Remark 5.45. The term Δt may be viewed as a discrete “derivative” of the value function vt with respect to the possible stock price changes. In financial language, a hedging strategy based on a derivative of the value process is often called a Delta hedge.

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Remark 5.46. Let H = h(ST) be a discounted claim which depends on the terminal value of S by way of an increasing function h. For instance, h can be the discounted payoff function h(x) = (x − K)+/(1 + r)T of a European call option. Then

is also increasing in x, and so the hedging strategy satisfies

In other words, the hedging strategy for H does not involve short sales of the risky asset.

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Exercise 5.5.2. Let T0 ∈ {1, . . . , T − 1} and K > 0. The payoff of forward starting call option has the form

Determine its arbitrage-free price and hedging strategy.

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