A Solution to Eliminate Drawbacks in the USD E- E-learning Environment

Assume given a riskless asset A with A(0) = 1 (we keep this assump-tion throughout this secassump-tion to simplify notaassump-tion), accruing interest at rate R, together with an arbitrary finite number of assets with prices S₁(n), . . . , Sd(n), which we consider at times n = 0, 1. We assume that for i= 1, . . . , d the prices Si(0) are known (non-random) and that at time 1 each Si(1) is a (positive) random variable defined on a finite sample space Ω = {ω1, . . . , ωM}, with a probability measure P defined on all the subsets ofΩ by the formula

P(A)=

ω∈A

P(ω)

where P(ωj) = pj ∈ (0, 1) for j = 1, . . . , M, andM

j=1pj = 1. A random variable is now a function

X:{ω1, . . . , ωM} → R with expectation given by

E(X) =

M j=1

pjX(ωj).

The random vector of prices (S₁(1), . . . , Sd(1)) is built of d random vari-ables, each taking at most M different values. In fact, we shall assume that altogether M different values appear at time 1. This ensures that all M elements ofΩ are ‘needed’, since if the random variables Si(1), i ≤ d, had fewer values than there are elements of the domain Ω, this would mean that for some ω1, ω2 all stock prices are the same and so some of the ω’s were redundant and could be removed from Ω. As a conse-quence of this natural non-degeneracy assumption, in the same way as in the binomial or trinomial model, any random variable X: Ω → R can be written in the form X = h(S1(1), . . . , Sd(1)) for some function h:R^d→ R.

These data constitute a finite single-step market model. The notion of a portfolio must be adjusted to incorporate d stocks: it is a (d+ 1)-vector

(x, y) = (x1, . . . , xd, y) whose coordinates indicate the stock holdings and risk-free investment respectively. The initial value of the portfolio (x, y) is V_(x,y)(0) = d

i=1xiSi(0) + yA(0), while its final value is the random variable

V_(x,y)(1)=

d i=1

xiSi(1)+ yA(1).

Note that fixing the initial value V(0) and the vector x of stock hold-ings of a portfolio automatically determines the bond holding y. This is true, in particular, for arbitrage opportunities: recall that a portfolio is an arbitrage opportunity if V(0) = 0, V(1) ≥ 0, and V(1) > 0 with a pos-itive probability, that is, in our setting, if V(1, ω) is positive for at least oneω ∈ Ω.

The first fundamental theorem of mathematical finance, which we now formulate and prove in this general setting, gives an equivalent condition for the absence of arbitrage.

First fundamental theorem

We consider a simple version of this key result, applicable to the single-step model. The general multi-step case will be considered in the next chapter.

First we note what is meant by a risk-neutral probability measure in this context:

Definition 2.33

The sequence Q= {q1, . . . , qM} of positive numbers is a risk-neutral prob-ability measure forΩ = {ω1, . . . , ωM} and stock prices {(Si(0), Si(1)) : i= 1, . . . , d} ifM

j=1qj= 1 and E^Q(Si(1))= (1 + R)Sⁱ(0) for all i, whereE^Q de-notes the expectation with respect to the probability Q(ωj)= qj> 0 for j ≤ M, and R denotes the riskless interest rate. Q is often called a (single-step) martingale probability for the stock price processes{Si: i= 1, . . . , d}.

Theorem 2.34

In a finite single-step market model there is no arbitrage if and only if there exists a risk-neutral probability measure.

Proof Recall that we assume A(0) = 1. First suppose that the model allows risk-neutral probabilities. Consider any portfolio with V(0) = 0.

Assume that V(1) ≥ 0 (otherwise this portfolio is not an arbitrage

opportunity). Compute

This implies that V(1) = 0 for all ω since zero expectation of a non-negative random variable forces this random variable to be identically zero, hence there are no arbitrage opportunities.

For the converse implication, assume that the pricing model does not allow arbitrage. Define the discounted gains of the portfolio

(x, y) = (x1, x2, . . . , xd, y) as the random variable Gxwith values

Gx(ω) =

forω ∈ Ω. Note that y does not appear, since the discounted gains of A are zero by definition. Gxis well-defined for any vector x inR^d.

The set

is a vector space, as it is built of discounted portfolio gains. It can be identified with a subspace of R^M, writing Gx = (Gx(ωj))j≤M, because Ω = {ω1, . . . , ωM} has M elements.

We verify the following property of the elements of W: if Gx(ω) ≥ 0 for allω then Gx(ω) = 0 for all ω. To see this, take x = (xi)i≤d such that Gx(ω) ≥ 0 for all ω. Ensure that the portfolio (x, y) has zero initial wealth V(0) = 0 by supplementing the xi by a suitable y. The discounted final

value of this portfolio is

The risk-neutral probabilities we seek belong to the following subset of R^M:

We shall construct a vector Q in A with positive coordinates, and orthogo-nal to the subspace W. The coordinates of such a vector give the required probabilities qj, since orthogonality means that the Euclidean inner prod-uct of (q₁, . . . , qM) and (Gx(ω1), . . . , Gx(ωM)) vanishes for each Gx in W, that is, withE^Qas in the definition of risk-neutrality,

EQ(Gx)= we shall also use later, when the multi-step version of the theorem will be presented. as above, note that A is obviously convex. It is compact, as it is closed

and bounded (Heine–Borel theorem!). Since Gx ≥ 0 implies Gx = 0, it follows that A∩ W = ∅. Now take (a1, ..., aM) in A with aj = 1 for some j (and so with the remaining coordinates necessarily zero) hence the lemma, applied toR^M, provides (z1, ..., zM) with zj > 0 for all j = 1, . . . , M and

M

j=1zjGx(ωj)= 0 for all Gxin W. So we need only take qj= 1

M i=1zi

zj

for j≤ M to ensure thatM

j=1qj= 1, while all qj> 0 and

M j=1

qjGx(ωj)= 0

for all Gxin W. This completes the proof of the theorem.  Remark 2.36

We have shown that for finite single-step models the existence of a risk-neutral probability assignment or martingale probability is equivalent to the requirement that the model does not allow arbitrage. We shall show in Chapter 3 that this remains true for multi-step finite market models in general. Our proof there will depend heavily on the single-step case, and crucially on the finiteness of the market model, that is, finiteness of the sample spaceΩ.

Next we turn our attention to the question of completeness.

Second fundamental theorem

The trinomial model is an example of an incomplete market model, the defining feature being the lack of replicating portfolios for some derivative securities. The binomial model, on the other hand, allows for such a repli-cation of all claims and is an example of a complete market model. We re-state the definition more generally. Recall that our market model con-tains d risky securities, whose prices we denote, for ease of notation, as a random vector S(1) = (S1(1), . . . , Sd(1)). The value at 1 of the portfo-lio (x, y) is the random variable V(x,y)(1) = x, S(1) + yA(1), where ., . denotes the inner product inR^d.

Definition 2.37

An arbitrage-free market model is complete if for each derivative security H = h(S(1)) there exists a replicating portfolio.

Any random variable H is a derivative security in our market model, since its value is completely determined by the price vector S(1) and it can be written in the form H = h(S(1)) for some h: R^d → R, which is a consequence of the non-degeneracy assumption made at the beginning of this section. In particular, for anyω ∈ Ω, 1{ω}is a derivative security. We now show that in a complete market model all risk-neutral probabilities coincide.

Theorem 2.38

An arbitrage-free market model is complete if and only if there is exactly one risk-neutral probability.

Proof No arbitrage implies the existence of a martingale probability, so we need to show that we cannot have more than one such probability in a complete market. Suppose we have two martingale probabilities Q₁  Q₂. For any ω ∈ Ω the random variable 1_{ω} is a derivative security. By completeness of the market model there exists a replicating portfolio, given by (x, y) such that

x, S(1) + y(1 + R) = 1{ω}. Take the expectation with respect to Qk, k = 1, 2:

EQ_k(x, S(1) + y(1 + R)) = EQ_k(1_{ω})= Qi(ω) and note that the left-hand side is independent of k, since

EQ_k(x, S(1) + y(1 + R)) = x, EQ_k(S(1)) + y(1 + R)

= x, S(0) (1 + R) + y(1 + R) so that

Q₁(ω) = Q2(ω).

Butω ∈ Ω was arbitrary, so the probabilities Q1, Q2coincide..

For the converse implication, assume Q is the unique martingale proba-bility and suppose there is a derivative D, which cannot be replicated. Our goal will be to construct a martingale probability Q₁different from Q.

Consider the set of values at time 1 of all portfolios: W = {V(x,y)(1) : (x, y) ∈ R^d+1}. Since none replicates D, we conclude that D  W. We treat W as a vector subspace ofR^Mby identifyingR^Mwith the set of all random variables onΩ = {ω1, . . . , ωM}. The subspace W contains the vector 1 = (1, . . . , 1) because if x = 0 and y = _1+R¹ then V_(x,y)(1)= 1 for all scenarios.

Now equipR^Mwith the inner productw, v Q =M

j=1wjvjQ(ωj) and use

the following version of Lemma 2.35 with the subspace W and the compact henceEQ(Z)= 0, where the expectation is taken under Q.

Now we define a new measure, which will turn out to be a martingale probability different from Q

Q₁(ωj)= for at least one coordinate it is different from 1 since Z is non-zero, thus Q₁  Q. Finally, we check the martingale property of each Si(for i≤ d)

We can now apply the fundamental theorems to option pricing in com-plete models satisfying the No Arbitrage Principle. Suppose the replicating portfolio is (x, y) = (x1, . . . , xd, y)

V_(x,y)(1)= H(1), so that H(0) = V(x,y)(0)

since otherwise we have an arbitrage opportunity. The expectation with respect to the unique risk-neutral probability gives

EQ(V_(x,y)(1))= EQ(x, S(1) + y(1 + R))

=

d i=1

xiEQ(Si(1))+ y(1 + R)

= (1 + R) ^d

i=1

xiSi(0)+ y

= (1 + R)V(x,y)(0)= (1 + R)H(0)

where we have used the martingale property of all stock prices. Since we have replication we obtain

H(0)= 1

1+ REQ(H(1)).

Finally we note that the sub- and super-replicating prices coincide in com-plete models.

Theorem 2.40

Completeness implies H^sub= H^super= H, say.

Proof If (x, y) = (x1, . . . , xd, y) is a replicating portfolio, V(x,y)(1) = H(1) and (x, y) is any super-replicating portfolio, V(x,y)(1)≥ H(1), then

V_(x,y)(1)≥ V(x,y)(1). The No Arbitrage Principle implies

V_(x,y)(0)≥ V(x,y)(0)

so taking the infimum over super-replicating portfolios we have inf

(x,y){V(x,y)(0)} ≥ V(x,y)(0).

But the replicating price belongs to the set of all super-replicating prices V_(x,y)(0)∈ {V(x,y)(0) : (x, y) super-replicating}

so the infimum is attained and inf

(x,y){V(x,y)(0)} = V(x,y)(0).

The argument for sub-replicating portfolios is similar and leads to V_(x,y)(0)= sup

(x,y)

{V(x,y)(0)},

where the supremum is taken over all sub-replicating portfolios.