A model of the nancial market

In order to apply the theory of the previous chapters to mathematical nance, one must construct a mathematical model of the nancial market. There are many possible ways to do this, but the following general model is a good foun-dation to build on.

The nancial market model is based on a probability space (Ω,F, P) con-sisting of a space of possible scenarios of the world Ω, a σ-algebra F, and a probability measureP on the measurable space(Ω,F).

The nancial market consists ofN + 1assets: N risky assets (stocks) and one bond (the bank). The assets each have a price process S_n(t, ω), for n = 0, ..., N, ω ∈ Ωand t ∈[0, T], where S₀(t, ω)is the price process of the bond.

The price processes S_n, n = 1, ..., N, are stochastic processes. We denote by S(t, ω) = (S₀(t, ω), S₁(t, ω)..., S_N(t, ω)), the composed price process of all the assets. Here, the timet∈[0, T]where the nal timeT may be innite. Though the nal time may be innite, one often assumes T < ∞ in mathematical nance.

Let (Ft)_0≤t≤T be a ltration. Usually, one assumes that the price process S is adapted to the ltration(Ft)t(S being adapted to(Ft)tmeans thatSn(t) is Ft-measurable for all t ∈ [0, T]), and it is very common to let (Ft)t be the ltration generated by the price process S. Actually, the price process is often assumed to be an (Ft)t-semimartingale. The precise denition of a semimartingale will not be discussed in this thesis, but the main point is that whenSn is a semimartingale, one can form the Itô-integral with respect to this stochastic process. See for example Shilling [41], or Øksendal [27], for more on ltrations andσ-algebras.

One often assumes that S₀(t, ω) = 1 for all t ∈ [0, T], ω ∈ Ω. This corre-sponds to having divided through all the other prices byS₀, and hence turning the bank into the numeraire of the market. The altered market is a discounted (or normalized) market. This text mainly considers discounted markets, but whether the market is discounted or not will be mentioned when necessary. To simplify notation, the price processes in the discounted market are denoted by S as well. Note that for a given probability space(Ω,F, P), the price processS describe the market. Therefore the market is sometimes denoted simply byS.

An example of a security market as the one above is the model used in Øksendal [27]:

S0(t, ω) = 1 + Z t

0

r(s, ω)S0(s, ω)ds (4.1) and forn= 1, ..., N

Sn(t, ω) =xn+ Z t

0

µn(s, ω)ds+ Z t

0

σn(s, ω)dB(s, ω) (4.2) or, written in brief form

dS₀(t) =r(t)S₀(t)dt, S₀(0) = 1 (4.3) and forn= 1, ..., N

dS_n(t) =µ_n(t)dt+σ_n(t)dB(t), S_n(0) =x_n. (4.4) whereB(t)is a D-dimensional Brownian motion.

Here, the process r called the interest rate process, µn is called the drift process (of asset n) and σ_n is called the volatility process (of asset n). Note that sometimes, as above, the dependence on ω is suppressed for notational convenience. The Brownian motion B may be more than one-dimensional, in

4.2. A MODEL OF THE FINANCIAL MARKET 51 general, B is D-dimensional. Therefore, σ_n is also D-dimensional (for all n).

IfD >1, the stochastic integral (Itô-integral)σ_n(t)dB(t) :=PD

i=1σ_nⁱ(t)dB_i(t), where σ_nⁱ(t) and B_i(t)denotes the i'th components of σ_n(t)and B(t) respec-tively. This Brownian motion driven market model will be central in this text.

As mentioned, the space Ω represents the possible scenarios of the world, and the probability measureP gives the probabilities for each of the measurable subsets ofΩto occur. As in real nancial markets, we are interested in modeling how information reveals itself to the investors, and that is where the ltration (Ft)_0≤t≤T is useful. This ltration represents what the investors know at timet. For instance, if(Ft)tis the ltration generated by the price processS, the agents in the market know the asset prices at any timet, but nothing more. For a nite Ωand discrete time, using ltrations to model information can be explained by the bijection between partitions andσ-algebras (theσ-algebra consists of every union of elements in the partition). The nested chain of partitions is a convenient way to model how the world scenarios reveal themselves, see Figure 4.1. When Ω is innite, the correspondence between partitions and ltrations disappears, so the use of a ltration is seen as a generalization of the nite Ωcase.

Trading of assets is essential in real nancial markets, so one must also model the concept of a portfolio. A portfolio (or a trading strategy) is anN+ 1 -dimensional, predictableS-integrable stochastic processH(t, ω) = (H₀(t, ω), H1(t, ω), ..., HN(t, ω)). Hn(t, ω) represents the amount of asset n held by the investor at timetin stateω∈Ω. In general, a predictable stochastic process is dened as a stochastic process which is measurable (when the process is viewed as a function fromR+×ΩintoR) with respect to a specialσ-algebraPR, called the predictableσ-algebra, on the product spaceR+×Ω. PRis constructed from predictable rectangles. A predictable rectangle is a subset ofR+×Ωof the form (s, t]×F, where s < tand F ∈ Fs, or of the form {0} ×F0, where F0 ∈ F0. PRis theσ-algebra generated by the collection of all predictable rectangles. In discrete time, being predictable means that H(t,·) is Ft−1-measurable for all t ∈ {1,2, ..., T}, whereHn(t, ω)represents the amount of asset number nheld from time t−1to timet. Hence, predictability means that the investor has to choose how much to hold of asset n between times t−1 and t based on what she knows at time t−1, that is F_t−1. The abstract notion of a predictable process is a generalization of this concept. Note that all predictable processes are adapted. However, it turns out that it does not matter whether one considers predictable or adapted trading strategies as long as the price processSis not a jump processes.

A portfolio H is called admissible if there exists a constant C < 0 such that Rt

0H(u)dS(u) := Rt

0H(u)·dS(u) > C P-almost surely for all t ∈ [0, T] (where the stochastic processes H(t, ω) = (H0(t, ω), H1(t, ω), ..., HN(t, ω))and S(t, ω) = (S0(t, ω), S1(t, ω), ..., SN(t, ω))are viewed as vectors in R^N+1, and · denotes the standard Euclidean inner product). The family of admissible trading strategies will be denoted by H.

The value of a portfolioH at timetis the random variable denoted byX^H(t)

and dened

X^H(t, ω) :=

N

X

n=0

H_n(t, ω)S_n(t, ω). (4.5) That is, the value of the investor's portfolio at a time t is the sum of the amount she owns of each asset, times the price of that asset at time t. When the underlying portfolio is clear, we sometimes denote X = X^H, to simplify notation.

A portfolio H is called self-nancing if it is an S-integrable process which satises

X^H(t) =X^H(0) + Z t

0

H(u)·dS(u)for all0≤t≤T

A portfolio being self-nancing means that no money is taken in or out of the system. That is, all trading of assets after time t = 0 is nanced by the price changes in the market, which aect the value of the portfolio.

An arbitrage in a nancial market is, roughly speaking, a riskless way of making money. More formally, one says that the market (S_t)_0≤t≤T has an arbitrage if there exists an admissible portfolioH such that

(i) X^H(0) = 0,

(ii) X^H(T)≥0almost surely, (iii) P(X^H(T)>0)>0.

If there was an arbitrage in the market, all investors would want to execute the arbitrage, and (possibly) cash in a riskless prot. Economic principles then suggest that the demand for the arbitrage opportunity would cause the prices to adjust such that the arbitrage disappears. Therefore, an arbitrage is a sign of lack of equilibrium in the market.

For a potential investor, it is interesting to determine whether a market has an arbitrage or not. In order to do this, the notion of equivalent martingale measure is useful.

An equivalent martingale measure is a probability measure Q on the mea-surable space(Ω,F)such that

(i) Qis equivalent toP, and

(ii) The price process S is a martingale with respect to Q (and w.r.t. the ltration (Ft)t).

Denote byM^e(S)the set of all equivalent martingale measures for the market S. Then one has the following theorem, which is Lemma 12.1.6 in Øksendal [27]

(for the proof of this theorem, see [27]).

4.2. A MODEL OF THE FINANCIAL MARKET 53 Theorem 4.2 Let the market be dened as in equations (4.3) and (4.4). If the set of equivalent martingale measures in nonempty, i.e. M^e(S)6=∅, then there is no arbitrage.

IfΩ is nite and time is discrete, one can attempt to nd equivalent mar-tingale measures by solving the set of linear equations (4.6) for Q

EQ[S(t)|Fu] = S(u)for all0≤u≤t≤T (4.6) When this system is solved, one must check if any of the solutions are probability measures on(Ω,F)equivalent toP (i.e. such thatP andQhave the same null sets).

It turns out that one can have either M^e(S) = ∅, M^e(S) = {Q} (a one-element set) or |M^e(S)| = ∞. Why is this? Assume there are at least two distinct equivalent martingale measures Q, Q^∗ ∈ M^e(S). Let λ ∈(0,1), then Q¯ = λQ+ (1−λ)Q^∗ ∈ M^e(S): Clearly, Q¯ is a probability measure. It is a martingale measure because EQ¯[S_t|Fu] =E_{λQ+(1−λ)Q}∗[S_t|Fu] =λE_Q[S_t|Fu] + (1−λ)E_Q^∗[S_t|Fu] =λS_u+ (1−λ)S_u=S_uforu < t. Finally,Q¯ is equivalent to P because ifAis aP-null set, then,Ais aQ- andQ^∗-null set, and therefore also a Q¯-null set. Conversely, ifA is aQ¯-null set, then it must be both aQ- and a Q^∗-null set. Hence, sinceQandQ^∗are equivalent toP,Amust also be aP-null set. Note that this also proves that M^e(S)is a convex set, see Denition 2.1 (actually, M^e(S) being convex implies that M^e(S) is either the empty set, a one-element set or an innite set).

In mathematical nance, one usually assumes thatM^e(S)6= ∅, since this (from Theorem 4.2) guarantees that the market is arbitrage free. Actually, Theorem 4.2 can be proved to "almost" hold with if and only if. If the market satises a condition similar to the no arbitrage condition called "no free lunch with vanishing risk" (abbreviated NFLVR), then there exists an equivalent mar-tingale measure (see Øksendal [27]). This is sometimes called the fundamental theorem of asset pricing.

Assume M^e(S) 6= ∅, so the market has no arbitrage from Theorem 4.2.

A contingent claim (or just claim) B is a nancial contract where the seller promises to pay the buyer a random amount of money B(ω) at time T, de-pending on which state of the world, ω ∈ Ω occurs. Formally, a claim is a lower bounded, FT-measurable random variable. If all contingent claims can be replicated by trade in the market, the market is called complete. More for-mally, this means that for any claimB there exists a self-nancing, admissible trading strategy H such that B(ω) = H(0)·S(0) +RT

0 H(t, ω)dS(t, ω)for all ω ∈ Ω. This can be shown to be equivalent toM^e(S) = {Q} (this is due to Harrison and Pliska (1983), and Jacod (1979), see Øksendal [27]). If there exists contingent claims which cannot be replicated by trade, the market is called in-complete. In this case,|M^e(S)|=∞. In the real world, nancial markets tend to be incomplete, because in order to have a complete market one must, among other things, have full information and no trading costs. This is not a realistic situation. However, incomplete markets are mathematically more dicult to handle than complete markets.