Multi-step general models
4.5 A general multi-step model
The multi-step models we have discussed so far all involve a market with a single underlying stock. We now analyse derivative securities defined on an underlying market containing several risky assets (stocks) and a single money market account. As we saw for trinomial models, where market completeness requires two underlying stocks, pricing and hedging deriva-tives in such a model proceeds much as in the single-stock case, and it is often useful to treat the vector of stock prices as a single entity. With this proviso, much of theory we have developed so far will generalise to this more general setting quite simply, and we begin by reviewing the main concepts.
We still assume that time runs in a discrete way, just as in the binomial model. With h being the length of the time step we use n = 0, 1, . . . , N to indicate the time instant nh. The basic market consists of the money market account and d risky assets (stocks) with prices S1(n), . . . , Sd(n), which comprise the components of the Rd-valued function S(n) and are discrete strictly positive random variables defined on the finite sample spaceΩ = {ω1, . . . , ωM}. The dynamics of the single risk-free asset (the money market account) are again determined by the risk-free rate R, which is assumed constant, so A(n) = A(0)(1 + R)n. A portfolio is the vector
(x, y) = (x1, . . . , xd, y) with coordinates representing positions in the corre-sponding securities. A strategy is a sequence of portfolios
(x(n), y(n)) = (x1(n), . . . , xd(n), y(n)), n = 1, . . . , N.
These areRd+1-valued functions (i.e. random vectors), except for the initial portfolio, which is determined at time 0 and so is deterministic. We shall write (x, y) for this sequence.
Later, this market will be extended by adding derivative securities and the portfolio vectors will contain additional coordinates zk, reflecting posi-tions in these assets.
The value of a strategy at time n is
V(x,y)(n)=
d j=1
xj(n)Sj(n)+ y(n)A(n), for n = 1, 2, . . . , N,
V(x,y)(0)=
d j=1
xj(1)Sj(0)+ y(1)A(0).
We shall again assume throughout that the strategies we consider are predictable, so the random vectors (x(n+ 1), y(n + 1)) are Fn-measurable.
This condition reflects the convention that the portfolio at time n+ 1 is constructed at time n using the funds and the information about stock prices available at that time.
The key ideas developed for the single-stock model also apply in this more general model. A strategy is self-financing if
V(x,y)(n)=
d j=1
xj(n+ 1)Sj(n)+ y(n + 1)A(n) for n = 0, 1, . . . , N − 1.
We state the version of the No Arbitrage Principle which applies in the model as follows:
Definition 4.25
A strategy (x, y) is an arbitrage opportunity in the underlying market if its value process satisfies V(x,y)(0)= 0, V(x,y)(n)≥ 0 for all n and for some n there is anω such that V(x,y)(n, ω) > 0. For the market extended by adding some derivative securities the definition is similar.
The construction given for the single-stock model in Proposition 4.24 also applies in the multi-stock setting: the only change is to replace the stock holdings x(n) by the vector x(n).
Assumption 4.26No Arbitrage Principle
Arbitrage opportunities do not exist in any market model.
In the First Fundamental Theorem below we shall prove that this is equivalent to the existence of a martingale probability; that is a probability Q with Q(ω) > 0 for all ω in Ω and such that for each j = 1, . . . , d, the discounted stock price process{ Sj(n) : n= 0, 1, . . . , N} is a Q-martingale.
The following definition is both intuitive and useful.
Definition 4.27
The gains process G(x,y)= (G(x,y)(n))n≤ Ngenerated by the strategy (x, y) is defined by
G(x,y)(0)= 0, G(x,y)(n)=
n k=1
d
j=1
xj(k)ΔSj(k)+ y(k)ΔA(k)
for n= 1, . . . , N.
It follows immediately that the strategy (x, y) is self-financing if and only ifΔV(x,y)(n)= ΔG(x,y)(n) for n≥ 1. Since V(x,y)(n)= V(x,y)(0)+n
i=1ΔV(x,y)(i) is a telescoping sum, and similarly for G, the strategy is self-financing if and only if
V(x,y)(n)= V(x,y)(0)+ G(x,y)(n) for n= 0, 1, . . . , N.
In self-financing and predictable strategies the risk-free position is a sec-ondary variable as the following theorem shows.
Theorem 4.28
Given V(0) and a predictable sequence x= (x1(n), . . . , xd(n)) there exists a unique predictable sequence y(n) such that
• V(x,y)(0)= V(0),
• the strategy (x, y) is self-financing.
Proof This is proved by induction. First, let
y(1)= V(0)− x1(1)S1(0)− · · · − xd(1)Sd(0)
A(0) ,
which ensures that V(x,y)(0)= V(0). Thus y(1) is non-random, as is neces-sary for y(n) to be predictable. By definition, we have therefore determined V(x,y)(1) uniquely, and the initial induction step is complete.
V(x,y)(1)= x1(1)S1(1)+ · · · + xd(1)Sd(1)+ y(1)A(1).
Next, for the induction step, assume that y(n) is known; thus so is V(x,y)(n).
We again define the next term in such a way that the self-financing condi-tion is guaranteed:
y(n+ 1) = V(x,y)(n)− x1(n+ 1)S1(n)− · · · − xd(n+ 1)Sd(n)
A(n) .
Then y(n+ 1) is Fn-measurable since for j= 1, 2, . . . , d both the xj(n+ 1) and Sj(n) are; the former by hypothesis, the latter by definition of theFn.
This completes the induction step.
Recall the definition of discounted prices; in fact, for any asset X we set
X(i) = X(i) (1+ R)i. Then the successive differences are
Δ X(i)= X(i)
(1+ R)i − X(i− 1) (1+ R)i−1 which is not the same as
ΔX(i) = ΔX(i)(1 + R)! −i= X(i)
The discounted gains process will be denoted by G x(n)= vectors inRd. The vector Δ S(i) has the discounted price increments Δ Sj(i) as its components. As the discounted increment in A is zero, there is now no dependence on y, only on x.
We now verify that for self-financing strategies the sum of the initial investment and discounted gains gives the discounted value process.
Theorem 4.29
If a strategy (x, y) is self-financing then
V(x,y)(n)= V(x,y)(0)+ Gx(n).
Proof Fix n≤ N and suppose (x1(n), . . . , xd(n), y(n)) is self-financing. We write
V(x,y)(i+ 1) = 1
(1+ R)i+1( x(i + 1), S(i + 1) + y(i + 1)A(i + 1))
= x(i + 1), S(i + 1) + y(i + 1)A(0), and similarly
V(x,y)(i)= 1
(1+ R)i( x(i + 1), S(i) + y(i + 1)A(i))
= x(i + 1), S(i) + y(i + 1)A(0).
Subtracting gives, for each i= 0, 1, . . . , n − 1
Δ V(x,y)(i+ 1) = V(x,y)(i+ 1) − V(x,y)(i)= x(i + 1), Δ S(i + 1)
= Δ Gx(i+ 1), so that with k= i + 1 we obtain
V(x,y)(n)− V(x,y)(0)=
n k=1
Δ V(x,y)(k)=
n k=1
Δ Gx(k)= Gx(n).
A strictly positive price process is often called a numeraire, i.e. a bench-mark security against which the price changes of other securities can be compared.
Exercise 4.6 Given an Fn-adapted sequence of d-vectors with Zj(n)> 0 for all n = 0, 1, . . . , N, j = 0, 1, . . . , d with (x, y) and (S, A) as above, show that the strategy (x,y) is self-financing for the price process (S, A) if and only if it is self-financing for the price process (Z1S1, . . . , ZdSd, Z0A).
The exercise verifies that the self-financing property of trading strategies is invariant under a change of numeraire.
Exercise 4.7 Explain why the result of the previous exercise shows that the converse of Theorem 4.29 also holds.
To discuss martingale properties in general we introduce a filtration of fieldsFnas before. Then we have the familiar definition:
Definition 4.30
A probability Q is risk-neutral (or a martingale probability) if the dis-counted stock prices are martingales
EQ( Sj(n+ 1)|Fn)= Sj(n) all j,
where for each n, Fn is generated by Sj(k), k ≤ n, j = 1, . . . , d (i.e. the smallest field for which all Sj(k), k ≤ n, are measurable).
We proved that if a predictable strategy (x, y) is self-financing, then V(x,y)(n)= V(x,y)(0)+ Gx(n).
We shall prove that under a risk-neutral probability V(x,y)(n) is a martin-gale as well, thus generalising Theorem 4.20. To this end we formulate the appropriate theorem using general notation since it can be used in various applications.
Theorem 4.31
If Mj(n) are martingales and the processes Hj(n) are predictable, j = 1, . . . , d, then the process X(n) defined by
X(n)= X(0) +
n k=1
d j=1
Hj(k)ΔMj(k)= X(0) +
n k=1
H(k), ΔM(k)
is a martingale, where X(0)= x0is an arbitrary real number.
This right-hand side of the above equation is sometimes called a mar-tingale transform or discrete stochastic integral.
Proof See page 108.
We will also need the following result.
Theorem 4.32
For M(n) to be a martingale it is sufficient that for each predictable process H(n)
E N
n=1
H(n)ΔM(n))
= 0.
Proof See page 109.