3 Arbitrage pricing theory in discrete time.

Transcription

1 3 Arbitrage pricing theory in discrete time. Orientation. In the examples studied in Chapter 1, we worked with a single period model and Gaussian returns; in this Chapter, we shall drop these assumptions and investigate very simple discrete-time models, beginning with single-period models, and later moving on to multi-period models. We shall continue with the notation already introduced: there are d risky assets, the price of the k th at (integer) time t being denoted S k t, and we shall sometimes also suppose that there is a riskless zero th asset. We use the notations S t (S 1 t,...,s d t ) T, St (S 0 t,s 1 t,...,s d t ) T. Where we are going in the first part of this Chapter is to show that in any reasonable market there exists a positive process ζ such that (ζ t S t ) t 0 is a martingale: ζ t S t = E t [ζ T S T ] for all 0 t T (3.1) where E t denotes conditional expectation given what is known at time t. To try to understand what this is saying, suppose we take t =0andmakethe benign assumption that E 0 = E; then (3.1) says S 0 = E[S T ζ T /ζ 0 ], (3.2) which is a pricing identity, akin to (1.13). In fact, we can arrive informally at (3.2) by very similar arguments. To do this, consider an agent who maximises his expected utility of wealth at time T, max E[U(X T )], X T A T where A T is the set of all attainable wealths which can be obtained by trading from his initial time-0 wealth. Suppose that the optimum is attained at XT. This agent now considers whether to buy a small amount ε of a contingent claim Y to be paid at time T, at price επ. So now he is considering the slightly perturbed optimisation problem max E[U(X T + ε(y π))], X T A T... or is he? In fact, this cannot be correct, because the price επ is paid at time 0, and cannot necessarily be passed forward to time period T ; all that can be done is that the price at time 0 is converted into an asset at time T by the agent selling an asset which is in the market. Thus we shall suppose that there is some (so-called) numeraire asset N which is in the market, and the agent sells the appropriate number of units of N. This means that the agent has the optimisation problem max E[U(X T + ε(y πn T /N 0 ))], X T A T 35

2 and the now-familiar first-order argument leads us to conclude that E[U (X T )Y ]=πe[u (X T )N T /N 0 ]. Comparing this with (3.2), we see that marginal-expected-utility pricing prompts us to look for ζ T of the form ζ T = U (XT ). This is exactly what we are going to do; when we prove (3.1) we will actually exploit the optimal portfolio of a CARA agent as a mathematical tool to construct ζ... but this will all become clear later. (End of orientation.) Once we drop the assumption that S 1 has a multivariate Gaussian distribution, an undesirable possibility arises. Definition 2 Assume St 0 if =1for all t 0. A portfolio θ is called an arbitrage P[ θ S 1 0] = 1 = P[ θ S 0 0], and P[ θ ( S 1 S 0 ) > 0]> 0. (3.3) Remarks. The assumption S 0 t = 1 is not restrictive. It would suffice that S 0 (or any other asset N among our d+1 assets), remains strictly positive. We can then quote discounted prices in terms of N and apply the above definition to S/N. An arbitrage is a way to make money without risk; the portfolio costs nothing (or less than nothing) at time 0, is worth at least something at time 1, and is not identically zero! It is hard to believe in a model which allows arbitrage; we reject all such on economic grounds. It turns out, however, that the absence of arbitrage has an equivalent (and extremely useful) mathematical characterisation. To state this, we need some more terminology. Definition 3 A probability Q is absolutely continuous with respect to P if there is an integrable non-negative function f such that for all events A Q(A) = fdp. The function f is referred to as the density of Q with respect to P, and the notation f = dq dp is used. When f>0 P-almost-surely, we say that Q and P are equivalent. Remarks. The Radon-Nikodym Theorem states that Q is absolutely continuous with respect to P if and only if every P-null event (that is, an event A for which P(A) = 0) is also Q-null. 36 A

3 Theorem 1 Assume that S 0 t =1for all t 0. Then the following are equivalent: (i) There is no arbitrage; (ii) There exists some probability Q equivalent to P such that When this condition holds, we may take for some θ R n. E Q [S 1 ]=S 0 (3.4) dq dp exp( θ S S 1 2 ) (3.5) Proof. Write X S 1 S 0 for brevity. We shall without loss of generality make the non-degeneracy assumption that P(θ X = 0)< 1 for all non-zero θ R n, for otherwise X lies in some proper subspace of R d, and there are linear dependencies among the assets. We could then discard redundant assets and reduce to a set for which the non-degeneracy assumption holds. (ii) (i) :Ifθ were an arbitrage, we shall have from (3.4) that E Q [θ S 1 ]=θ S 0, (3.6) and the left-hand side is non-negative, the right-hand side is non-positive, so both must be zero. But this means that P(θ S 1 =0)=1=P(θ S 0 =0),soθ is not an arbitrage. (i) (ii) : We must prove the existence of some equivalent martingale measure assuming that there is no arbitrage. This is somewhat more involved, but we will actually construct such a measure, using the principle of maximisation of expected utility; the result (1.7) is in effect what we need. For this, define the function θ ϕ(θ) E exp( θ X 1 2 X 2 ) E exp( 1. 2 X 2 ) This function is finite-valued 9, non-negative, continuous, convex and differentiable. If inf θ ϕ(θ) is attained at some θ, then by differentiating we learn that E[exp( θ X 1 2 X 2 ) X] =0, 9 The interpretation of the proof in terms of a utility-maximisation argument is far more direct if we had simply used ϕ 0 (θ) =E exp( θ X), for then we are literally maximising the CARA utility of θ X. The snag is that this expectation need not be finite for all θ, whereas the definition given for ϕ is certain to be finite-valued. If ϕ 0 were finite-valued, we could have used this instead of ϕ. 37

4 so defining Q by dq dp = c 1 exp( θ X 1 2 X 2 ) c = E exp( θ X 1 2 X 2 ) gives (3.4). So the only thing that could go wrong is that the infimum inf θ ϕ(θ) isnot attained. We shall now prove that this can only happen if there is arbitrage, in contradiction of our hypothesis; it follows then that the infimum is attained, and we do have an equivalent martingale measure. If we consider the sets F α {θ R d : θ =1,ϕ(αθ) 1}, (α 0) we see that these are closed subsets of a compact subset of R d, and it is not hard to see 10 that F β F α for all 0 α β. By the Finite Intersection Property, either the intersection α F α is non-empty, or for some α, F α =. If the infimum is not attained, then it is less than 1 = ϕ(0) and there exist a k such that ϕ(a k ) decrease to the infimum; these a k cannot be bounded, else some subsequence would converge to a point where the infimum is attained, so we must have a sequence of points tending to infinity where ϕ is less than 1, and so α F α is non-empty. Thus there is some unit vector a such that ϕ(ta) = E exp( ta X 1 2 X 2 ) E exp( X 2 ) for all t 0, and this can only happen if P[a X<0] = 0. Thus a X = a (S 1 S 0 ) 0, (3.7) and with positive P-probability (non-degeneracy!) this inequality is strict. We therefore take a portfolio consisting of a in the risky assets, and a S 0 in the riskless asset; at time 0 this is worth nothing, and at time 1 it is worth a X. Because of (3.7), this portfolio is an arbitrage. Remarks. The assumption that S 0 is identically 1 is restrictive, asymmetric and unnecessary; the notion of arbitrage for any (d + 1)-vector S of assets does not require this, and in fact we can deduce a far more flexible form of the above result by convexity of ϕ and the fact that ϕ(0) =

5 Corollary 3 (Fundamental Theorem of Asset Pricing, 0). Let ( S t ) t {0,1} be a (d +1)-vector of asset prices, and assume: Assumption (N): Among the assets S 0,...,S d, there is one which is strictly positive. Select a strictly positive asset N from the d +1 assets. Then the following are equivalent: (i) There is no arbitrage; (ii) There exists some probability Q equivalent to P such that E Q [ S 1 /N 1 ]=S 0 /N 0. (3.8) The probability Q is referred to as an equivalent martingale measure (or sometimes an equivalent martingale probability.) Proof. It is evident that θ is an arbitrage for S if and only if it is an arbitrage for S, where we define S t i St/N i t. The result follows by applying Theorem 1 to S (assume without loss of generality that N = S 0.) Remarks. (i) The strictly positive asset N used above is referred to as a numeraire. We have often considered a situation where there is a single riskless asset (referred to variously as the money-market account, the bond, the bank account,..) in the market, and it is very common to use this asset as numeraire. It turns out that this will serve for our present applications, but there are occasions when it is advantageous to use other numeraires. Note that the Fundamental Theorem of Asset Pricing does not require the existence of a riskless asset. (ii) Note that the Fundamental Theorem of Asset Pricing does not make any claim about uniqueness of Q when there is no arbitrage. This is because situations where there is a unique Q are rare and special; when Q is unique, the market is called complete. We shall have more to say about this presently. (iii) Theorem 3 tells us in a single-period setting that when there is no arbitrage, there exists an equivalent martingale measure. The meaning of the term equivalent has been defined and you will recall from the last Chapter what a martingale is. We now consider a multi-period setting t {1,..., T } and first say what arbitrage means in this context. The intuition is very clear: you set up a portfolio at 39

6 time 0 which costs nothing (or less), and whose value at time T is 0 a.s. and > 0 with some positive probability. Re-balancing the portfolio at intermediate periods should be allowed as long as (a) this is done without peaking into the future and (b) no rich uncle helps out. Definition 4 Assume St 0 =1for all t 0. (As remarked after the definition of a 1-period arbitrage this merely means we work with discounted prices.) An arbitrage in the multi-period setting t {1,..., T } is a previsible process ( θ t ) t=1,...,t such that P[ θ T S T 0] = 1 = P[ θ 1 S 0 0], and P[ θ T S T θ 1 S 0 > 0]> 0. (3.9) Moreover, ( θ) is self-financing by which we mean for all times t =1,..., T 1, ( θ t+1 θ t ) S t =0 (3.10) Note that if we consider ( θ) as a gambling strategy for the (not necessarily fair) game ( S) then the P&L (short for profit and loss) is given by T θ t ( S t S t 1 )= θ T S T θ 1 S 0 (3.11) t=1 and the equality holds precisely because of the self-financing condition. Theorem 2 (Fundamental Theorem of Asset Pricing). Let ( S t ) t {0,...,T } be a (d +1)-vector of asset prices, and assume: Assumption (N): Among the assets S 0,...,S d, there is one which is strictly positive. Select a strictly positive asset N from the d +1 assets. Then the following are equivalent: (i) There is no arbitrage; (ii) There exists some probability Q equivalent to P such that ( ) St is a Q-martingale. (3.12) N t t Z + The probability Q is referred to as an equivalent martingale measure. 40

7 (iv) We have just proved a very general form of the Fundamental Theorem of Asset Pricing in discrete time, though only in the single-period situation. Its extension to the multi-period situation is not essentially difficult, though there are some technical points to be handled 11 to give the result in its simplest and strongest form. There is an analogous result in continuous time, but this is quite deep and subtle 12 ; the first subtlety is in framing the definition of arbitrage correctly! We shall not dwell on the details of extending to the multi-period case in a general context, but shall for the rest of this chapter consider only very simple and explicit models where we can characterise the equivalent martingale measure completely, and perform calculations. (v) To link the statement of Theorem 2 with the discussion at the beginning of this Chapter, we need to observe that if we define Z t dq, dp Ft then Z is a P-martingale, and for any Q martingale M the product ZM is a P-martingale. The proofs of these facts are left as simple exercises. We then define ζ t = Z t, N t and the pricing expression (3.1) is seen to amount to the same as (3.12). Aside: axiomatic derivation of the pricing equation. For this little aside, we use the language of measure-theoretic probability, but this is not essential. We shall show how the pricing expression (3.1) can be derived very quickly from four simple axioms which a family of pricing operators should naturally obey. Suppose that we have pricing operators (π tt ) 0 t T for contingent claims; if Y is some F T measurable contingent claim to be paid at time T, the time-t market price will be π tt (Y ), which may be random, but must be F t -measurable. We shall assume that the pricing operators (π tt ) 0 t T satisfy certain axioms: (A1) Each π tt is a bounded positive linear operator from L (F T )tol (F t ); (A2) If Y L (F T ) is almost surely 0, then π 0T (Y ) is 0, and if Y L (F T )is non-negative and not almost surely 0, then π 0T (Y ) > 0; 11.. relating to measurable selection of maximising portfolios in the case of non-uniqueness. A paper by LCG Rogers discussed this in detail 12 Due to Delbaen & Schachermayer. 41

8 (A3) For 0 s t T and each X L (F t )wehave π st (Xπ tt (Y )) = π st (XY ); (A4) For each t 0 the operator π 0t is bounded monotone-continuous - which is to say that if Y n L (F t ), Y n 1foralln, andy n Y as n, then π 0t (Y n ) π 0t (Y )asn. Axiom (A1) says that the price of a non-negative contingent claim will be nonnegative, and the price of a linear combination of contingent claims will be the linear combination of their prices - which are reasonable properties for a market price. Axiom (A2) says that a contingent claim that is almost surely worthless when paid, will be almost surely worthless at all earlier times (and conversely) - again reasonable. The third axiom, (A3), is a consistency statement; the market prices at time s for XY at time T,orforX times the time-t market price for Y at time t, should be the same, for any X which is known at time t. The final axiom is a natural continuity condition which is needed for technical reasons. Let s see where these axioms lead us. Firstly, for any T>0wehavethatthe map A π 0T (I A ) defines a non-negative measure on the σ-field F T, from the linearity and positivity (A1) and the continuity property (A4). Moreover, this measure is absolutely continuous with respect to P, in view of (A2). Hence there is a non-negative F T -measurable random variable ζ T such that π 0T (Y )=E[ζ T Y ] for all Y L (F T ). Moreover, P[ζ T > 0] > 0, because of (A2) again. Now we exploit the consistency condition (A3); we have π 0t (Xπ tt (Y )) = E[Xζ t π tt (Y )] = π 0T (XY )=E[XY ζ T ]. Since X L (F t ) is arbitrary, we deduce that π tt (Y )=E t [Yζ T ]/ζ t, which shows that the pricing operators π st are actually given by a risk-neutral pricing recipe, with the state-price density process ζ. 42