3. The Discount Factor

Transcription

1 3. he Discount Factor Objectives Eplanation of - Eistence of Discount Factors: Necessary and Sufficient Conditions - Positive Discount Factors: Necessary and Sufficient Conditions Contents 3. he Discount Factor 3.1 he Law of One Price and the Eistence of a Discount Factor 3.2 he Absence of Arbitrage and the Eistence of a Positive Discount Factor 3.3 An Alternative Pricing Equation Financial heory, js he Discount Factor 22

2 3.1 Law of One Price and the Eistence of a Discount Factor In this chapter we investigate the role of the discount factor in a more general setting. Instead of deriving a specific discount factor from investor s preferences via an optimal decision, we search random discount factors (random variables) that generate present prices from future random payoffs as ( ) = E( m ). p We look for assumptions that ensure the eistence (uniqueness) of such discount factors Space he payoff space X is the set of all payoffs investors can purchase or sell. he state space R S forms the maimum permissible market space. he markets are complete if the payoff space equals state space, i.e. X = R S. On the contrary, markets are incomplete if the payoffs form a proper subset of the state space, i.e. X R S. he payoffs space contains primitive assets as well as all portfolios formed from these assets. Assumptions A.3.1: Free portfolio formation For any arbitrary portfolios in the payoff space, 1, 2 X, and for all a,b R, implies a 1 + b 2 X. A.3.1 his assumption is not innocuous because it implies investor s preferences such that these portfolios will be formed. Moreover, it rules out short sales restrictions and transaction costs. State 3 X 2 1 X State 1 State 1 Figure 3.1: and state spaces of incomplete markets Assumptions A.3.2: Law of one Price ( ) = ap( 1 ) + bp( 2 ) A.3.2 p a 1 + b 2 he law of one price makes sure that the repackaging does not yield profits. (No happy meal!) Financial heory, js he Discount Factor 23

3 heorem 3.1: Eistence of a Discount Factor Given assumptions A.3.1 (free portfolio formation) and A.3.2 (law of one price), there eists a unique payoff X such that p( ) = E( ). (3.1) he assumptions A.3.1 and A.3.2 imply that the price functions on the payoff space X are parallel hyperplanes as in Figure 3.2. However, they are defined on the payoff space X which may be a subspace of the entire state space. Related to the right hand eample of Figure 3.1 X can be interpreted as the plane of a larger space which is laid flat on the page. Geometric proof: Prices of different levels can be presented as decreasing linear functions with slope pc( 1) pc( 2). Price = 2 θ Price = 1 Price = 0 Figure 3.2: Eistence of a Discount Factor he payoff can be presented by a vector along the line from the origin perpendicular to the price lines. As the vector is orthogonal to the price-zero line each payoff along the pricezero line has the price zero. 0 = p( ) = E ( ) (3.2) Each payoff along the price 1 line has the same inner product. We just have to choose the right length of the vector in order to obtain 1 = p( ) = E ( ). (3.3) he length of the vector must be = 1 prices payoffs on any other price lines p ( cosθ). Of course we have a payoff that ( ) = E( ). Financial heory, js he Discount Factor 24

4 Algebraic proof: X is a N-dimensional vector space, generated by portfolios of N basis payoffs. Moreover, = payoffs (row vectors). he vector p = 1, 2 N is a N-dimensional vector of s 1-dimensional basis p 1, p 2 p N contains their prices. If the vector c = c 1,c 2 c N embodies portfolio weights the payoff space is defined as X = c Since we look for a discount X the latter must be of the form Moreover, we choose so that it prices the basis assets N. We demand p ( ) = E ( ) = E S 1 hus, we need the weights c = E N N c = E S N 1 S = c. { }. c. (3.4) p. (3.5) If E is nonsingular, a unique c eists. By assumption A.3.2, the singularity is guaranteed if we prune redundant rows. Because of (3.5), the discount we are looking for is 1 S = c = E N N p = p E N N. (3.6) he discount is an element of the payoff space X because it is a linear combination of the basis payoffs. he discount prices every element of the payoff space. In particular it prices any portfolio of basis payoffs N c N as ( ) = E p c 1 S ( ) c = E 1 S S N c Obviously, from linearity follows p( c ) = c p. Consequences: = p E N N E N N c = p 1. here is a unique discount X. here may be other discounts m X. c. (3.7) ( ). If 2. If markets are incomplete there are infinite many discounts satisfying p = E m ( ), we have p = E ( m + ε) ( ) = 0. his construction creates all possible discounts. a discount satisfies p = E m, i.e. E ε 3. Every discount (stochastic variable) that satisfies p = E m m = +ε, with E ε ( ) = 0. for any ε orthogonal to ( ) can be constructed as 4. In an economy with complete markets there is nowhere to go orthogonal to the payoff space. herefore, the discount is unique. Financial heory, js he Discount Factor 25

5 Figure 3.3 gives an eample of a one-dimensional payoff space X in a two-dimensional state space. In this case the there is a whole line of possible discounts. Space X space of discount factors m = + ε α Figure 3.3: Many Discounts in Incomplete Markets is the projection of any possible discount on the payoff space X. he pricing implications of any possible discount m on a payoff space X are the same as those of the projection of m on X. his discount is known as the mimicking portfolio for m: ( ( ) + ε ) p( ) = E( m ) = E proj m X ( ) = E proj m X hus, we can conclude that the eistence of a discount is guaranteed by the assumptions A.3.1 (free portfolio formation) and A.3.2 (law of one price). Both assumptions are quite restrictive. heorem 3.2: Discount Factor and the Law of One Price he eistence of a discount factor implies the law of one price. Proof: If = y + z, then E m ( ) = E m ( y + z) = E m y + E m z. (3.8) 3.2 No arbitrage and the Eistence of a Positive Discount Factors In this section we will analyze the central role of the no arbitrage condition. he latter is demands the absence of arbitrage. Assumption A.3.3: Absence of Arbitrage 0 p ( ) > 0 A.3.3 We assume that a pricing function p ( ) on the elements of the payoff space X is free of arbitrage if all payoffs 0 (non-negative in all states, and positive in at least one state) have a positive price p ( ) > 0. Financial heory, js he Discount Factor 26

6 he graph shows a payoff with an arbitrage opportunity, because > 0 and p( ) = < 0. Of course, is not the strictly positive. p = -1 p = 0 p = +1 Figure 3.4: Eistence of Arbitrage and non-positive Discounts heorem 3.3: Positive discounts imply no arbitrage p( ) = E( m ), and m > 0 (positive discount in all states), imply no arbitrage. Proof: For m > 0, any X, with 0, has positive evaluation in some states ( ) ( s) > 0. hus, the price of nonnegative payoffs is positive, p( ) = E( m ) > 0. m s Assumption A.3.4: Complete Markets Markets are complete if the payoff space equals the state space, i.e. X = R S. A.3.4 heorem 3.4: Eistence of a Unique Positive Discount he assumptions A.3.2 (law of one price), A.3.3 (no arbitrage) and A.3.4 (complete markets) imply the eistence of a unique positive discount m > 0, such that p X = R S. Proof: From A.3.2 (law of one price) follows the eistence of such that p ( ) = E( m), for all ( ) = E( ). Suppose that 0 (non-positive in all states and negative in some states). hen form a payoff that is 1 in the states with negative, and zero in all other states. For this payoff we have 0, and p < 0. his violates the assumption A.3.3. ( ) = pc( s) ( s) s: ( s)<0 It is more difficult to proof the eistence of a positive discount in case of incomplete markets. Financial heory, js he Discount Factor 27

7 heorem 3.5: Eistence of Positive Discounts Assumptions A.3.2 (law of one price) and A.3.3 (no arbitrage) imply the eistence of strictly positive discounts m > 0, such that p ( ) = E( m ), for all X. Proof: We form vectors of pairs of prices and payoffs p( ), M of feasible pairs of prices and payoffs as M = p( ), ; X ( ) R S +1, and define the set { }. (3.9) Because of A.3.2, M is a linear space. his in turn implies that any linear combination of elements of the set M, m 1,m 2 M are elements of M as well, i.e. am 1 + bm 2 M. m > 0 α p = 1 p = 2 X_ Figure 3.5: Non-positive Discounts: he eample shows that the discount is neither unique nor strictly positive. A.3.3 (absence of arbitrage) implies that the discounts m are not elements of the positive S orthant, i.e. m R +1 +, since 0 implies p( ) < 0. hus M is a hyperplane that intersects the positive orthant at the origin of the coordinate system. We can create a linear function F :R S +1 R, such that F ( p,) = 0 for ( p,) M, and S F ( p,) > 0 for ( p,) R +1 +, ecept the origin. Since we can present any linear function by an inner product of perpendicular vectors, there is a vector ( 1,m) such that F ( p,) = ( 1,m) ( p,) = p + m. (3.10) he epected value of (3.10) is ( ) ( p,) E 1,m Finally, since F p, = p ( ) + E( m ). (3.11) S ( ) > 0 for ( p,) R +1 +, m must be positive. Financial heory, js he Discount Factor 28

8 p = 1 p = 2 m > 0 α X_ Figure 3.6: Strictly positive Discount: Each particular choice of m > 0 induces an arbitragefree etension of the prices on X to all contingent claims. Some Consequences of theorem 3.4: 1. he theorem says that a positive discount factor eists. 2. It does not say that the discount factor is unique. In figure 3.5 any discount through and perpendicular to X prices the assets. Again we ( ) = E ( m + ε) ( ) = 0. All discount factors that lie in the have p for all E ε positive orthant are positive, and thus satisfy the theorem. here are many of them. Only in complete markets the discount factor is unique! 3. he theorem does not say that all discount factors are positive. In figure 3.5 all discounts outside the positive orthant are not strictly positive. Since they satisfy p they deserve as discount. his applies e.g. to X in figure 3.5. ( ) = E( m ) 4. he theorem says that we can etend the pricing function defined on X to all possible payoffs R S that do not offer arbitrage opportunities in the larger space of payoffs. Moreover, it says that there is a pricing function p ( ) = E( m ) defined over total R S that assigns the same prices on X and displays no arbitrage over total R S. Graphically this means that we can paint price panels in a way that they intersect X in the right places and assign positive prices to the payoffs in the positive orthant as done in figure 3.6. However, all discount factors m > 0 that satisfy these conditions generate such an etension. Of course there are infinitely many of them! hus, there is no unique discount. 5. We can think of the strictly positive discount factors m > 0 as contingent claims prices. If payoffs and prices offer no arbitrage opportunities we can think of the prices as generated by a complete-market, contingent-claims economy. However, if there is no absence of arbitrage there are infinitely many strictly positive discounts that generate such pricing Financial heory, js he Discount Factor 29

9 system the contingent claims prices are not defined uniquely. In fact there are many of them. 6. he absence of arbitrage is a very weak characterization of the economy (preferences and market equilibrium) that allows the use of the p ( ) = E( m ) formalism with m > An Alternative Formula Sometimes it is convenient to epress the stochastic discount = p E from (3.6) as deviations from its mean. It can be epressed as E ( ) = p E( ) E( ) ( ) is the covariance matri of the payoffs, defined as = E E( ) E ( { ) } = E We could substitute E inverse matri E E. (3.12) E ( ) E( ). (3.13) ( ) by E( ) E( ) + in order to derive formulas (3.12), but the ( ( ) E( ) + ) is not very useful. However, we can derive formula (3.12) by postulating a discount that is a linear function of the shocks to the payoffs, E ( ), i.e. = E ( ) + E( ) b. (3.14) In addition, we choose b to ensure that prices the basis payoffs as p = E or ( ) = E E( ) + E( ) p E( ) E( ) = E( ) E( ) E( ) where b is defined as b = p E( ) E( ) b, (3.15) b = b, (3.15 ). (3.16) his representation of the discount factor is of particular interest if we apply the discount to price-zero (ecess return) payoffs. For any given risk free rate of return (3.12) modifies to = 1 R f 1 R f E R e ( ) R e E( R ) e, and the covariance matri to = cov R e ( ). We will use both representations of the discount factor further on. Financial heory, js he Discount Factor 30