ContingentClimsMrkets未定權(quán)益市場(chǎng)

1、Ch3. Contingent Claims MarketsBrief introductionIn the frame of complete market, we look forward to see the equation p=E(mx) more intuitive.The structure is as follows: 3.1 Contingent Claims 3.2 Risk-Neutral Probabilities 3.3 Investors Again 3.4 Risk Sharing 3.5 State Diagram and Price Function 3.1

2、Contingent ClaimsA contingent claim is a security that pays one dollar (or one unit of the consumption good) in one state s only tomorrow. pc(s) is the price today of the contingent claim. （狀態(tài)價(jià)格）Complete marketIn a complete market investors can buy any contingent claim.They dont necessarily have to

3、be faced with explicit contingent claims; they just need enough other securities to span or synthesize all contingent claims.（有幾種狀態(tài)，就有幾種相互無(wú)法復(fù)制的證券，就是完全市場(chǎng)）x(s) denote an assets payoff in state of nature s Since the asset is a bundle of contingent claims , its price must then equal the total value of t

4、he contingent claims. p(x) =pc(s)x(s) (3.1)It is easier to take expectations rather than sum over states. p(x) = (s) (pc(s)/ (s) x(s), where (s) is the probability that state s occurs. Then define m as the ratio of contingent claim price to probability，m(s) = pc(s)/ (s)Conclusion about discount fact

5、orNow we can write the bundling equation as an expectation, p = (s) m(s) x(s) = E(mx)If there are complete contingent claims, a discount factor exists, and it is equal to the contingent claim price divided by probabilities. Expand to infinite spaceIn general, we posit states of nature that can take

6、continuous (uncountably infinite) values in a space . In this case, the sums become integrals, and we have to use some measure to integrate over . Thus, scaling contingent claims prices by some probability-like object is unavoidable. 3.2 Risk- neutral probabilities DefineThe *(s) are positive, less

7、than or equal to one and sum to one, so they are a legitimate set of probabilities. Then we can rewrite the asset pricing formula as:We use the notation E* to remind us that the expectation uses the risk neutral probabilities *instead of the real probabilities .*(s) = (m(s) / E(m) (s)*gives greater

8、weight to states with higher than average marginal utility mrisk aversion is equivalent to paying more attention to unpleasant states, relative to their actual probability of occurrence. Application to report ones reasonable subjective probabilities.We can also think of the discount factor m as the

9、derivative or change of measure from the real probabilities to the subjective probabilities *連續(xù)時(shí)間在完全市場(chǎng)中，兩者的風(fēng)險(xiǎn)源相同。From (1.35),we have 風(fēng)險(xiǎn)中性定價(jià)在風(fēng)險(xiǎn)中性世界，風(fēng)險(xiǎn)價(jià)格必須等于0，即3.3 Investors choice The investor starts with a pile of initial wealth y and a state-contingent income y(s). He purchases contingent claims to

10、 each possible state in the second period. His problem is then Eliminating the Lagrange multiplier , Coupled with p = E(mx), we obtain the consumption-based model again. marginal rates of substitutionThe investors first order conditions say that marginal rates of substitution between states tomorrow

11、 equals the relevant price ratio, 邊際替代率相對(duì)價(jià)格比（經(jīng)概率調(diào)整）Economics behind this approach to asset pricing (figure 3.1)3.4 Risk Sharing In complete markets, the prices are the same for all investors. 如果信息是透明的，每個(gè)人都知道客觀概率，則 marginal utility growth should be the same for all investorsIf investors have the same

12、 homothetic utility function (for example, power utility), then consumption itself should move in lockstep.It means that shocks to consumption are perfectly correlated across individuals. It doesnt say that expected consumption growth should be equal; it says that consumption growth should be equal

13、ex post. In a complete contingent claims market, all investors share all risks, so when any shock hits, it hits us all equally (after insurance payments). Pareto-optimal risk sharing. Suppose a social planner wished to maximize everyones utility given the available resources. For example, with two i

14、nvestors i and j, he would maximize first order condition This simple fact has profound implications: It shows you why only aggregate shocks should matter for risk prices. Any idiosyncratic income risk will be equally shared, and so 1/N of it becomes an aggregate shock. Then the stochastic discount

15、factors m that determine asset prices are no longer affected by truly idiosyncratic risks. Much of this sense that only aggregate shocks matter stays with us in incomplete markets as well. Sub-markets for risk sharing:Insurance marketbond marketstock marketReasons for individual consumptions not mov

16、e in lockstep:The real economy does not yet have complete markets or full risk sharing.Different utility functions.Different values of individual impatient coefficients.Different 3.5 State Diagram and Price FunctionThink of the contingent claims price pc and asset payoffs x as vectors in RS, where e

17、ach element gives the price or payoff to the corresponding state,pc = pc(1) pc(2) pc(S) x = x(1) x(2) x(S) Figure 7 is a graph of these vectors in RS 圖7狀態(tài)價(jià)格與回報(bào)P=0(超額收益率RfP=2P=1（收益率）狀態(tài)1回報(bào)狀態(tài)2回報(bào)1/Rfpc線的斜率在經(jīng)概率調(diào)整后的狀態(tài)偏好中性世界中，pc(1)=pc(2),因此pc線是45度線。在現(xiàn)實(shí)生活中，投資者對(duì)不同狀態(tài)的偏好不同。投資者越愛(ài)好（經(jīng)概率調(diào)整）某種狀態(tài)的PAYOFF，該狀態(tài)的PC就越高。在上

18、圖中，pc(1)0 and pc 0. m 0 means the realization of m is positive in every state of nature, or, equivalently every element of the vector m is positive. The set of payoffs with any given price lie on a (hyper)plane perpendicular to the contingent claim price vector. Since the price of the payoff x must

19、be given by its contingent claim value, p(x) =pc(s)x(s)Interpreting pc and x as vectors, this means that the price is given by the inner product of the contingent claim price and the payoff.If two vectors are orthogonal if they point out from the origin at right angles to each other then their inner

20、 product is zero. The set of all zero price payoffs must lie on a plane orthogonal to the contingent claims price vector, as shown in figure 7.The inner product of two vectors x and pc equals the product of the magnitude of the projection of x onto pc times the magnitude of pc. where means the lengt

21、h of the vector x Since all payoffs on planes (such as the price planes in figure 7) that are perpendicular to pc have the same projection onto pc, they must have the same price. Planes of constant price move out linearly, and the origin x = 0 must have a price of zero. If payoff y = 2x, then its price is twice the price of x, p(y) =pc(s) y(s) = pc(s) 2x(s) = 2p(x) We can think of p(x) as a pricing function, a map from the state space or payoff space