oneperiodmodel衍生證券的定價(jià)與保值廈門(mén)大學(xué)

1、Part 4Chapter 11YulinYDepartment of Finance,Xiamen UniversityMain line:1 A partial-equilibrium one-period model2 A general intertemporal equilibrium model of the asset market, includes three models(model 1 is based on a constant interest rate assumption, model 2 is a no-riskless-asset case, model 3

2、is the general model). A partial-equilibrium one-period modelWe follows the warrant pricing approach used in Chapter 7, that is, investors choose among three assets: the warrant, the stock of the firm and a riskless asset to form optimal portfolios which maximize their expected utility. Consider an

3、economy made up of only one firm with current value , and there exists a “representative man” acts so as to maximize the expected utility of wealth at the end of a period of length, that is, Define a random variable Z by and assume its probability distribution is known at present, more importantly,

4、is independent of the particular structure of the firm, this is consistent with the MM(Modigliani-Miller) theorem.Define as the current value of the i th type of security issued by the firm. The different types of securities are distinguishable by their terminal value . For a debt issue(i=1), Becaus

5、e each of the securities appears separately in the market, so: and Define , so we can rewrite as a maximization under constraint: The corresponding first-order conditions are:This can be rewritten in terms of util-prob distributions Q as: Where and is a new multiplier related to .dQ is independent o

6、f the functions by the assumption that the value of the firm is independent of its capital structure, so is a set of integral equations linear in the , and we can rewrite as *Suppose the firm issues just one type of security-equity, then Substituting in , we have From , we can see that the expected

7、return on all securities in util-prob space must be equated. If U was linear, then dQ=dP and would imply the result for the risk-neutral case. Hence, the util-prob distribution is the distribution of returns adjusted for risk.Some examplesExample 1:Firm issues two types of securities, debt and equit

8、y with current value and respectively. From and , we have : Suppose or for then as .In the limit, the debt becomes riskless, so will be replaced by r. Another useful form of is Since in equilibriumSo, . This is identical to the warrant pricing equation derived in Chapter 7. This equation can also be

9、 derived directly from the terminal value of equity in the same way as debt.Example 2:Firms capital structure made up from three types of securities: debt, equity(N shares outstanding with current price per share of S, i.e. ), warrants (exercise price is ). Assume there are n warrants outstanding wi

10、th current market value per warrant of W, i.e. . Because the warrant is a junior security to the debt, the current value of the debt will be the same as in the first example. The current value of the equity will be Where .Rewrite as In equilibrium, . So from we have If we define normalized price of

11、the firm as And define the normalized price of a warrant as , then can be rewritten as which is of the same form as equation (7.24). Example 3:Firms capital structure contains two securities:convertible bonds with a total terminal claim on the firm of either B dollars or alternatively the bonds can

12、be exchanged for a total of n shares of equity; and N shares of equity with current price per share of S dollars. So, , andWhere is determined to be .By inspection of this equation, we have the well-known result that the value of a convertible bond is equal to its value as a straight bond plus a war

13、rant with exercise price .Example 4:A “dual” fund: it issues two types of securities to finance its assets: namely, capital shares(equity) which are entitled to all the accumulated capital gains(in excess of the fixed terminal payment); and income-shares(a type of bond) which are entitled to all the

14、 ordinaryincome in addition to a fixed terminal payment.Let be the instantaneous fixed proportion of total asset value earned as ordinary income, V be the current asset value of the fund and Z the total return on the fund. Let be the current value of the income shares with terminal claim on the fund

15、 of B dollars plus all interest and dividends earned, be the current value of the capital shares.So, from definition, we have AndWhere .The current value of the capital shares can be less than the current net asset value of the capital shares, defined tobe V-B, becauseIf , that is, then, . A general

16、 intertemporal equilibrium modelConsider an economy with K consumers investors and n firms with current value .Each consumer acts so as to Define , where is the number of shares and is the price per share at time t.Assume that expectations about the dynamics of the prices per share in the futures ar

17、e the same for all investors and can be described by the stochastic differential equation:Further assume that one of the n assets (the nth one) is an instantaneously riskless asset with instantaneous return : For simplicity, we assume that and are functions only of .From , divide both side by and su

18、bstitute for ,then The accumulation equation for the kth investor can be written as Where is his wage income and is the fraction of his wealth invested in the ith security.So, his demand for the ith security can be written as Where is the number of shares of the ith security demanded by investor k.

19、Substituting for , we can rewrite as From the budget constraint, and from , we have i.e. the net value of shares purchased must equal the value of savings from wage income.According to Chapter 4 and 5, the necessary optimality conditions for an individual who acts to maximize his expected utility ar

20、e (10) subject to . From (10), we can derive m+1 first-order conditions (11) (12)Equation (12) can be solved explicitly for the demand functions for each risky security as where the are the elements of the inverse of the instantaneous variance-covariance matrix of returns , and .Applying the Implici

21、t Function Theorem to (11), we haveThe aggregate demands are (13)If it is assumed that the asset market is always in equilibrium, then ,where M is the total value of all assets. So, from (14) Let be the price per “share” of the market portfolio and N be the number of “shares”, i.e. . Then, N and are

22、 defined by From and , then And from (13), we get (15)Define .Divide equation (14) by M and substitute for We can rewrite (15) as (16)And from this we can determine From (13), we can solve for the yields on individual risky assets in matrix-vector form: (17)Since in equilibrium, , it can be rewritte

23、n in scalar form as (18) Multiplying both side by and summing from 1 to m, we have (19)Hence, from (18) and (19), if we know the equilibrium prices, then the equilibrium expected yields of the risky assets and the market portfolio can be determined. The equilibrium interest rate can be determined fr

24、om (11).Model 1: A constant interest rate assumption With a constant interest rate, the ratios of an investors demands for risk assets are the same for all investors. Hence, the “mutual-fund” or separation theorem holds, and all optimal portfolios can be represented as a linear combination of any tw

25、o distinct efficient portfolios(we can choose them to be the market portfolio and the riskless asset). By combining (18) and (19) we have (20)With a slightly different interpretation of the variables, (20) is the equation for the Security Market Line of CAPM.If it is assumed that the are constant ov

26、er time, then from (16), is log-normally distributed.We can integrate the stochastic process for to get conditional on where is a normal variate with zero mean and variance .Similarly, we can integrate (16) to get where is a normal variate with zero mean and variance .Define the variables Then consi

27、der the ordinary least-squares regression , if Model 1 is the true specification, then the following must hold: ; ; is a normal variate with zero mean and a covariance with the market return of zero.Reconsider the first example where firms capital structure consists of two securities: equity and deb

28、t, and it is assumed that the firm is enjoined from the issue or purchase of securities prior to the redemption date of the debt namely,So, (21)Let be the current value of the debt with years until maturity and with redemption value at that time of B.Let be the current value of equity and the dynami

29、cs of the return on equity can be written as (22) (20)Like every security in the economy, the equity of the firm must satisfy (20) in equilibrium, hence, (23)By Itos lemma, substitute dV from (21), we get: (24)Comparing (24) with (22), it must be that (25) (26)And also, in equilibrium, (27)Substitut

30、ing for and from (23) and (27), we have the Fundamental Partial Differential Equation of Security Pricing (28) subject to the boundary conditionThe solution is (29)Z is a log-normally distributed random variable with mean and variance of , and is the log-normal density function.(28) is the Fundament

31、al Partial Differential Equation of Asset Pricing because all the securities in the firms capital structure must satisfy it. And each securities are distinguished by their terminal claims.Equation (29) can be rewritten in general form as (30)Although (30) is a kind of discounted expected value formu

32、la, one should not infer that the expected return on F is r. From (23),(26) and (27), the expected return on F can be written as(28) was derived by Black and Scholes (1973) under the assumption of market equilibrium when pricing option contracts, but it actually holds without this assumption.Conside

33、r a two-asset portfolio which contains the firm as one security and any one of the security in the firms capital structure as the other.Let P be the price per unit of this portfolio, the fraction of the total portfolios value invested in the firm and the fraction in the particular security chosen fr

34、om the firms capital structure.So,Suppose is chosen such that Then the portfolio will be perfectly hedged and the instantaneous return on the portfolio will be certain(equal to r),that is, . So . Then, as was done previously, we can arrived at (28). Nowhere was the market equilibrium assumption need

35、ed!Remarks:Although the value of the firm follows a simple dynamic process with constant parameters, the individual component securities follow more complex processes with changing expected returns and variances. Thus, in empirical examinations using a regression, if one were to use equity instead o

36、f firm values, systematic biases would be introduced. Model 2: The “no riskless asset” caseIf there exists uncertain inflation and there are no future markets in consumption goods or other guaranteed “purchasing power” securities available, there will be no perfect hedge against future price changes

37、, i.e. no riskless asset exists.Follow the same procedure as in section 11.4, we can derive analogous equilibrium conditions, namely, (31) (32)The nth security must satisfy (31) in equilibrium (33)Solve and G in (32),(33) and substitute them into (31), we have (34)In a similar fashion to the analysi

38、s in Model 1, we get the Fundamental Partial Differential Equation for Security Pricing ,where If security n is a zero-beta security, i.e. ,then ,and (34) can be rewritten as where . Model 3: The general modelIn this model, the interest rate varies stochastically over time.In section 11.4, we have (

39、35) (36)So, (37) Solve (36) and (37) for and , and substitute them into (35), we have (38)where and Theorem 11.1(Three “Fund” Theorem)Given n assets satisfying the conditions of the model in section 11.4, there exist three portfolios (“mutual funds”) constructed from these n assets such that all ris

40、k-averse individuals, who behave to maximize their expected utilities, will be indifferent from these three funds.Further, a possible choice for the three funds is the market portfolio, the riskless asset, and a portfolio which is (instantaneously)perfectly correlated with changes in the interest rate.Since in this model, the interest rate varies stochastically, we can determine the term structure from this model, and nowhere in the model is it necessary to introduce concepts such as liquidity, transactions costs,time horizon or habit to explain the existence of a