ne-erimel衍生证券的定价与保值-厦门大学

1、,Part 4 Chapter 11 Yulin Y Department of Finance, Xiamen University,Main line: 1 A partial-equilibrium one-period model 2 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,

2、 model 3 is the general model)., A partial-equilibrium one-period model,We 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.,C

3、onsider an 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 im

4、portantly,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

5、=1), ,Because 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

6、is independent of 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 t

7、hat the expected 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 examples,Example 1: Firm issues two types of securi

8、ties, debt and equity 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 equilibrium,So, . This is identical to the warrant pricing equation derived in Chapter 7. Thi

9、s equation can also be 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

10、 warrants outstanding with 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 d

11、efine normalized price of 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 a

12、lternatively the bonds can be exchanged for a total of n shares of equity; and N shares of equity with current price per share of S dollars.,So, , and Where 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

13、as a straight bond plus a warrant 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

14、) which are entitled to all the ordinary,income 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 share

15、s with terminal claim on the fund of B dollars plus all interest and dividends earned, be the current value of the capital shares. So, from definition, we have,And Where . The current value of the capital shares can be less than the current net asset value of the capital shares, defined to,be V-B, b

16、ecause If , that is, then, ., A general intertemporal equilibrium model,Consider 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

17、of the prices per share in the futures are 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

18、only of .,From , divide both side by and substitute 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

19、 of the ith security demanded by investor k.,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 individua

20、l who acts to maximize his expected utility are (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-covarianc

21、e matrix of returns , and . Applying the Implicit Function Theorem to (11), we have,The 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

22、 be the number of “shares”, i.e. . Then, N and are 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

23、form: (17) Since in equilibrium, , it can be rewritten 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 determin

24、ed. The equilibrium interest rate can be determined from (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 portfoli

25、os can be represented as a linear combination of any two 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 Marke

26、t Line of CAPM. If it is assumed that the are constant over 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 w

27、ith zero mean and variance . Define the variables,Then consider 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

28、 capital structure consists of two securities: equity and debt, 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 t

29、hat time of B. Let be the current value of equity and the dynamics 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

30、), it must be that (25) (26) And also, in equilibrium, (27) Substituting for and from (23) and (27), we have the Fundamental Partial,Differential Equation of Security Pricing (28) subject to the boundary condition The solution is (29) Z is a log-normally distributed random variable with mean and var

31、iance of , and is the log-normal density function.,(28) is the Fundamental 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 gener

32、al form as (30),Although (30) is a kind of discounted expected value formula, 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 pric

33、ing option contracts, but it actually holds without this assumption. Consider 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 val

34、ue invested in the firm and the fraction in the particular security chosen from 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 previous

35、ly, we can arrived at (28). Nowhere was the market equilibrium assumption needed!,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, i

36、n empirical examinations using a regression, if one were to use equity instead of firm values, systematic biases would be introduced.,Model 2: The “no riskless asset” case,If there exists uncertain inflation and there are no future markets in consumption goods or other guaranteed “purchasing power”

37、securities available, there will be no perfect hedge against future price changes, 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)

38、,(33) and substitute them into (31), we have (34) In a similar fashion to the analysis 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 model,In th

39、is model, the interest rate varies stochastically over time. In section 11.4, we have (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

40、 exist three portfolios (“mutual funds”) constructed from these n assets such that all risk-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

41、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