On%20the%20Pricing%20of%20Contingent%20Claims%20and%20the%20Modigliani-Miller%20Theorem

1
On the Pricing of Contingent Claims and the
Modigliani-Miller Theorem

• Chen Miaoxin

2

• Introduction

3
Two assumptions

• The theory of portfolio selection in
  continuous time has as its foundation two
  assumptions
• (a) the capital markets are assumed to be open at
  all times, and therefore economic agents have the
  opportunity to trade continuously
• (b) the stochastic processes generating the state
  variables can be described by diffusion processes
  with continuous sample paths (Chs 5 and 8).
• If these assumptions are accepted, then the
  continuous-time model can be used to derive
  equilibrium security prices (Ch. 15).

4
Black and Scholes(1973)

• B-S used the continuous-time analysis to
  derive a formula for pricing common-stock
  options.
• The resulting formula expressed in terms of
  the price of the underlying stock does not
  require as inputs expected returns, expected cash
  flows, the price of risk, or the covariance of
  the returns with the market. In effect, all these
  variables are implicit in the stocks price.

5
Contingent claim

• The essential reason that the B-S pricing
  formula requires so little information as inputs
  is that the call option is a security whose value
  on a specified future date is uniquely determined
  by the price of another security (the stock). As
  such, a call option is an example of a contingent
  claim.
• The same analysis could be applied to the
  pricing of corporate liabilities generally where
  such liabilities were viewed as claims whose
  values were contingent on the value of the firm.
• Moreover, whenever a securitys return
  structure is such that it can be described as a
  contingent claim, the same technique is
  applicable.

6
Structure of this Ch.

• In section 13.2, Merton derives a general
  formula for the price of a security whose value
  under specified conditions is a known function of
  the value of another security.
• In section 13.3, the MM theorem that the
  value of the firm is invariant to its capital
  structure is extended to the case where there is
  a positive probability of bankruptcy.
• In section 13.4, the author introduces the
  applications of contingent-claims analysis in
  corporate finance.

7

• A General Derivation of A Contingent-Claim Price

8
Assumptions 1-2

• 1 Frictionless markets There are no
  transactions costs or taxes. Trading takes place
  continuously in time. Borrowing and short-selling
  are allowed without restriction. The borrowing
  rate equals the lending rate.
• 2 Riskless asset There is a riskless asset whose
  rate of return per unit time is known and
  constant over time. Denote this return rate by r.

9
Assumptions 3

• 3 Asset 1 There is a risky asset whose value
  at any point in time is denoted by V(t). The
  dynamics of the stochastic process generating
  V() over time are assumed to be describable by a
  diffusion process with a formal stochastic
  differential equation representation of

10
Assumptions 4

• 4 Asset 2 There is a second risky asset whose
  value at any date t is denoted by W(t) with the
  following properties
• For 0tltT, its owners will receive an
  instantaneous payout per unit time,
• For any t (0tltT)
• For tT
• Asset 2 is called a contingent claim, contingent
  on the value of Asset 1.

11
Assumptions 5-6

• 5 Investor preferences and expectations It is
  assumed that investors prefer more to less. It is
  assumed that investors agree upon s2, but it is
  not assumed that they necessarily agree on a.
• 6 Other There can be as many or as few other
  assets or securities as one likes.

12
Derivation

• If it is assumed that the value of Asset 2
  can be written as a twice continuously
  differentiable function of the price of Asset 1
  and time, then the pricing formula for Asset 2
  can be derived by the same procedure as that used
  in Merton (Section 12.2 and 12.3) to derive the
  value of risky debt.

13
Derivation

• If W(t)FV(t),t for 0tT and for
  then, to avoid arbitrage, F must satisfy the
  linear partial differential equation
• To solve (13.1), boundary conditions must be
  specified. From Assumption 4, we have that
• (13.1) together with (13.2a)-(13.2c) provide the
  general equation for pricing contingent claims.

14
The boundary conditions of B-S

• While the function f, g, and h are required
  to solve for F, they are generally deducible from
  the terms of the specific contingent claim being
  priced.
• For example, the original case examined by
  B-S is a common-stock call option with an
  exercise price of E dollars and an expiration
  date of T. If V is the value of the underlying
  stock, then the boundary conditions can be
  written as

15
Proposition

• Suppose there exists a twice continuously
  differentiable solution to (13.1) and (13.2).
  Because the derivation of (13.1) used the
  assumption that the pricing function satisfies
  this condition, it is possible that some other
  solution exists which does not satisfy this
  differentiability condition.
• The following alternative derivation is a
  direct proof that if a twice continuously
  differentiable solution to (13.1)and (13.2)
  exists, then to rule out arbitrage, it must be
  the pricing function.

16
Proof

• Let F be the formal twice continuously
  differentiable solution to (13.1) with boundary
  conditions (13.2).
• Consider the continuous-time portfolio
  strategy where the investor allocates the
  fraction w(t) of his portfolio to Asset 1 and
  1-w(t) to the riskless asset.
• Moreover, let the investor make net
  withdrawals per unit time (e.g. for
  consumption) of C(t).

17
Proof

• If C(t) and w(t) are right-continuous
  functions and P(t) denotes the value of the
  investors portfolio, then the author has shown
  elsewhere (Equation 5.14) that the dynamics for
  the value of the portfolio, P, will satisfy the
  stochastic differential equation

18
Proof

• Suppose we pick the particular portfolio
  strategy with
• And the consumption strategy
• Substituting from(13.5) and (13.6) into (13.4),we
  have that

19
Proof

• Since F is twice continuously differentiable,
  we can use ITO lemma (Ch. 5) to express the
  stochastic process for F as
• But F satisfies (13.1). Hence, we can rewrite
  (13.8) as
• ?

20
Proof

• Let Q(t)FV(t),t. Then from (13.7)and
  (13.9), we have that
• But (13.10) is a nonstochastic differential
  equation with solution
• For any time t and where Q(0) )P(0)-FV(0),0.
  Suppose that the initial amount invested in the
  portfolio, P(0), is chosen equal to
  FV(0),0.Then from (13.11) we have that

21
Proof

• By construction, the value of Asset 2, W(t),
  will equal F at the boundaries V and ,
  and at the termination date T. Hence, from
  (13.12), the constructed portfolios value P(t)
  will equal W(t) at the boundaries. Moreover, the
  interim payments or withdrawals available to
  the portfolio strategy, D2V(t),t, are identical
  to the interim payments made to Asset 2.
• Therefore, if W(t)gtP(t) or W(t)ltP(t), there
  will be an opportunity for intertemporal
  arbitrage. Hence, W(t) must equal FV(t),t

22
Advantages

• Unlike the original derivation, this alternative
  derivation doesnt assume that the dynamics of
  Asset 2 can be described by an ITO process, and
  therefore it does not assume that Asset 2 has a
  smooth pricing function.
• Indeed, the portfolio strategy described by
  (13.5) and (13.6) involves only combinations of
  Asset 1 and the riskless asset, and therefore
  does not even require Asset 2 exists! The
  connection between the portfolio strategy and
  Asset 2 is that, if Asset 2 exists, then the
  price of Asset 2 must equal FV(t),t or there
  will be an opportunity for intertemporal
  arbitrage.

23

• On The Modigliani-Miller Theorem With
  Bankruptcy

24
Introduction

• In Section 12.5, Merton proved that, in the
  absence of bankruptcy costs and corporate taxes,
  the MM theorem obtains even in the presence of
  bankruptcy.
• The method of derivation used in the previous
  section provides an immediate alternative proof.

25
Proof

• Let there be a firm with two corporate
  liabilities (a) a single homogeneous debt issue
  and (b) equity. The debt issue is promised a
  continuous coupon payment per unit time, C, which
  continues until either the maturity date of the
  bond, T, or the total assets of the firm reach
  zero.
• The firm is prohibited by the debt indenture from
  issuing additional debt or paying dividends. At
  the maturity date, there is a promised principal
  payment of B to the debtholders. In the event
  that the payment is not made, the firm is
  defaulted to the debtholders, and the equity
  holders receive nothing.
• So VL(t)S(t)D(t). In the event that the total
  assets of the firm reach zero, VL(t)S(t)D(t)0.
  Also, by limited liability, D(t)/ VL(t)1

26
Proof

• Consider a second firm with initial assets and an
  investment policy identical with those of the
  levered firm. However, the second firm is
  all-equity financed with total value equal to
  V(t).
• Let the second firm have a dividend policy that
  pays dividends of C per unit time either until
  date T or until the value of its total assets
  reaches zero (i.e. V0).
• Let the dynamics of the firms value be as
  posited in Assumption 3 where D1(V,t)C for Vgt0
  and D10 for V0

27
Proof

• Let F(V,t) be the formal twice-continuously
  differentiable solution to (13.1) subject to the
  boundary conditions F(0,t)0 F(V,t)/V1 and
  FV(T),TminV(T),B.
• Consider the dynamic portfolio strategy of
  investing in the all-equity firm and the riskless
  asset according to the rules (13.5) and (13.6)
  of Section 13.2 where C(t)C. If the total
  initial amount invested in the portfolio, P(0),
  is equal to FV(0),0, then from (13.12),
  P(t)FV(t),t.

28
Proof

• Because both the levered firm and the all-equity
  firm have identical investment policies including
  scale, it follows that V(t)0 if and only if
  VL(t)0. And on the maturity date T, VL(T)V(T).
• By the indenture conditions on the levered firms
  debt, D(T)minVL(T),B. But since V(T) VL(T)
  and P(T)FV(T),T, it follows that P(T)D(T).
  Moreover, since VL(T)0 if and only if V(t)0, it
  follows that P(t)F(0,t)D(t)0 in that event.
• Thus, by following the prescribed portfolio
  strategy, one would receive interim payments
  exactly equal to those on the debt of the levered
  firm. On a specified future date T, the value of
  the portfolio will equal the value of the debt.
  Hence, to avoid arbitrage or dominance, P(t)D(t).

29
Proof

• The proof for equity follows along similar lines.
  Therefore, p(t)S(t).
• If one were to combine both portfolio strategies,
  then the resulting interim payments would be C
  per unit time with a value at the maturity date
  of V(T). That is , both strategies together are
  the same as holding the equity of the unlevered
  firm. Hence, fV(t),tFV(t),tV(t). But it was
  shown that fV(t),tFV(t),tS(t)D(t)VL(t).
  Therefore, VL(t)V(t).

30
Comments

• While the proof was presented in the traditional
  context of a firm with a single debt issue, the
  proof goes through in essentially the same
  fashion for multiple debt issues or for hybrid
  securities such as convertible bonds, preferred
  stock , or warrants.

31
MM theorem

• The MM theorem holds that for a given investment
  policy the value of the firm is invariant to the
  choice of financing policy. It does not imply
  that the choice of financing policy will not
  influence investment policy and thereby affect
  the value of the firm.
• Stiglitz(1972) and Merton (1973a) managers of
  firms with large quantities of debt outstanding
  may choose to undertake negative
  net-present-value projects that reduce the market
  value of the firm but increase the market value
  of its common stock.
• Merton(1990b)discussion of the conflict between
  debtholders and equityholders over the investment
  and financing policies of the firm.

32
MM theorem

• If the choice of securities issued by the firm
  can alter the tax liabilities of the firm, or if
  there are bankruptcy costs, then the MM theorem
  no longer obtains. But, as noted in Merton
  (1982a,1990a), the contingent-claims pricing
  technique can still be used to value corporate
  liabilities.
• To do so, redefine V(t) as the pre-tax and
  pre-bankruptcy-cost value of the firm at time t
  and include as explicit liabilities of the firm
  both the governments tax claim and the
  deadweight bankruptcy-cost claim. Because these
  additional noninvestor liabilities, like those
  held by investors, are entitled to specified
  payments that depend on the fortunes of the firm,
  the previous analyses of this section and Section
  13.2 apply. Hence, for a fixed investment policy,
  the redefined investor-plus-noninvestor value
  of the firm will be invariant to the firms
  choice of financing policy.

33
MM theorem

• Although the total investor-plus-noninvestor
  value of the firm does not depend on the
  financing choice, the allocation of that value
  between investor and noninvestor components does.
  Thus, for a given investment policy, the firms
  financing policy does matter to its management,
  debtholders, and stockholders.

34

• Applications of Contingent-Claims Analysis in
  Corporate Finance

35
Advantages of CCA

• (a) the relatively weak assumptions required for
  its valid application make CCA robust
• (b) the variables and parameters required as
  inputs in the valuation equation are either
  directly observable or reasonable to estimate
• (c) there are several computationally feasible
  numerical methods for solving the partial
  differential equations for prices
• (d) the generality of the methodology permits
  adaptation to a wide range of finance
  applications.

36
Applications of CCA

• 1 The applications of CCA to the pricing of
  corporate liabilities
• Masulis(1976) analyze the effects on debt prices
  of changes in the firms investment policy.
• Galai and Brennan and Schwartz(1976b) and
  Merton(1974) analyze the valuation of debt with
  call provisions.
• Brennan and Schwartz(1977c,1980),
  Ingersoll(1977), and Merton (1970b)the pricing
  of convertible bonds.
• Baldwin(1972) and Emanuel(1983b) evaluate
  preferred stock.

37
Applications of CCA

• 2 The applications of CCA to the evaluation of
  loan guarantees and deposit insurance
• Gatto, Geske, Litzenberger, and Sosin(1980)
  study the pricing of mutual-fund insurance.
• Kraus and Ross (1982) apply the continuous-time
  model to the problem of determining the fair rate
  of profit for a property-liability insurance
  company.
• Bodie(1990) applies CCA to evaluate inflation
  insurance.

38
Applications of CCA

• 3 The applications of CCA to the financial
  analysis of corporate employee-compensation
  plans, such as the evaluation of executive stock
  options and the evaluation of both explicit and
  implicit labor contracts that provide wage floors
  and employment guarantees, including tenure.

39
Applications of CCA

• 4 The applications of CCA to capital-investment
  decisions and corporate strategy.
• Traintis and Hodder(1990) CCA can be used to
  evaluate the benefits of a more broadly trained
  work force against the cost of the higher wages
  that must be paid whether this extra training is
  used or not.
• Myers and Majd(1983) analyze the value of the
  option to abandon a project.
• Myers(1977) CCA can be used to evaluate the
  option to choose when to initiate a project.
  Myers points out that recognition of this option
  is important to the proper evaluation of a firms
  growth opportunities.

40

• Thanks to
• Wang Baohe