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Stochastic Processes in Models of Financial Markets: the Valuation Equation of Finance and its Application
SESSIO~
Copyrighl © IFAC 3rd Svmposium
16
- STOCHASTIC PROCESSES
Control of Distri buted Pa rameter Svstt"ms
T oulouse . Fra nce. 1982
.
STOCHASTIC PROCESSES IN MODELS OF FINANCIAL MARKETS: THE VALUATION EQUATION OF FINANCE AND ITS APPLICATIONS D. Gabay Laboratoire d'Analyse Num eriqu e, Universite P. et M . Curie and INRIA , Domaine de Voluceau , B .P. 105, Le Ch esnay Cedex 78153, France
Abstract. This paper presents a brief survey of the continuous-time model of financial market used to develop the theory of contingent claims valuation. New results are given on the existence, uniqueness and regularity of the solution of the resulting evolution problems. Proofs and computational methods will be given in [6J. I. INTRODUCTION
In this paper we present a general and unifying framework for many recent contributions of "continuous-time analysis" to the development of financial economic theory. Assuming that trading takes place continuously in time and that the underlying stochastic variables follow diffusion processes (see Merton [ 17 J for a discussion of the relevance of this assumption) it is possible under ideal assumptions on the market mechanisms to derive a theory of contingent claims pricing using only observable variables. Black and Scholes in their seminal paper [2J inaugurated this approach to establish that the price of an "european" option on a stock must satisfy a parabolic partial differential equation. They actually found an explicit formula for its solution which has been widely used to test the validity of the ideal market assumptions on the many available option prices time series. The same concept has also been used for the pricing of corporate liabilities, forward contr.a c.t s, insurance contracts ... (see Smith .[ 18J for a survey). In the next section we describe a general model for the financial market and derive the valuation equation of finance. Section 3 is devoted to its mathematical analysis ; this problem has been approached by Gleit [7J but we give here stronger results. Because of space limitation we defer the proofs to a forthcoming report [6 J where numerical methods will also be proposed. In the conclusion (Section 4) we briefly state the optimal stopping problem associated to optimal conversion strategies for callable contingent claims like american options or convertible bonds.
2. THE VALUATION OF CONTINGENT CLAIMS. 2.1. The model: definitions, notations and assumptions. Following Merton [ 16 J we consider a financial market where trading takes place conti395
nuously over a time interval [O,TJ. For simplicity we also assume that it is "frictionless" : there are no transaction costs nor taxes ; borrowing and shortselling are allowed without restriction and the borrowing rate equals the lending rate. There can be as many assets or securities as one wishes but we are interested in only three of them. First, there is a riskless asset, called bond thereafter, yielding a known instantaneous rate of return per unit time pe t» ~ O. We normalize prices by taking as unit the value of the bond at time 0 ; clearly its value at time t is B(t) = exp ( J tp (S)dS). Next, there is a risky asset which p~ovides to its holder a known instantaneous net payout d(x,t) per unit time when the asset price is x at time t. We assume that the price X(t) of the risky asset follows a continuous diffusion process, solution of the stochastic differential equation dX(t)= y (X(t),t)X(t)dt+O(X(t),t)X(t)dW(t)(2.la) X(O) = Xo > 0
(2.lb)
in (2.1) W is a standard Wiener process on a probability space ( n ,~,p), y (x,t) and o (x,t) denote respectively the instantaneous expectation and standard deviation at time t of the rate of return on the risky asset per unit time when X(t) = x (net of the payout rate 6 (x,t) = d(x,t)/x). Notice that the asset holder has a limited liability since X(t) ~ 0 ; if there exists an instant T € [O,T J such that X(T) = 0, then X(t) = 0 for all t € [T,TJ. We assume that y and 0 are bounded, continuously differentiable functions on Q= [ O,+oo) x [ O,T J ; this guarantees that (2.1) has a continuous solution on ( ~ ,~,P) which is unique in the sense of probability law. The agents of the economy are assumed to hold subjective probabilities on the states of the world equivalent to P, i.e. they only have to agree on the "impossible" events a such that pea) = 0 ; they also agree on the values
D. Gaba y
396
of a (but not necessarily of y). The information available at time t to every agent is described by the a -algebra J t generated by the random variables W(s) for s E [ O,t J ; we assume that the uncertainty is resolved at the "maturity" date T, i.e. J T c J. We are finally interested in another type of asset : a contingent claim Y is an asset which provides at maturity T a payout depending on the actual state of the world. It is thus a random variable on ( Si ,;;) and can be written Y = h[ X(T) J where h is a non-negative JT-measurable function (see Harrison-Kreps [8 J). We also assume that the holder of Y receives before maturity a known instantaneous payout f(x,t) ~ per unit time depending upon the value x of the risky asset at time t ~ T. To avoid an explicit description of the preferences and the behaviours of the agents we make the following assumptions on the functioning of the financial market : partial equilibrium analysis can be applied (i.e. each agent can buy or sell any amount of an asset at the market price without influencing it) and there are no arbitrage opportunities (i.e. all riskless securities must have the same rate of return pet»~.
°
2.2. The hedging portfolio and the valuation equation. We now turn to the key problem of finding the 'rational' price for the contingent claim of all times t E [O,T J . Clearly it depends upon the value of the risky asset and we write it U(t) = u[X(t),t J ; at maturity we have U(T)= u[X (T) ,TJ = h[ X(T) J . We assume that u is a twice continuously differentiable function on Q = JO,+oo[xJO,T[ (this will be established in §3) ; !to's formula yields dU(t)=u [X(t),t Jdt+u [ X(t),t JdX(t)+ 1 t x 2 2 2 u [X(t) ,tJ a [X(t) ,tJ X(t) dt = xx 1 2 2 (u + 2 a X uxx + yXux)dt+ a xu dW(t) t x c g[X(t) ,t Jdt+ s[X(t) ,tJdW(t). Following Merton's generalizations of the Black-Scholes model [13 J[ 16 J we consider at each time t E [O,T J a continuously adjusted portfolio TI(t) consisting of the contingent claim of value U(t) dollars, a short position of Vet) dollars in the risky asset (i.e. V(t)/X(t) shares) and a short position of -(U(t)-V(t» dollars in bonds so that the net investment is alwa ys zero ; this is achieved by using the proceeds of payouts, short sales and borrowing to finance the long position on the contingent claim as allowed by the frictionless market assumption. We assume that {Vet) } is adapted to { J } . The instantaneous return on the portfolio ts
~~~~[dX(t)+d[X(t),t Jdt J
1
-[U(t)-V(t)J p (t)dt
[g+f- pU-( y+o- p)V Jdt+ [s-a VJdW(t). We choose a hedging strategy V*(t)=s[X(t),tJ/ o= ux[X(t),t JX(t) which eliminates the stochastic
2
2
-ut[X(t) ,t J - r[X(t) ,tJ X(t) uxx[X(t) ,t ] + +( 6[ X(t) ,tJ-p (t»X(t)u}X(t) ,t J + + p (t)u[X(t) ,t J = f[X(t) ,tJ ':,e have thus established that the price u(x,t) of the contingent claim at date t when the r:sky asset has price x is solution of the following Cauchy problem for the parabolic partial differential equation:
-~
t
o 2x 2u
- l2
hex)
~(x,T)=
xx
+ ( o- p)xu + pu=f in Q x '
(2.2a)
Vx
(2.2 b)
E
[O,+ooJ
addition we specify the boundary condition (at the boundary point x=O)
ill
°
~
u(O,t)
'it
E
[O,T]
(2.2c)
which expresses that a contingent claim on a bankrupt asset is worthless. We shall refer to (2.2) as the valuation equation of finance and we shall give in Section 3 conditions insuring the existence and uniqueness of its solution u(x,t). 2.3. Interpretation and remarks. The valuation equation (2.2) depends only upon observable quantities : the price of the risky asset x, the rate of interest p et), the dividend rate policy o (x,t), the payout to the contingent claim f(x,t), its value at maturity hex) and the variance of the rate of return on the risky asset o (x,t)2 (which can be estimated from time-series data). Given the assumptions of the model it is natural that the price of the contingent claim does not depend upon the risk preferences of investors and the supplies of the different assets in the market. In addition it does not depend upon the expected rate of return y on the risky asset, which is a SUbjective judgement for each agent; notice that key to the derivation of (2.2a) is a spanning assumption according to which the returns over time to the contingent claim can be perfectly replicated by continuous combinaisons of the risky asset and the bond (see Kreps [ 10 J for further comments). These observations are emphasized by the probabilistic interpretation of the solution of the valuation equation : according to Friedman [5 J , under some regularity conditions specified in §3, we have the following explicit formula for the (unique) solution of (2.2)
dTI (t)=dU(t)+f[X(t),t Jdt -
perturbation ; the resulting hedging portfolio TI*(t) is riskless and its (realized) return must be zero to avoid arbitrage profits, which yields
:
U(X,t)=E {J TeXP(-J s p ()' )d ). )f[Z(S) ,s Jds + T t t (2.3a) + exp(- p (). )d ). )h[Z(T) J} J t
where Z(s) is the unique strong solution on
St oc hasti c Process e s in Models of Financial Harke ts [t,TJ of the stochastic differential equation on (rl ,J,P) dZ(s)=[ p (s)- 6 (Z(s) ,s) JZ(s)ds +
(2.3b)
a (Z(s),s)Z(s)dW(s) , Z(t)
=
x.
(2.3c)
This result can be interpreted as follows. The rational price u(x,t) of the contingent claim at time t when the risky asset has price x is given by its value in a risk-neutral world (expected present value of the stream of income over the period [ t,TJ plus its maturity payout) where the expected total return on all assets must be the same ( p- 6 as in (2.3b)+ 6 = P ). See also Cox-Ross [4J for further comments on (2.3). Notice that (2.2a,b) is simply the backward differential equation for u(x,t) defined by (2.4). This probabilistic formulation is ver y helpful to find the pricing function u(x,t) when the risky asset follows a process which is not a diffusion : Cox-Ross [3J used it to include the possibility of jumps (see also Merton [ISJ ) ; recently Harrison-Pliska [9J have used it to extend the theory of contingent claim valuation to a large class of semi-martingales. 2.4. Examples of applications.
=
2
u(X,T-,) = xN( £n(x)+( p+a /2), - e
, N(
av,
£n(x)+( p- a 2/2)~)
Assuming that the value of the firm (total assets) X(t) follows the diffusion (2.1), the value of the debt issue u(x,t) is solution of the valuation equation (2.2) with 6 (x,t) z [d(x,t)+f(t)J/x and hex) = Min{x,D}. Other applications to financial economics are described in Smith [18J and the references therein (see also [6J for applications to project analysis, energy policy •.• ). 3. MATHEMATICAL ANALYSIS OF THE VALUATION EQUATION. In the last section we have established that the value of the contingent claim u(x,t), as a function of the stock price x and the time t, must satisfy on Q = JO,+oo[ x JO,T [ the partial differential equation 2 au I 2 2 a u - -- - - a (x t) x --- + at 2 ' ax2 au (3.la) [ 6 (x,t)- p (t) J x ax + p (t)u ;t(x,t)
f (x, t)
Q
E
together with the Cauchy condition Yx
u (x, T) = h (x)
[0, +00)
E
(3. I b)
and the boundary condition
European options were the first examples treated in the seminal paper by Black and Scholes [2 J . An european call (respectively put) option is a security, issued by an individual, giving to its owner the right to purchase from (respectively sell to) the issuer a share of stock at a given exercise price E at the maturity date T. By homogeneity there is no loss of generality in taking E=I ; assuming that the stock price X(t) follows the diffusion process (2.1), the call price function is the solution of the valuation equation (2.2) for f(x,t) 0 and hex) = (x-I)+ (the call is actually exercised only if the stock price at maturity exceeds the exercise price). Assuming that p and a are constant and that the stock does not pay any dividend (6=0), Black and Scholes derived the analytic solution of (2.2)
-p
397
(2.4)
ah where , i s the time to maturity and N the standard normal distribution function. While options are relativel y unimportant financial instruments, both Black and Scholes [2J and Merton [13 J [14 J recognized that the same approach could be applied for pricing corporate debt. Suppose that a corporation has two classes of outstanding claims : a debt (company bonds) and a residual claim (common stocks). The bond issue is promised a continuous coupon payment per unit time f(t). At time T, there is a promised payment D to the debt holders ; if payment cannot be made, the firm is the defaulted to the debt holders. The first is also pa y ing dividends d(x,t) to the stock holders.
u(O,t)
~
0
;tt
E
[ O,T J
(3.lc)
for consistency we assume that h(O)=f(O,t)=O. Problem (3.1) is a linear parabolic equation on the unbounded domain Q, with unbounded coefficients and its degenerates on the boundary x-o. Notice also that if f(x,t)=6(x,t).x = d(x,t) and hex) = x (i.e. the stock and the contingent claim provide the same flow of income), then u(x,t) = x is a solution to (3.1) (as expected from the economic viewpoint since the stock and the claim are equivalent) ; thus we are looking for solutions which may be unbounded. 3.1. Existence and uniqueness of the solution. Regularity. The degeneracy problem can be treated by a change of variable: let y=log x and O(y,t)= u(exp y,t) ; we similarly denote bye, 0, f, the resulting functionals on ~ x [O,TJ and by 6 the resulting functional on~. If u satisfies equations (3.1) 0 must verify ao I 2 a 20 a2 ao _ f - ~t - -2 a --- + (o-p + -- ) "" + pu a ay 2 2 ay on Q = ~ x J O,T[, (3.2a) O(y,T) = hey) lim
O(y,t) = 0
;ty
E
(3.2b)
IR ,
Yt
E
[O,T].
(3.2c)
y-+-<'"
By definition of the risky asset
e(y,t)~ao>O
Y(y,t) E Q ; (3.2) is then a non-degenerate Cauchy problem. To treat the problem on the unbounded domain we introduce the following functional spaces : let ~ a positive scalar m(y) = exp(-~/) ; for p ~ 2 we define
Q
D. Gabay
398
f: f~mP(y) Iv(Y,~IPdy
~~ I
Q
dt <+oo}
2 I and the Sobolev space III ' ,p (Q) = 2 IJ av a v av p V v, ..,,-, --2' ..,,- EL (Q) • In these definitions {I ay ay at IJ the partial derivatives are taken in the sense of distributions. Notice that if ii E LP(Q) the A
A
}
IJ
condition (3.2c) is satisfied; notice also that ii(y,t) = exp y belongs to 1ll~,I,P for
°
~ hex) ~
\Ix E [ 0,+0:> [
x
using a maximum principle, we can then establish that the solution u of ( 3.1) satisfies
o
$
u(x,t)
x
~
V(x,t)
E
Q,
i.e. the contingent claim has limited liability and its value never exc e eds the value of the risky ass e t. Other natural economic properties can also be derived directly from the equation (see [6 ] ).
IJ > 1.
4. CONCLUSIONS We first establish an existence result which also provides the correct framework for the discrete approximation of (3.2) and its numerical solution. Theorem 3.1. Suppose that p E CO[O,T],
&E f
If -
~~
CO(Q) n LOO (Q) , ELP(Q) and hE IJ -
6 E Loo (Q).
,
wIJ2 'P(R)
then there exists
a unique solution ii E III 2, I ,p (Q) of the Cauchy IJ
problem (3.2). Proof : This result derives from techniques and propositions of Bensoussan-Lions [I] . See [ 6] for details. Provided the problem data are sufficiently regular, we can show that (3.2) has a classical solution in the space of Holder-continuous functions 2 2 x 1 I av a v Cl. 0. /2 C +CI. ,I+CI. /21'ID .....' [O,T])={v VEC '''''-t' EC' } a ax2 with
a ,o. /2 { 1 = w sup
C
y'ry tdO,T] sup
t'rt
J w(y',t)-w(y,t) l< +00 , Iy'-yl o.
Iw(y,t ')-w(y,t) 1< + oo} a 2 It'-t l /
In addition to a brief survey of the theor y of contingent c laims valuation this paper presents some precis e results on the valuation equation of finan c e ; this effort must be viewed as a necessar y first step to the conception of effic ient numeric al methods for its solution (see [6 ] ). We also believe that our analysis applies to more general models where rates of non-negative variables are described by diffusion processes, a situation which arises naturally in economics, management science or population dynamic s for example. The more general of callable claims, like american options or convertible bonds, where the owner can choose the time 8 E [ t,T ] when the exercises its right in order to maximize 8 U(X,t; 8)=E {J eX P (- r; (A)d A)f [Z (S)'S ] ds + r8
t
Jt
(4. I)
ex p (-) p (A)d A)h[Z( 8) ]} t
is an optimal stopping time problem and leads to a free-boundar y problem [12 ] . Using the framework of variational inequalities of Bensoussan-Lions [I ] , similar existence results can be e stablished and numerical methods can be adapted. These problems will be treated in [6 ] .
y Theorem 3.2. In addition to the assumptions of theorem 3.1, let A
a a /2
f E C'
a a /2 a E C' for o.=I-I/p. If 2+ex A
OR x [O,T] and hEC
exists a unique ~ (3.2).
-.-
solut~on
A
OR), then there
2+0. 1+0. /2
UEC '
ORx[O,T ] )
Proof : This regularity result follows from the embedding theorems for
2,I,p and rather dif-
IllIJ
ficult Schauder-type estimates established by Lad yzenskaya, Solonnikov, Ural'tseva [11 ] . See [6 ] for more details. 3.2. Properties of the solution. Suppose that f and h are such that f and h satisf y the conditions of theorem 3.2 and in addit ion
o~
f(x,t)
~
x o (x,t)
v (x, t)
E
Q,
ACKNOWLEDGEMENTS The author is grateful to Douglas Breeden, John Cox, David Kr eps and Rob ert Herton for their encouragements to stud y this problem. Special thanks are due to my fri ends Italo Dolcetta and Umberto Hosco for their assistance and companionship during this work.
St oc ha s t i c Pr oc es s e s in Models of Financ ial Narkets
S. REFERENCES [IJ A.BENSOUSSAN et J.L.LIONS, Applications des inequations variationnelles en contr61e stochastique (Paris : Dunod) 1978. [ 2J F . BLACK and M.SCHOLES, "The pricing of options and corporate liabilities", J. Pol.Econ.81 (1973), 637-654. [3J J.C.COX and S.A.ROSS, "The valuation of options for alternative stochastic processes", J.Fin.Econ.3 (1976) 145-166. [4J J.C.COX and S.A.ROSS, "A survey of some new results in financial option pricing theory", J.Finance 31 (1976),383-402. [SJ A.FRIEDMAN, Stochastic differential equations and applications, Vol. I (New York: Academic Pres s) 1975. [ 6 J D.CABAY, The valuation equation of finance : mathematical analysis and computational methods (in preparation). [7] A. CLEIT, "Valua t ion of general cont ingent claims", J .Fin. Econ. 6 (1978) 71-87. [ 8 J J;M.HARRISON and D.M.KREPS, "Martingales and arbitrage in multiperiod securities markets", J.E.T.20 (1979) 381-408. [ 9J J.M.HARRISON and S.R. PLISKA, "Martingales and stochastic integrals in the theory of continuous trading", Stochastic Processes and their Applications II (1981), 215-260. [IO J D.KREPS, "Three e ssays on capital markets", IMSSS Tech.R.N° 298, Stanford University (1979) • [ IIJ O.A.LADYZENSKAYA, V.A.SOLONNIKOV, N.N.URAL'TSEVA, Linear and Quasilinear equations of parabolic t ype, American Mathematical Societ y (Providence, R.I.) 1968. [ 12J H.P.McKEAN Jr.,"Appendix : a free boundary problem for the heat equation arising from a problem in mathematical economics", Industrial Management Rev.6 (1965), 32-39. [13 J R.C.MERTON, "Theory of rational option pricing", Bell J. Econ.Mgt.4 (1973) 141-183. [14 J R.C.MERTON, "The pricing of corporate debt: the risk structure of inter est rates", J.Finance 29 (1974 ) 449- 470 . [IS J R.C.MERTON, "Option pricing when underlying stock returns are discontinuous", J.Fin.Econ.3 (1976) 125-144. [16J R.C . MERTON, "On the pricing of contingent claims and the Modigliani-Miller theorem", J.Fin.Econ.S (1977) 241-249. [ 17] R.C.MERTON, "On the mathematic s and economics assumptions of continuous-time financial model" in W.F.Wharpe ed., Financial Economics : Essa ys in honor Of Paul Cootner (Englewood Cliffs Prentice Hall) forthcoming. [18 J C. W.SMITH Jr., "Option pric ing a review", J.Fin.Econ.3 (19 76) 3-51.
399
Copyrighl © IFAC 3rd Svmposium
16
- STOCHASTIC PROCESSES
Control of Distri buted Pa rameter Svstt"ms
T oulouse . Fra nce. 1982
.
STOCHASTIC PROCESSES IN MODELS OF FINANCIAL MARKETS: THE VALUATION EQUATION OF FINANCE AND ITS APPLICATIONS D. Gabay Laboratoire d'Analyse Num eriqu e, Universite P. et M . Curie and INRIA , Domaine de Voluceau , B .P. 105, Le Ch esnay Cedex 78153, France
Abstract. This paper presents a brief survey of the continuous-time model of financial market used to develop the theory of contingent claims valuation. New results are given on the existence, uniqueness and regularity of the solution of the resulting evolution problems. Proofs and computational methods will be given in [6J. I. INTRODUCTION
In this paper we present a general and unifying framework for many recent contributions of "continuous-time analysis" to the development of financial economic theory. Assuming that trading takes place continuously in time and that the underlying stochastic variables follow diffusion processes (see Merton [ 17 J for a discussion of the relevance of this assumption) it is possible under ideal assumptions on the market mechanisms to derive a theory of contingent claims pricing using only observable variables. Black and Scholes in their seminal paper [2J inaugurated this approach to establish that the price of an "european" option on a stock must satisfy a parabolic partial differential equation. They actually found an explicit formula for its solution which has been widely used to test the validity of the ideal market assumptions on the many available option prices time series. The same concept has also been used for the pricing of corporate liabilities, forward contr.a c.t s, insurance contracts ... (see Smith .[ 18J for a survey). In the next section we describe a general model for the financial market and derive the valuation equation of finance. Section 3 is devoted to its mathematical analysis ; this problem has been approached by Gleit [7J but we give here stronger results. Because of space limitation we defer the proofs to a forthcoming report [6 J where numerical methods will also be proposed. In the conclusion (Section 4) we briefly state the optimal stopping problem associated to optimal conversion strategies for callable contingent claims like american options or convertible bonds.
2. THE VALUATION OF CONTINGENT CLAIMS. 2.1. The model: definitions, notations and assumptions. Following Merton [ 16 J we consider a financial market where trading takes place conti395
nuously over a time interval [O,TJ. For simplicity we also assume that it is "frictionless" : there are no transaction costs nor taxes ; borrowing and shortselling are allowed without restriction and the borrowing rate equals the lending rate. There can be as many assets or securities as one wishes but we are interested in only three of them. First, there is a riskless asset, called bond thereafter, yielding a known instantaneous rate of return per unit time pe t» ~ O. We normalize prices by taking as unit the value of the bond at time 0 ; clearly its value at time t is B(t) = exp ( J tp (S)dS). Next, there is a risky asset which p~ovides to its holder a known instantaneous net payout d(x,t) per unit time when the asset price is x at time t. We assume that the price X(t) of the risky asset follows a continuous diffusion process, solution of the stochastic differential equation dX(t)= y (X(t),t)X(t)dt+O(X(t),t)X(t)dW(t)(2.la) X(O) = Xo > 0
(2.lb)
in (2.1) W is a standard Wiener process on a probability space ( n ,~,p), y (x,t) and o (x,t) denote respectively the instantaneous expectation and standard deviation at time t of the rate of return on the risky asset per unit time when X(t) = x (net of the payout rate 6 (x,t) = d(x,t)/x). Notice that the asset holder has a limited liability since X(t) ~ 0 ; if there exists an instant T € [O,T J such that X(T) = 0, then X(t) = 0 for all t € [T,TJ. We assume that y and 0 are bounded, continuously differentiable functions on Q= [ O,+oo) x [ O,T J ; this guarantees that (2.1) has a continuous solution on ( ~ ,~,P) which is unique in the sense of probability law. The agents of the economy are assumed to hold subjective probabilities on the states of the world equivalent to P, i.e. they only have to agree on the "impossible" events a such that pea) = 0 ; they also agree on the values
D. Gaba y
396
of a (but not necessarily of y). The information available at time t to every agent is described by the a -algebra J t generated by the random variables W(s) for s E [ O,t J ; we assume that the uncertainty is resolved at the "maturity" date T, i.e. J T c J. We are finally interested in another type of asset : a contingent claim Y is an asset which provides at maturity T a payout depending on the actual state of the world. It is thus a random variable on ( Si ,;;) and can be written Y = h[ X(T) J where h is a non-negative JT-measurable function (see Harrison-Kreps [8 J). We also assume that the holder of Y receives before maturity a known instantaneous payout f(x,t) ~ per unit time depending upon the value x of the risky asset at time t ~ T. To avoid an explicit description of the preferences and the behaviours of the agents we make the following assumptions on the functioning of the financial market : partial equilibrium analysis can be applied (i.e. each agent can buy or sell any amount of an asset at the market price without influencing it) and there are no arbitrage opportunities (i.e. all riskless securities must have the same rate of return pet»~.
°
2.2. The hedging portfolio and the valuation equation. We now turn to the key problem of finding the 'rational' price for the contingent claim of all times t E [O,T J . Clearly it depends upon the value of the risky asset and we write it U(t) = u[X(t),t J ; at maturity we have U(T)= u[X (T) ,TJ = h[ X(T) J . We assume that u is a twice continuously differentiable function on Q = JO,+oo[xJO,T[ (this will be established in §3) ; !to's formula yields dU(t)=u [X(t),t Jdt+u [ X(t),t JdX(t)+ 1 t x 2 2 2 u [X(t) ,tJ a [X(t) ,tJ X(t) dt = xx 1 2 2 (u + 2 a X uxx + yXux)dt+ a xu dW(t) t x c g[X(t) ,t Jdt+ s[X(t) ,tJdW(t). Following Merton's generalizations of the Black-Scholes model [13 J[ 16 J we consider at each time t E [O,T J a continuously adjusted portfolio TI(t) consisting of the contingent claim of value U(t) dollars, a short position of Vet) dollars in the risky asset (i.e. V(t)/X(t) shares) and a short position of -(U(t)-V(t» dollars in bonds so that the net investment is alwa ys zero ; this is achieved by using the proceeds of payouts, short sales and borrowing to finance the long position on the contingent claim as allowed by the frictionless market assumption. We assume that {Vet) } is adapted to { J } . The instantaneous return on the portfolio ts
~~~~[dX(t)+d[X(t),t Jdt J
1
-[U(t)-V(t)J p (t)dt
[g+f- pU-( y+o- p)V Jdt+ [s-a VJdW(t). We choose a hedging strategy V*(t)=s[X(t),tJ/ o= ux[X(t),t JX(t) which eliminates the stochastic
2
2
-ut[X(t) ,t J - r[X(t) ,tJ X(t) uxx[X(t) ,t ] + +( 6[ X(t) ,tJ-p (t»X(t)u}X(t) ,t J + + p (t)u[X(t) ,t J = f[X(t) ,tJ ':,e have thus established that the price u(x,t) of the contingent claim at date t when the r:sky asset has price x is solution of the following Cauchy problem for the parabolic partial differential equation:
-~
t
o 2x 2u
- l2
hex)
~(x,T)=
xx
+ ( o- p)xu + pu=f in Q x '
(2.2a)
Vx
(2.2 b)
E
[O,+ooJ
addition we specify the boundary condition (at the boundary point x=O)
ill
°
~
u(O,t)
'it
E
[O,T]
(2.2c)
which expresses that a contingent claim on a bankrupt asset is worthless. We shall refer to (2.2) as the valuation equation of finance and we shall give in Section 3 conditions insuring the existence and uniqueness of its solution u(x,t). 2.3. Interpretation and remarks. The valuation equation (2.2) depends only upon observable quantities : the price of the risky asset x, the rate of interest p et), the dividend rate policy o (x,t), the payout to the contingent claim f(x,t), its value at maturity hex) and the variance of the rate of return on the risky asset o (x,t)2 (which can be estimated from time-series data). Given the assumptions of the model it is natural that the price of the contingent claim does not depend upon the risk preferences of investors and the supplies of the different assets in the market. In addition it does not depend upon the expected rate of return y on the risky asset, which is a SUbjective judgement for each agent; notice that key to the derivation of (2.2a) is a spanning assumption according to which the returns over time to the contingent claim can be perfectly replicated by continuous combinaisons of the risky asset and the bond (see Kreps [ 10 J for further comments). These observations are emphasized by the probabilistic interpretation of the solution of the valuation equation : according to Friedman [5 J , under some regularity conditions specified in §3, we have the following explicit formula for the (unique) solution of (2.2)
dTI (t)=dU(t)+f[X(t),t Jdt -
perturbation ; the resulting hedging portfolio TI*(t) is riskless and its (realized) return must be zero to avoid arbitrage profits, which yields
:
U(X,t)=E {J TeXP(-J s p ()' )d ). )f[Z(S) ,s Jds + T t t (2.3a) + exp(- p (). )d ). )h[Z(T) J} J t
where Z(s) is the unique strong solution on
St oc hasti c Process e s in Models of Financial Harke ts [t,TJ of the stochastic differential equation on (rl ,J,P) dZ(s)=[ p (s)- 6 (Z(s) ,s) JZ(s)ds +
(2.3b)
a (Z(s),s)Z(s)dW(s) , Z(t)
=
x.
(2.3c)
This result can be interpreted as follows. The rational price u(x,t) of the contingent claim at time t when the risky asset has price x is given by its value in a risk-neutral world (expected present value of the stream of income over the period [ t,TJ plus its maturity payout) where the expected total return on all assets must be the same ( p- 6 as in (2.3b)+ 6 = P ). See also Cox-Ross [4J for further comments on (2.3). Notice that (2.2a,b) is simply the backward differential equation for u(x,t) defined by (2.4). This probabilistic formulation is ver y helpful to find the pricing function u(x,t) when the risky asset follows a process which is not a diffusion : Cox-Ross [3J used it to include the possibility of jumps (see also Merton [ISJ ) ; recently Harrison-Pliska [9J have used it to extend the theory of contingent claim valuation to a large class of semi-martingales. 2.4. Examples of applications.
=
2
u(X,T-,) = xN( £n(x)+( p+a /2), - e
, N(
av,
£n(x)+( p- a 2/2)~)
Assuming that the value of the firm (total assets) X(t) follows the diffusion (2.1), the value of the debt issue u(x,t) is solution of the valuation equation (2.2) with 6 (x,t) z [d(x,t)+f(t)J/x and hex) = Min{x,D}. Other applications to financial economics are described in Smith [18J and the references therein (see also [6J for applications to project analysis, energy policy •.• ). 3. MATHEMATICAL ANALYSIS OF THE VALUATION EQUATION. In the last section we have established that the value of the contingent claim u(x,t), as a function of the stock price x and the time t, must satisfy on Q = JO,+oo[ x JO,T [ the partial differential equation 2 au I 2 2 a u - -- - - a (x t) x --- + at 2 ' ax2 au (3.la) [ 6 (x,t)- p (t) J x ax + p (t)u ;t(x,t)
f (x, t)
Q
E
together with the Cauchy condition Yx
u (x, T) = h (x)
[0, +00)
E
(3. I b)
and the boundary condition
European options were the first examples treated in the seminal paper by Black and Scholes [2 J . An european call (respectively put) option is a security, issued by an individual, giving to its owner the right to purchase from (respectively sell to) the issuer a share of stock at a given exercise price E at the maturity date T. By homogeneity there is no loss of generality in taking E=I ; assuming that the stock price X(t) follows the diffusion process (2.1), the call price function is the solution of the valuation equation (2.2) for f(x,t) 0 and hex) = (x-I)+ (the call is actually exercised only if the stock price at maturity exceeds the exercise price). Assuming that p and a are constant and that the stock does not pay any dividend (6=0), Black and Scholes derived the analytic solution of (2.2)
-p
397
(2.4)
ah where , i s the time to maturity and N the standard normal distribution function. While options are relativel y unimportant financial instruments, both Black and Scholes [2J and Merton [13 J [14 J recognized that the same approach could be applied for pricing corporate debt. Suppose that a corporation has two classes of outstanding claims : a debt (company bonds) and a residual claim (common stocks). The bond issue is promised a continuous coupon payment per unit time f(t). At time T, there is a promised payment D to the debt holders ; if payment cannot be made, the firm is the defaulted to the debt holders. The first is also pa y ing dividends d(x,t) to the stock holders.
u(O,t)
~
0
;tt
E
[ O,T J
(3.lc)
for consistency we assume that h(O)=f(O,t)=O. Problem (3.1) is a linear parabolic equation on the unbounded domain Q, with unbounded coefficients and its degenerates on the boundary x-o. Notice also that if f(x,t)=6(x,t).x = d(x,t) and hex) = x (i.e. the stock and the contingent claim provide the same flow of income), then u(x,t) = x is a solution to (3.1) (as expected from the economic viewpoint since the stock and the claim are equivalent) ; thus we are looking for solutions which may be unbounded. 3.1. Existence and uniqueness of the solution. Regularity. The degeneracy problem can be treated by a change of variable: let y=log x and O(y,t)= u(exp y,t) ; we similarly denote bye, 0, f, the resulting functionals on ~ x [O,TJ and by 6 the resulting functional on~. If u satisfies equations (3.1) 0 must verify ao I 2 a 20 a2 ao _ f - ~t - -2 a --- + (o-p + -- ) "" + pu a ay 2 2 ay on Q = ~ x J O,T[, (3.2a) O(y,T) = hey) lim
O(y,t) = 0
;ty
E
(3.2b)
IR ,
Yt
E
[O,T].
(3.2c)
y-+-<'"
By definition of the risky asset
e(y,t)~ao>O
Y(y,t) E Q ; (3.2) is then a non-degenerate Cauchy problem. To treat the problem on the unbounded domain we introduce the following functional spaces : let ~ a positive scalar m(y) = exp(-~/) ; for p ~ 2 we define
Q
D. Gabay
398
f: f~mP(y) Iv(Y,~IPdy
~~ I
Q
dt <+oo}
2 I and the Sobolev space III ' ,p (Q) = 2 IJ av a v av p V v, ..,,-, --2' ..,,- EL (Q) • In these definitions {I ay ay at IJ the partial derivatives are taken in the sense of distributions. Notice that if ii E LP(Q) the A
A
}
IJ
condition (3.2c) is satisfied; notice also that ii(y,t) = exp y belongs to 1ll~,I,P for
°
~ hex) ~
\Ix E [ 0,+0:> [
x
using a maximum principle, we can then establish that the solution u of ( 3.1) satisfies
o
$
u(x,t)
x
~
V(x,t)
E
Q,
i.e. the contingent claim has limited liability and its value never exc e eds the value of the risky ass e t. Other natural economic properties can also be derived directly from the equation (see [6 ] ).
IJ > 1.
4. CONCLUSIONS We first establish an existence result which also provides the correct framework for the discrete approximation of (3.2) and its numerical solution. Theorem 3.1. Suppose that p E CO[O,T],
&E f
If -
~~
CO(Q) n LOO (Q) , ELP(Q) and hE IJ -
6 E Loo (Q).
,
wIJ2 'P(R)
then there exists
a unique solution ii E III 2, I ,p (Q) of the Cauchy IJ
problem (3.2). Proof : This result derives from techniques and propositions of Bensoussan-Lions [I] . See [ 6] for details. Provided the problem data are sufficiently regular, we can show that (3.2) has a classical solution in the space of Holder-continuous functions 2 2 x 1 I av a v Cl. 0. /2 C +CI. ,I+CI. /21'ID .....' [O,T])={v VEC '''''-t' EC' } a ax2 with
a ,o. /2 { 1 = w sup
C
y'ry tdO,T] sup
t'rt
J w(y',t)-w(y,t) l< +00 , Iy'-yl o.
Iw(y,t ')-w(y,t) 1< + oo} a 2 It'-t l /
In addition to a brief survey of the theor y of contingent c laims valuation this paper presents some precis e results on the valuation equation of finan c e ; this effort must be viewed as a necessar y first step to the conception of effic ient numeric al methods for its solution (see [6 ] ). We also believe that our analysis applies to more general models where rates of non-negative variables are described by diffusion processes, a situation which arises naturally in economics, management science or population dynamic s for example. The more general of callable claims, like american options or convertible bonds, where the owner can choose the time 8 E [ t,T ] when the exercises its right in order to maximize 8 U(X,t; 8)=E {J eX P (- r; (A)d A)f [Z (S)'S ] ds + r8
t
Jt
(4. I)
ex p (-) p (A)d A)h[Z( 8) ]} t
is an optimal stopping time problem and leads to a free-boundar y problem [12 ] . Using the framework of variational inequalities of Bensoussan-Lions [I ] , similar existence results can be e stablished and numerical methods can be adapted. These problems will be treated in [6 ] .
y Theorem 3.2. In addition to the assumptions of theorem 3.1, let A
a a /2
f E C'
a a /2 a E C' for o.=I-I/p. If 2+ex A
OR x [O,T] and hEC
exists a unique ~ (3.2).
-.-
solut~on
A
OR), then there
2+0. 1+0. /2
UEC '
ORx[O,T ] )
Proof : This regularity result follows from the embedding theorems for
2,I,p and rather dif-
IllIJ
ficult Schauder-type estimates established by Lad yzenskaya, Solonnikov, Ural'tseva [11 ] . See [6 ] for more details. 3.2. Properties of the solution. Suppose that f and h are such that f and h satisf y the conditions of theorem 3.2 and in addit ion
o~
f(x,t)
~
x o (x,t)
v (x, t)
E
Q,
ACKNOWLEDGEMENTS The author is grateful to Douglas Breeden, John Cox, David Kr eps and Rob ert Herton for their encouragements to stud y this problem. Special thanks are due to my fri ends Italo Dolcetta and Umberto Hosco for their assistance and companionship during this work.
St oc ha s t i c Pr oc es s e s in Models of Financ ial Narkets
S. REFERENCES [IJ A.BENSOUSSAN et J.L.LIONS, Applications des inequations variationnelles en contr61e stochastique (Paris : Dunod) 1978. [ 2J F . BLACK and M.SCHOLES, "The pricing of options and corporate liabilities", J. Pol.Econ.81 (1973), 637-654. [3J J.C.COX and S.A.ROSS, "The valuation of options for alternative stochastic processes", J.Fin.Econ.3 (1976) 145-166. [4J J.C.COX and S.A.ROSS, "A survey of some new results in financial option pricing theory", J.Finance 31 (1976),383-402. [SJ A.FRIEDMAN, Stochastic differential equations and applications, Vol. I (New York: Academic Pres s) 1975. [ 6 J D.CABAY, The valuation equation of finance : mathematical analysis and computational methods (in preparation). [7] A. CLEIT, "Valua t ion of general cont ingent claims", J .Fin. Econ. 6 (1978) 71-87. [ 8 J J;M.HARRISON and D.M.KREPS, "Martingales and arbitrage in multiperiod securities markets", J.E.T.20 (1979) 381-408. [ 9J J.M.HARRISON and S.R. PLISKA, "Martingales and stochastic integrals in the theory of continuous trading", Stochastic Processes and their Applications II (1981), 215-260. [IO J D.KREPS, "Three e ssays on capital markets", IMSSS Tech.R.N° 298, Stanford University (1979) • [ IIJ O.A.LADYZENSKAYA, V.A.SOLONNIKOV, N.N.URAL'TSEVA, Linear and Quasilinear equations of parabolic t ype, American Mathematical Societ y (Providence, R.I.) 1968. [ 12J H.P.McKEAN Jr.,"Appendix : a free boundary problem for the heat equation arising from a problem in mathematical economics", Industrial Management Rev.6 (1965), 32-39. [13 J R.C.MERTON, "Theory of rational option pricing", Bell J. Econ.Mgt.4 (1973) 141-183. [14 J R.C.MERTON, "The pricing of corporate debt: the risk structure of inter est rates", J.Finance 29 (1974 ) 449- 470 . [IS J R.C.MERTON, "Option pricing when underlying stock returns are discontinuous", J.Fin.Econ.3 (1976) 125-144. [16J R.C . MERTON, "On the pricing of contingent claims and the Modigliani-Miller theorem", J.Fin.Econ.S (1977) 241-249. [ 17] R.C.MERTON, "On the mathematic s and economics assumptions of continuous-time financial model" in W.F.Wharpe ed., Financial Economics : Essa ys in honor Of Paul Cootner (Englewood Cliffs Prentice Hall) forthcoming. [18 J C. W.SMITH Jr., "Option pric ing a review", J.Fin.Econ.3 (19 76) 3-51.
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