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OPTIMAL BANK PORTFOLIO CHOICE
UNDER FIXED-RATE DEPOSIT INSURANCE
by Anlong Li
Anlong Li is a Ph.D. candidate in operations
research at the Weatherhead School of
Management at Case Western Reserve University,
Cleveland, Ohio, and is a research associate
at the Federal Reserve Bank of Cleveland. The
author wishes to thank Peter Ritchken and
James Thomson for their valuable input and
Andrew Chen, Ramon DeGennaro, Richard
Jefferis, and Joseph Haubrich for their
helpful comments.
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated
to stimulate discussion and critical comment.
The views stated herein are those of the
author and not necessarily those of the
Federal Reserve Bank of Cleveland or of the
Board of Governors of the Federal Reserve
System
.
August 1991
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Abstract
This paper analyzes the optimal investment decisions of insured
banks under fixed-rate deposit insurance.
value,
In the presence of charter
trade-offs exist between preserving the charter and exploiting
deposit insurance.
Allowing banks to dynamically revise their asset
portfolios has a significant impact on both the investment decisions and
the fair cost of deposit insurance.
can
be
solved
analytically
for
The optimal bank portfolio problem
constant
charter
value.
The
corresponding deposit insurance is shown to be a put option that matures
sooner than the audit date.
An efficient numerical procedure is also
developed to handle more general situations.
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1. Introduction
The current system of fixed-rate deposit insurance in the United
States gives insured banks the incentive to take on riskier investments
than they otherwise would. To relate the cost of deposit insurance to
a bank's investment risk, Merton (1977) shows that deposit insurance
grants a put option to the insured bank. Under this model, banks tend
to take on extremely risky projects to exploit the put option. As a
result, fixed-rate deposit insurance is apt to be underpriced for
high-risk-taking banks and overpriced for low-risk-taking banks.
Implementation of option models for valuing deposit insurance can be
found in Marcus and Shaked (1984) and ROM and Verma (1986).
In reality, not all banks take extreme risks. Being in business is
a privilege and is reflected in a firm's charter value or growth option.
Extreme risk-taking may lead a bank into insolvency, forcing it out of
business by regulators. The charter value comes from many sources, such
as monopoly rents in issuing deposits, economies of scale, superior
information in the financial markets, and reputation.
Taking into account the charter value, Marcus (1984) shows that
banks either minimize or maximize their risk exposure as a result of the
trade-offs between the put option value and the charter value. Under a
different setting, Buser, Chen, and Kane (1981) show that the trade-offs
reestablish an interior solution to the capital structure decision.
They also argue that capital requirements and other regulations serve as
additional implicit constraints to discourage extreme risk-taking.
Almost all models of deposit insurance assume that banks' asset
risk is exogenously given. With the exception of the discussion in
Ritchken et al. (19911, the flexibility for banks to dynamically adjust
their investment decisions has been mostly ignored.
However, their
model allows only a finite number of portfolio revisions between audits.
In this paper, I establish a continuous-trading model to identify
how an equity-maximizing bank dynamically responds to flat-rate deposit
insurance schemes and how this affects the actuarially fair value of
deposit insurance.
Since investment decisions are carried out by
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optimizing the investment portfolio, I model the problem as the optimal
control of a diffusion process. Upon obtaining the optimal portfolio,
the actuarially fair cost of insurance can be easily calculated.
In this model, I use the traditional dynamic programming approach
(Fleming and Rishel [I97511. The disadvantage of this approach is that
it often reduces the problem to an intractable partial differential
equation (PDE) where analytical soiutions are rare. Merton*s (1971
application to the optimal consumption problem is among the few
cases in which analytical solutions are obtained. Fortunately, in this
problem the resulting PDE can be explicitly solved provided that the
charter value is constant. Even though I assume lognormal price to
warrant an analytical solution, general price distributions can be
easily built into the model.
The dynamic programming procedure can also be carried out
numerically by lattice approximation. This is especially attractive
when more realistic assumptions are made. As the bank changes its
portfolio risk over time, the most common binomial model is no longer
path-independent, and the problem size grows exponentially with the
number of partitions. This difficulty is resolved by using a trinomial
lattice. The lattice is set up in such a way that the decision variable
is incorporated into the transition probabilities rather than into the
step size.
This paper is organized as follows: Section 2 formulates the model
and summarizes the results under no portfolio revision. Section 3
solves the optimal portfolio problem under continuous portfolio
revision. The value of deposit insurance is derived based on the
optimal portfolio decisions.
Section 4 presents the trinomial
approximation of controlled diffusion process. Section 5 extends the
model to more general situations, and section 6 concludes the paper.
The proof of the main results can be found in the appendix.
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2.
The Static Model
- No Portfolio Revision
Investment Opportunities:
Assume that financial markets are complete.
The bank can invest in both riskless bonds (earning rate r) and a
portfolio of risky securities that follows a geometric Wiener process
Capital and Liability:
The bank's
initial asset X(0) consists of
For simplicity, I asshe no net
capital K(O1 and deposit base D(0).
external cash inflows into the deposit base, no capital injections, and
Because all
no dividend payments during the time interval [O,TI.
deposits are insured, I assume that deposits earn the riskless rate r.
Let L(t1 be the liability at time t; then
Investment Decisions:
Management decides at time zero to put a fraction
q of its assets in risky securities and the remaining in riskless bonds.
Without portfolio revision, q is fixed before the audit.
The market value of the assets at time t is
where
is the standard normal
random variable with density and
distribution function n 0 and N O , respectively.
Auditing and Closure Rules:
The regulator conducts an audit at time T:
If the bank is solvent, i.e., the market value of its assets exceeds its
liabilities, it claims the residual X(T)
-
L(T) and keeps its charter.
If the bank is insolvent, the regulator takes over and equityholders
receive nothing.
at time T.
Let C(T1 represent the charter value of a solvent bank
C(T) is assumed to be a constant fraction of total
liabilities. Define
C(t) = fL(t),
0
< f < 1.
Let V(t;ql be the equity value at time t under policy q.
Then
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X(T)
V
T =
{0
-
if X(T) > L(T)
L(T) + G(T)
(5)
otherwise.
The equity value at time 0 can be obtained by using standard option
pricing
V(0.q)
techniques,
=
t
qX(OIN(dl)-[L(O)-C(0)-(1-q)X(0)1N(d2)
if (l-q)X(O)<L(O)
X(0)-L(O)+G(O)
otherwise ,
( 61
where
On behalf of the shareholders, management will maximize the equity
value by choosing the optimal fraction q* such that
V(O,q*) = max
9
{
V(0,q)
I.
This optimization problem can be solved analytically.
insolvent banks are treated separately.
(7
Solvent and
Even though an initially
insolvent bank would be an unusual case, it is included to complete the
analysis.
Theorem 1.
optimal.
I summarize these results in theorems 1 and 2.
For an insolvent bank without portfolio revisions, q* = 1 is
Consequently, the value of the deposit insurance1 is
where
The value of deposit insurance always refers to the actuarially fair
cost of deposit insurance.
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Theorem 2.
For a solvent bank without portfolio revision, the optimal
policy is
where
Consequently, the value of deposit insurance is
Theorems 1 and 2 show that without revision opportunities between
audits, banks always take extreme positions. Regardless of the charter
value, an insolvent bank always takes the riskiest position. With a
small charter value, a solvent bank may be better off by taking the
riskiest position so as to maximize the value of the deposit insurance.
Only solvent banks with a sufficiently large capital-deposit ratio m or
a relatively high charter value will invest in fiskless bonds.
2
The value of insurance for an insolvent bank, or for a solvent bank
with f < 1
-
H(m1, is the same as in Merton (1977) where the charter
value is zero.
When f
2
1
-
H(m), risk-taking is discouraged and the
insurance has no intrinsic value.
J
This can be shown from the fact that H(m1 is an increasing function
of m with H(-1) = 0 and H(m) = 1.
To be precise, when f = 1
- H(m1,
a bank is indifferent between q = 0
(preserving the charter) and q = 1 (exploiting the insurance).
the bank's
However,
actual decision on q does affect the value of insurance.
This discontinuity in the insurance value is one of the drawbacks of
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3. Continuous Portfolio Revision
In this section, I assume that banks can revise their investment
portfolios continuously over time at no cost.
Let X(t) be the market
value of the assets and q = q(t,X(t)) be the fraction of risky assets in
the portfolio at time t E [O,Tl. Then X(t) follows a diffusion process
where V(t) is a standard Brownian motion.
The liability and charter
value are given by equations (2) and (41, respectively.
purposes, one can substitute p with r in equation (11).
be the maximum equity value of the bank at time t.
-r (T-t 1
J(t,X(t)) = max Et [J(TnXT)e
'I
For valuation
Let J(t,X(t))
Then
I.
(12)
It has the boundary condition
J(T.X(T)
=
- L(T)
{ X(T)
0
+
C(T)
if X(T)
2
L(T)
(13)
otherwise.
We are interested in the maximum equity value J(O,X(O)) for any
given X(O1 = Xo at time zero and the corresponding optimal policy qf(t)
for all t E [O,Tl. This problem is solved by using dynamic programing.
The results are presented in the following theorem.
Theorem 3.
Let
7
be the solution of the following equation4
Suppose the asset value at time t is X(t).
Under the assumptions of
section 2 and continuous portfolio revision, the optimal decision q*(t)
and the corresponding equity value J(t,X(t)) are as follows.
static models.
4
If the solution is negative, simply let
7
= 0.
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( I ) ~f t E [r,T) and X(t1
(21
If t
E [r,TI
2
L(t), then qe(t) = 0 , and
and X(t) < L(t1, then q*(tl = 1, and
where
r1 =
31
If t
E
ln[X(t)/L(t)l + cr2(~-t1/2
m
[O.t), then $(t) = 1, and
J(t.X(t)) = C(t) X
~
+ X(t1N(r31
N
-
+
L ( ~- ) N ( T ~ . - ~ ~ . P ) I
[L(t1-C(t)lN(r4)
where
and N(x,y,pI is the standard cumulative bivariate normal
distribution with correlation coefficient p.
5
In summary, the optimal policy is
q*
;to
1
if t
E [T,T)and X(t1 r L(t1
if t e [O,r1 or X(t1 < L(t1.
Theorem 3 clearly illustrates the trade-offs between preserving the
-
-
Actually, when t
E [t,T1 and ~ ( t >) ~ ( t ) any
,
q
is optimal as long as
q is set at 0 when X(t1 hits the solvency curve ~ ( t 1 .
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charter and exploiting the deposit insurance. The deposit insurance is
essentially a put option on the bank's assets that matures at the time
of the audit. The longer the time before an audit, the higher the value
of the deposit insurance. Prior to time r, the deposit insurance is
more valuable than the fixed charter value, and shareholders exploit the
deposit insurance by choosing q = 1. After time r, since the audit is
near, the deposit insurance is less valuable than the charter, and
shareholders will do their best to ensure that the market value of the
bank's assets remains above the solvency curve L(t) in order to preserve
its charter. Figure 1 shows this optimal policy where the riskless rate
is set to zero.
1
X(t) (Asset Value)
q =
0
-- L(0)
L(t)
-
(Time
I
>
t
Figure 1. Optimal Portfolio Policies
The critical time r is uniquely determined by equation (14) for any
0 a
f s 1. To see this, rewrite equation (14) with 13 = 6 / 2 :
Since the left-hand side of (18) decreases from
to
+a,
a positive 6 is uniquely determined.
+a,
to 1 as /3 goes from 0
We can also show that r is
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increasing in c and decreasing in f. However, it depends on neither the
the riskless interest rate r nor the banks' capital-deposit ratio m.
As an example, consider an audit period of one year.
Suppose the
volatility of the risky assets is c = 10 percent annually, and the
charter value is f = 10 percent of the deposit base.
(14) yields r = 0.293.
0.834.
Solving equation
If f drops to 5 percent, r will increase to
If there is no charter at all, r equals T, the audit date.
To obtain the value of the deposit insurance I(O1, note that the
equity value comes from three sources:
namely,
the initial capital
K(O), the deposit insurance I(O1, and the charter value C(O1.
where P{X(T)kL[T))
That is,
is the probability that the bank passes the audit.
Following the same argument as in the proof of theorem 3, we have
where the 7's are evaluated at time t = 0.
Substituting this into
equation (191, we have the actuarially fair value of deposit insurance
for a bank with continuous revision opportunities
where 7 and 7 are evaluated at time t = 0.
3
4
This insurance value can be viewed as a put option on the bank's
assets with maturity r instead of T.
This clearly explains the impact
of the charter value and the continuous portfolio revision on the value
of deposit insurance.
Since
+ <
T as long as f > 0, the deposit
insurance is less valuable in the presence of charter value.
Compared
to the static model, the insurance value in equation (20) is continuous
in terms of charter value and capital-asset ratio.
Even for very highly
capitalized banks, as long as r > 0, the insurance has a positive value.
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4.
Trinomial Approximation
For general terminal payoff functions other than the one in
equation (51, analytical solutions may not always exist, and numerical
procedures must be used to solve the optimal portfolio problem. Without
portfolio revisions, a simple binomial model can be used to approximate
the bank's asset value. However, when the portfolio is revised, the
resulting lattice becomes path-dependent.
To see this, partition the audit period [O,Tl into n subintervals
of equal length h = T/n. The asset portfolio may be revised at discrete
decision points tl= ih, i = O,l,...,n-1. Let q(tl,X(tl)) be the revised
fraction of risky investments at time t if the market value of the
1
bank's assets is X(tl).
Let q be initially set to qo. The portfolio is
revised at time tl by changing qo to ql at the up state and q2 at the
down state, respectively. The two-period binomial lattice looks like
where
for i = 0,1,2. Obviously, if uod1 # dou2, the lattice is path-dependent.
To overcome this difficulty, a path-independent lattice is first
set up as if there is no portfolio revision.
Then, when the portfolio
is revised to a new q value at a revision point, one changes only the
transition probabilities such that the drift and variance terms match
locally.
This suggests adding one more degree of freedom to the
lattice.
Consider the following trinomial lattice when the asset value
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at time tl is XI:
The transition probabilities are set to
Obviously,
As h
-+ 0 ,
zJ pJ = 1 .
The first and second local moments are
these moments converge to the true mean and variance of the
diffusion process X(t)
in equation (11).
This ensures that the
trinomial process converges to the process X(t1 in distribution.
To find the optimal policy q*, a dynamic programming procedure can
be applied to the trinomial lattice. A t the very end-nodes', payoff
values are given.
q;(h)
Working backward, at any node X
1*
an optimal policy
and equity value can be easily obtained. Under certain smoothness
conditions on the payoff function, as h + 0 , q;(hl will converge to the
optimal policy q*. The optimal policy of theorem 3 can be easily
confirmed using this procedure.
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5. A Second Look at the Charter Value
In sections 2 and 3, we adopted Marcus's
the charter value.
(1984) specification of
The bank either retains or loses the full charter
value depending on whether or not it is solvent at the audit date.
corresponds to the terminal payoff curve OBCD in figure 2.
This
However,
despite its simplicity, this specification is far from realistic.
For example, regulators may, for economic or political reasons,
choose to inJect additional funds into a slightly insolvent bank rather
than simply to close it.
Thus, the payoff curve OBCD in figure 2 should
stretch farther to the left.
As for the equityholders, if the market
value of the bank's assets is below the liability value Just before the
audit, it would be to the bank's advantage to inject additional funds in
order to preserve the charter.
It may do so as long as the charter
value exceeds the liability minus asset value.
This suggests the payoff
curve OAD of a call option with strike price L(T1
-
CCT).
In this case,
the charter can be viewed as part of the bank's tangible assets.
However, when a bank is close to insolvency, it may face financial
distress or bankruptcy costs, which would decrease the charter value.
Usually the charter value depends not only on the size of the deposit
base, but also on the soundness of the bank (such as the capital-deposit
ratio).
When this ratio drops below a certain level, a regulatory tax
is likely to be charged (Buser, Chen, and Kane [I9811 1.
Therefore, a
more reasonable payoff function would be somewhat like the OEFD curve in
figure 2.
For
a highly capitalized bank,
the charter value
proportional to the deposit base (the F-D segment).
is
As the bank lowers
its capital, the charter-deposit ratio decreases (the E-F segment). If
the capital is too low, the charter value is zero (the O-E segment).
After the payoff curve is specified, we can use the trinomial
approximation of section 4 to calculate the present value of bank equity
and the actuarially fair price of deposit insurance.
For demonstration
purposes, suppose the payoff curve has the following form:
if K(T) r 0
(241
V(T) =
otherwise,
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where K(T1 = X(T) - (1-f)L(T),a.8 r 0.
This payoff function contains many interesting special cases. When
f = 0 , it reduces to the case of Merton (19771. When 8 = ioo and a < +cv,
it reduces the OAD curve in figure 3 where an insolvent bank can inJect
additional funds at no extra cost in order to retain its charter. When
a = +OD and 8 < +a, it reduces to that of Marcus (19841, which
corresponds to the OBCD payoff curve in figure 2.
V(T1
(Equity Value)
Figure 2. Alternative Payoff Functions
Figure 3 shows the payoff function (241 for a = 1, 2, 4 and oo,
while 8 = 1. The corresponding optimal policies are shown in figure 4,
where the other parameters are T = 1, r = 0, Xo = Lo = 100, CT = 0.1, and
f = 0.05.
All of the optimal policies are similar to the one in theorem
3. Banks initially choose q = 1. After a critical time r, there is a
critical curve K(t1.
If asset value X(t1 is above K(t), q = 0 is
optimal; otherwise q = 1 is optimal. In contrast to theorem 3, the
critical curve K(t1 is no longer a straight line. It is interesting to
note that the larger the value of a, the larger the critical time r ,
because the charter value erodes faster as a increases.
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0
950
1000
1050
Figure 3. Some Specific Payoff Functions
T
X(t1
(Asset Value)
Figure 4. Optimal Policies Under the Payoff Functions in Figure 3
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6. Conclusion
This paper develops a stochastic control model to analyze the
investment decisions of a bank whose deposits are fully insured under a
I show how banks dynamically adjust their
fixed-rate insurance premium.
investment portfolios in response to market information and how this
flexibility affects both investment decisions and the value of deposit
insurance.
The optimal portfolio problem
is
solved analytically
assuming lognormal asset price and constant charter value.
For general
payoff patterns, an efficient numerical procedure is presented.
Under continuous portfolio revision I
show that, before some
critical time T, the bank always takes the riskiest position regardless
of its solvency situation.
The bank may act cautiously only between
time r and the audit date T.
The value of deposit insurance remains a
put option, but with maturity r instead of T.
This critical time r
depends on the charter value, on the volatility of the risky assets, and
on the time between audits.
This gives the regulators some guidelines,
at least in theory, on the timing of audits.
The major limitation of this model is the empirical difficulty in
specifying the charter value.
This is further complicated by other
factors such as transaction costs, asymmetric information, reputation,
and economic conditions.
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Appendix
Proof of Theorem 1.
Since X(0) < L(O), from equation (6) we have
The equity value V is increasing in q; q* = 1 is optimal.
Proof of Theorem 2.
For a solvent bank, when q
S
Q.E.D.
qmln= 1
L(O)/X(O),
the riskless bonds alone will be enough to pay off the obligation at
time T, and the bank will pass the audit with certainty.
V(0.q) = X(0)
-
In this case,
[L(O) - G(011.
When q > qmin, 1.e.. L O - 1 - 0 > 0, we have
Hence, the equity value V(0,q) is flat on interval [O, qmin] and convex
on interval [qmln,11.
in [O,qml,l.
The optimal policy q* is either 1 or any value
Therefore, from equations (6) and (7)
v(o,q*) = max { V(0,0), V(0,1) )
Q.E.D.
This leads to equation (10).
To prove theorem 3, a few lemma are necessary.
Lemma 1 is an
adaptation of Fleming and Rishel (1975, p. 124, theorem V.S.l).
Lemma 2
is a classic result (Bhattacharya and Waymire [1990, p. 321).
In the
rest of the proof, I use the shorthand notations J and f for J(t ,X(t1)
and f(s;t,X(t)), respectively, as long as no confusion arises.
Leuma 1.
(Sufficient optimality condition for discounted stochastic
dynamic programming)
Let X(t) be a diffusion process on [O,Tl
(A. 1)
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where p and c satisfy the linear growth and the Lipschitz conditions.
Let U ( t , X ) and J ( T , X ) be continuous and satisfy the polynomial growth
Let J ( t , X ) be the solution of the dynamic programming
condition.
equation
1
( A .2 )
r J = max ( J t + p ( X ) J x + z ( c ( ~ ) ) +
2H
~(,t , X ) )
with boundary value J ( T , X ( T ) ) .
t
E
[ O , T ) and continuous for t
If J ( . t , X ) is twice differentiable for
E
[ O , T l , then
( A . 3)
for any admissible policy q .
Lema 2.
Let X ( t 1 be a Brownian motion with drift p .
Let T
Z
first time the process reaches level z conditioned on X ( 0 ) = x .
be the
Then
the probability density and distribution functions of T are
z
f(t;x,z) =
L e m a 3.
(z-x)
f i c t2'3
expl-
(z - x - p t
2c2t
12
I
t > 0,
(A. 4 )
The functional J ( t , X ( t ) ) and the policy q* defined in theorem
3 is optimal if
(1)
when J x x is continuous at ( t . X ( t ) ) , the maximizing q is
( A .6 )
and
r J = J t + r X ( t ) J x + f ( c x ( t )2)J x x
if q* = 1
( A .7 )
rJ = Jt+ rX(t)Jx
if q* = 0
( A .8 )
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( 2 ) when J has a Jump at (t,X(t)),
(A.9a)
(A.9b)
q* = 0
where
Proof: Part (1) follows immediately from lemma 1 with p(X) = rX. To
show part (21, note that J(t,X(t)) is twice differentiable except when
X(t) = L(t) and t E (*,TI where J; > J; = 1 and J; = 'J XX = 0 ; when t E
(r,TI, J(t,X(t)) is convex for X(t) s L(t) and linear for X(t) r L(t)
(see the proof of theorem 3). To apply lemma 1 , add a smoothing term'P
to J such that J E ( t , ~ ( t ) ) = J(t,X(t)) + ~ ' ( t , ~ ( t ) )is twice
z L(t) for
differentiable, convex for X(t) 5 L(t), and concave for X(t) i
&
t E (z,T) and for any small number E > 0. For example, one such P is
( -&n7
pC =
{
AJL
if X(t)>L(t)+cn and t~(7.T)
AP
-rx(t)-L(t)+rin( X(t1-L(t)
&
cI
I 0
where
bf
if OsX(t)-L(t)sen and t~(r.7')
otherwise,
=
J'(L(~))
X
- J;(L(~)).
~(P'I =
-
Define
rpC + P:+
~x(~IP:.
Then for any admissible policy q,
-
-rJC + J: + r~(t)J: + &(q~(tlo)2JLx'x # ( f 1
s
- rJC
+ J:
+ r~(t)J: + & ( q * ~ ( t)o12fx
-
#(f 1
) J(t,X(t))
where q* is the policy in theorem 3. Therefore, J E ( t , ~ ( t ) =
+ pE(t ,X(t1) is the solution of the dynamic programming equation
&
rJC = max [Jt + rx(t)J: + k ( q ~ ( )ol2.fX
t
q
-
#(pF)]
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for t E (r,T-6) and any small number 6 > 0.
Let e.6
zero.
+ 0.
.
Applying lemma 1, we have
The last three terms on the right-hand side all go to
Then J(t,X(t))
*
k E [J(T,X(T))I
q
for any q.
and q- are optimal.
This implies J(t,X(t)l
Q. E.D.
We need to show that the given functional J(t,X(t))
and the corresponding policy q* satisfy the conditions in lemma 3.
Proof of Theorem 3.
Let t E (r,T1 and X(t)
Case 1.
k
L(t1.
When X(t1 > L(t), q*(t) =
0 and J(t,X(t)) in equation (15) together satisfy the conditions (A.6)
When X(t) = L(t), as we will show later, JX is
:
not continuous in X(t). However, from lemma 3, q = 0 is optimal if J
and (A.8) in lemma 3.
Since J
: = 1, we need only to show that J- > 1 at X(t) = L(t).
< J;.
X
First note that J(t,X(t)) is continuous at X(t) = L(t).
as X(t)?~(t),
rl+ -
f i / 2 and
manipulation yields J(t,X(t))
r2+
fi/2
+ G(t)
In fact,
in equation (16). Further
= J(t,X(t)).
Now differentiate
J(t,X(t)) in equation (161, and let ~(t)?~(t). Then
(A.10)
Since
a~-/at
X
=
C(t) n(m/2)
-i
n7 c(~-tl3I2 < 0, J-x
is strictly increasing in t.
Noting that J- = 1 at t = r , we have J- > 1 for all t E (=,TI.
X
Case 2.
X
Let t E (r,Tl, X(t) < L(t).
Differentiating equation
(16). and noting that X(t)n(rl) = L(t)n(r2), we have
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(A. 11)
J
XX
is obviously continuous.
To show that q (t) = 1 is optimal, we need
only to check that condition (A.7) in lemma 3 is satisfied.
goal, let Y(t) = ln[X(t)l
-
Toward this
rt; then
The first passage times are the same for the geometric Wiener process
X(t) to reach L(s) given X(t) at time t and for the Brownian motion Y(t)
to reach ln[L(s)l
-
rs given Y(t) = lnX(t)
-
rt at time t.
From lemma
2, the density function of this first passage time is
It is easy to show that J(t,X(t)) =
c
C(t)f(s;t,X(t))ds.
Since the
density function f satisfies the backward Kolmogorov equation
(A.12)
condition (A.7) can be easily checked:
= -rC(t) fds + [rC(tI fds
Jt
Jt
+
G(t) ftds]
Jt
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Case 3.
Let t E [O,T].
We first show that J ( t , X ( t ) )
in equation
(17) is the risk-neutral value of a contingent claim with terminal value
J(r,X(r))
at time
7.
To see this, let
(A. 1 3 )
2
where X ( T )
1/2
= X ( t ) e ( r u 12)
(7-t)W(7-t)
Substltutlng equations ( 1 5 )
and ( 1 6 ) into ( A . 1 3 ) .
where
rl
and 72 are evaluated at
7
rather than at t.
integrations above gives equation ( 1 7 ) .
1
-2 (
From ( A . 1 3 ) we have
cm
~ ( t ) o ) ~ ~ ~ ~ ( t , ~e(-rt (7-t)
) ) =
-2
-d z .
6
XX
Now we need only to check condition ( A . 7 )
Let p = p ( r , y ; t , X ( t ) )
Then equation ( A . 7 )
( 7 - t )~
r 0.
in order to show q ( t ) =
be the density function of the
lognormal price X ( r ) conditioned on X ( t ) .
~ ( t , x ( t ) ) = e- r
2
/2
~[X(T)CI~J~~(T,X
I (eT )
Since J ( r , X ( r ) ) r 0 from cases 1 and 2, J x x ( t , X ( t ) )
1 is optimal.
Carrying out the
Rewrite equation (17) as
( T , Y ) P ( T , Y ; f , X ( t 1 My-
(A. 14)
can be established by the fact that p ( - c , y ; t , X ( t ) )
satisfies the backward Kolmogorov equation ( A . 1 0 ) .
Q.E.D.
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