Description
In finance, an exchange rate (also known as a foreign-exchange rate, forex rate, FX rate or Agio) between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency.
ISG HS1 - 598
Minimizing the Probability of Ruin
in Exchange Rate Markets
A Professional Master’s Project Report
submitted to the Faculty
of the
WORCESTER POLYTECHNIC INSTITUTE
in partial fu?llment of the requirements for the
Degree of Master of Science
in
Financial Mathematics
by
Tyler A. Chase
April 2009
Professor Hasanjan Sayit, Project Advisor
This report represents the work of one or more WPI graduate students
submitted to the faculty as evidence of completion of a degree requirement.
WPI routinely publishes these reports on its web site without editorial or peer review.
Abstract
The goal of this paper is to extend the results of Bayraktar and Young (2006)
on minimizing an individual’s probability of lifetime ruin; i.e. the probability
that the individual goes bankrupt before dying. We consider a scenario in
which the individual is allowed to invest in both a domestic bank account and
a foreign bank account, with the exchange rate between the two currencies
being modeled by geometric Brownian motion. Additionally, we impose the
restriction that the individual is not allowed to borrow money, and assume
that the individual’s wealth is consumed at a constant rate. We derive for-
mulas for the minimum probability of ruin as well as the individual’s optimal
investment strategy. We also give a few numerical examples to illustrate
these results.
i
Contents
Abstract i
1 Introduction 1
2 Background 3
2.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Probability of Lifetime Ruin 11
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Minimum Probability of Ruin . . . . . . . . . . . . . . . . . . 15
4 Numerical Examples 34
5 Summary and Conclusion 37
6 Appendix: MATLAB and Maple Code 39
ii
1 Introduction
In the current state of the American economy, it is natural for individuals to
be concerned about their ?nancial well-being in the present as well as in the
future. In particular, some have reason to be concerned about the possibility
of bankruptcy during retirement. In a situation where the American dollar
is a bit weaker or more unstable, it is reasonable to suppose that some in-
dividuals may be interested in investing in a potentially more stable foreign
currency.
In this paper we begin to consider an extension of the work of Bayraktar
and Young (2006) in determining how an individual should invest her wealth
in order to minimize the probability that she ruins before death. We focus
on the scenario in which the individual’s rate of consumption is constant and
borrowing constraints are imposed. However, we look at a ?nancial market
model in which the individual has the option of investing some of her wealth
in a domestic bank account and some in a foreign bank account; the risk is
introduced by the random exchange rate between the two currencies.
The Foreign Exchange market is an interesting model to consider. For
some investors, the possibility of trading in a currency market can be more
appealing than trading in a stock exchange. The Foreign Exchange o?ers
high market liquidity, and has a high trading volume. Margins of pro?t are
lower than in other, possibly riskier markets, but there is still the potential
for signi?cant earnings. So while not every investor would be interested in
this market, its applications are relevant.
1
The most common criterion for optimization problems in ?nancial lit-
erature is the maximization of expected utility of consumption, and there
has been a substantial amount of work done on that subject. Bayraktar and
Young (2006) note that these methods generally depend on a subjective util-
ity function for consumption, whereas minimizing the probability of lifetime
ruin may be more appealing and comprehensible to individuals since that
criterion is more objective. And indeed, this technique has seen increased
application in recent years.
Our work closely follows that of Bayraktar and Young (2006), since our
market model is closely related. We consider only the “no-borrowing” case
with constant consumption; after that, it should not be di?cult to see how
the other cases would follow. Before presenting the main results, we review a
few of the de?nitions and theorems from probability and stochastic calculus,
that the reader may have a suitable reference. Later we give a few numerical
examples to demonstrate our results.
2
2 Background
2.1 Probability
To begin with, it would be helpful to establish the setting in which our work
takes place. Speci?cally, we will assume the existence of a continuous-time
?ltered probability space. We will give a brief de?nition of most of the
relevant fundamental concepts. While this is not strictly necessary, it helps
to assure that the reader is able to follow the reasoning presented in the
paper.
De?nition 2.1 (?-algebra). Let ? be a nonempty set, and let F be a collec-
tion of subsets of ?. F is a ?-algebra (also known as a ?-?eld) on ? if the
following conditions are satis?ed:
• ? ? F
• A ? F =? A
c
? F
• A
1
, A
2
, . . . ? F =?
?
i=1
A
i
? F
De?nition 2.2 (Probability measure). Let ? be a nonempty set, and F a
?-algebra of subsets of ?. A probability measure is a function P : F ? [0, 1]
such that
• P(?) = 1.
• If A
1
, A
2
, . . . ? F are disjoint, then P (
?
i=1
A
i
) =
?
i=1
P (A
i
).
3
De?nition 2.3 (Probability Space). A triple (?, F, P), consisting of a sam-
ple space ?, a ?-algebra F on ?, and a probability measure P on F, is called
a probability space.
De?nition 2.4 (Random Variable). Given a probability space (?, F, P), a
random variable is a function X : ? ? R with the property that for any
Borel set B ? B, the inverse image X
?1
(B) belongs to F.
De?nition 2.5 (Measurability). A random variable X is said to be measur-
able with respect to a ?-algebra G (or, X is G-measurable) if for any Borel
set B ? B the inverse image X
?1
(B) belongs to G.
These are the basic assumptions of most models in probability. To dis-
cuss results relying on stochastic calculus, we also would like to review the
concepts of ?ltrations and stopping times.
De?nition 2.6 (Filtration). Let (?, F, P) be a probability space, and let T be
a ?xed positive number. A continuous ?ltration is a collection of ?-algebras
with the following properties:
• ?t ? [0, T] , ?F
t
? F.
• s ? t =? F
s
? F
t
Moreover,
_
?, F, P, {F
t
}
0?t?T
_
is called a ?ltered probability space.
De?nition 2.7 (Stopping Time). A stopping time ? is a random variable
satisfying the following property: ?t ? [0, T], {? ? ? : ? (?) ? t} ? F
t
.
4
De?nition 2.8 (Stopped ?-algebra). Let
_
?, F, P, {F
t
}
0?t?T
_
be a ?ltered
probability space, and let ? be a stopping time. The stopped ?-algebra (stopped
at ?) is de?ned as:
F
?
= {A ? F : A ? {? ? t} ? F
t
}.
2.2 Stochastic Calculus
In addition to the material on probability theory, we also provide a reference
for fundamental de?nitions and theorems in stochastic calculus, as these are
necessary tools in the proofs of our results.
De?nition 2.9 (Stochastic Process). A continuous stochastic process is a
collection of random variables {X
t
, t ? [0, T]}. For each ? ? ?, X
t
(?) is a
deterministic function called the sample path, or trajectory.
De?nition 2.10 (Adaptedness). A stochastic process X
t
is said to be adapted
to a ?ltration {F
t
} if for every t, X
t
is F
t
-measurable.
De?nition 2.11 (C` adl`ag Process). A stochastic process X
t
is called c`adl`ag
if it has sample paths satisfying the following conditions almost surely:
• X
t
(?) is right-continuous, i.e. lim
t?a
+ X
t
(?) = X
a
(?) for all a.
• X
t
(?) has left-limits, i.e. lim
t?a
? X
t
(?) exists for all a.
The word “c` adl` ag” is a French acronym, standing for continue `a droite,
limit´ee `a gauche, literally “continuous on the left, limited on the right.”
5
De?nition 2.12 (Conditional Expectation). The conditional expectation of
a random variable X with respect to a ?-algebra G ? F, denoted E [X|G], is
itself a random variable with the following properties:
• E [X|G] is G-measurable.
• ?A ? G,
_
A
XdP =
_
A
E [X|G] dP
De?nition 2.13 (Martingale). Let
_
?, F, P, {F
t
}
0?t?T
_
be a ?ltered prob-
ability space. An adapted stochastic process M
t
is called a martingale with
respect to F
t
if it satis?es the following property:
E [M
t
|F
s
] = M
s
for all 0 ? s ? t (2.1)
De?nition 2.14 (Stopped Process). If X is a stochastic process and ? is a
stopping time, then we can de?ne the stopped process
X
t??
=
_
¸
_
¸
_
X
t
if t ? ?
X
?
if t > ?
(2.2)
In that case, the stochastic process is said to be “stopped” at time ?. The
stopped process is equal to the original process until time ?, and becomes
constant after that time, equal to the value of X
?
.
A similar notion is the “killed” process, which instead of taking on the
constant value X
?
at its killing time, takes a value ? (outside the range of
X) called the “co?n state”.
6
De?nition 2.15 (Brownian Motion). A standard Brownian motion (or Wiener
process) is a continuous stochastic process W
t
with independent increments
which are normally distributed: For all 0 = t
0
< t
1
< t
2
< . . . < t
m
, the
increments
W
t
1
?W
t
0
, W
t
2
?W
t
1
, . . . , W
t
m
?W
t
m?1
(2.3)
are independent and normally distributed with mean 0 and variance t
i
?t
i?1
.
De?nition 2.16 (It¯ o Integral). The It¯o integral
_
t
0
H
s
dW
s
of a c`adl`ag process
H is de?ned as:
_
t
0
H
s
dW
s
= lim
n??
n
i=1
H
t
i?1
_
W
t
i
?W
t
i?1
_
. (2.4)
where 0 = t
0
< t
1
< t
2
< . . . < t
n
is a partition of [0, t], growing ?ner as n
increases.
Theorem 2.17 (Properties of the It¯ o Integral). The It¯o integral I
t
=
_
t
0
H
s
dW
s
has the following properties:
• I
t
has continuous sample paths.
• I
t
is F
t
-adapted.
• I
t
is a martingale.
• I
t
has quadratic variation [I, I]
t
=
_
t
0
H
2
s
ds.
De?nition 2.18 (It¯ o Process). An It¯o Process is a stochastic process X of
7
the following form:
X
t
= X
0
+
_
t
0
µ
s
dW
s
+
_
t
0
?
s
ds (2.5)
where µ and ? are adapted stochastic processes, and X
0
is a nonrandom
initial value.
Alternatively, this can be written in the di?erential form:
dX
t
= µ
s
dW
s
+?
s
ds (2.6)
Also note that the quadratic variation of the It¯o process X is given by (in
integral and di?erential form):
[X, X]
t
=
_
t
0
µ
2
s
ds (2.7)
d [X, X]
t
= µ
2
t
dt (2.8)
Theorem 2.19 (It¯ o’s Formula). Let X
t
be an It¯o process and let f (t, x) be
a function for which the partial derivatives f
t
, f
x
, and f
xx
are de?ned and
continuous. Then
f (T, X
T
) = f (0, X
0
) +
_
T
0
f
t
(t, X
t
) dt +
_
T
0
f
x
(t, X
t
) dX
t
+
1
2
_
T
0
f
xx
(t, X
t
) d [X, X]
t
(2.9)
A Poisson process is a stochastic process which takes nonnegative integer
values characterized by a rate parameter ?. It is typically used to model the
8
number of events which occur in a given time interval.
De?nition 2.20 (Poisson Process). The Poisson process N
t
with rate pa-
rameter ? obeys a Poisson distribution with parameter ?t:
P (N
t
= k) =
(?t)
k
k!
e
??t
(2.10)
The Poisson process is an example of what is called a “pure jump” pro-
cess. It has stationary, independent increments. Moreover, the times between
successive jumps are independent and follow an exponential distribution with
parameter ? (i.e., an exponential distribution with mean 1/?).
Theorem 2.21 (Compensated Poisson Process). If N
t
is a Poisson process
with rate parameter ?, we de?ne the compensated Poisson process M
t
=
N
t
??t. M
t
is a martingale.
De?nition 2.22 (Jump Process). Let X
t
be an It¯o process, and let J
t
be a
“pure jump” process. That is, J
t
is an adapted, c`adl`ag process with ?nitely
many jumps on the interval (0, T], and is constant between jumps. We will
call a process of the following form a jump process:
Y
t
= X
t
+J
t
(2.11)
When discussing processes with jumps, It¯ o’s Formula takes a slightly
altered form.
Theorem 2.23 (It¯ o’s Formula for Jump Processes). Let Y
t
be a jump process,
and let f
be a function which is twice continuously di?erentiable. Then
9
we have the following:
f (Y
t
) = f (Y
0
) +
_
t
0
f
(Y
s
) dY
c
s
+
1
2
_
t
0
f
(Y
s
) d [Y, Y ]
c
s
+
0<s?t
[f (Y
s
) ?f (Y
s
?)] (2.12)
Here, the quantity Y
c
denotes the continuous part of the jump process
Y . If Y is given in the form of (2.11), then Y
c
= X.
10
3 Probability of Lifetime Ruin
We will be considering the problem of minimizing an individual’s probability
of ruin under the condition that borrowing is forbidden. In section 3.1 we
outline the ?nancial market model used in the analysis. In 3.2 we present
and prove the main results.
3.1 Model
We will assume a model in which an individual has the option of investing
in two assets: A domestic bank account B, and a foreign bank account F.
Each of these banks will have its own ?xed interest rate, and the exchange
rate X between the two currencies will be modeled by a geometric Brownian
motion. So our assets have the following dynamics:
dB = r
d
Bdt
dF = r
f
F dt (3.1)
dX = µX dt +?X dW
where ? and ? are constants, and W is a standard Brownian motion. We
will prove a lemma which states that the above formulation is equivalent to
a situation in which the individual is allowed to invest in the domestic bank
account and in a domestic risky asset whose price is given by
˜
F = FX.
11
Lemma 3.1. The model in (3.1) is equivalent to the following model:
dB = r
d
Bdt
d
˜
F = (µ +r
f
)
˜
F dt +?
˜
F dW (3.2)
where
˜
F = FX.
Proof. The proof follows from the multidimensional version of It¯o’s lemma.
The two-dimensional case is as follows: If U and V are It¯o processes, and if
f (t, u, v) is a function which is twice continuously di?erentiable, then
df (t, U, V ) =f
t
(t, U, V ) dt +f
u
(t, U, V ) dU +f
v
(t, U, V ) dV
+
1
2
f
uu
(t, U, V ) dUdU +
1
2
f
vv
(t, U, V ) dV dV (3.3)
+f
uv
(t, U, V ) dUdV
where f
u
denotes the ?rst partial derivative of f with respect to u, and so
on. Using f (t, u, v) = uv, this reduces to
df (t, U, V ) = 0dt +V dU +UdV + 0dUdU + 0dV dV + 1dUdV
= V dU +UdV +dUdV
12
Substituting U = X and V = F, we obtain
d
˜
F = FdX +XdF +dXdF
= F (µXdt +?XdW) +X (r
f
Fdt) + (µXdt +?XdW) (r
f
Fdt)
=
˜
F (µdt +?dW) +
˜
F (r
f
dt) + 0
And this can be rearranged as
d
˜
F = (µ +r
f
)
˜
Fdt +?
˜
FdW (3.4)
So the possibility of investing in the domestic bank B and the foreign bank
F is equivalent to investing in the domestic bank and a domestic asset
˜
F
with dynamics as given in (3.4).
We will henceforth assume that the individual may act by investing a
portion of her wealth in
˜
F, with the remainder invested in B. In this for-
mulation,
˜
F can be interpreted as a risky asset while B is a risk-free asset;
we assume that µ + r
f
> r
d
(indeed, we are not interested in the problem
otherwise, since no investor should invest in a risky asset with lower expected
return than the risk-free asset).
We will also assume that the individual’s total wealth is continuously
consumed at a constant rate c.
Let V
t
denote the wealth of the individual at time t, and denote by ?
t
the amount she invests in the risky asset
˜
F. Then the amount invested in the
risk-free asset B is V
t
??
t
. Therefore the wealth process obeys the following
13
dynamics:
dV
t
=
?
t
˜
F
t
d
˜
F
t
+
(V
t
??
t
)
B
t
dB
t
?cdt
? dV
t
=
?
t
˜
F
t
_
(µ +r
f
)
˜
F
t
dt +?
˜
F
t
dW
t
_
+
(V
t
??
t
)
B
t
r
d
B
t
dt ?cdt
This can be simpli?ed as:
_
¸
_
¸
_
dV
t
= [r
d
V
t
+ (µ +r
f
?r
d
) ?
t
?c] dt +??
t
dW
t
V
0
= v
(3.5)
We now wish to de?ne what is meant by “lifetime ruin”. We let ?
0
denote
the ?rst time that V = 0, and let ?
d
denote the individual’s time of death.
Lifetime ruin is de?ned as the event in which the wealth process reaches zero
before the individual dies, i.e. the event {?
0
< ?
d
}. Here we will assume that
?
d
follows an exponential distribution with parameter ?, so that the expected
value of ?
d
is 1/? (later, we will model this using a Poisson process with rate
parameter ?, since the time between jumps in the Poisson process follows an
exponential distribution).
The minimum probability of ruin will be denoted by ? (v) (the argu-
ment v indicates that this probability is conditional on V
0
= v). So we are
minimizing the probability that ?
0
< ?
d
, with respect to the set of admissi-
ble trading strategies (denoted by A). For this paper, we also impose the
restriction that 0 ? ?
t
? V
t
(i.e. no borrowing or short-selling is possible).
14
Therefore the probability ? (v) is given by:
? (v) = inf
?
P[ ?
0
< ?
d
| V
0
= v ] (3.6)
For each real number ?, we can de?ne a second-order di?erential operator
L
?
which is associated with the minimization problem. For each open subset
G of R
+
and for each h ? C
2
(G), de?ne the function L
?
h : G ? R as
follows:
L
?
h(v) = [r
d
v + (µ +r
f
?r
d
) ? ?c] h
(v) +
1
2
?
2
?
2
h
(v) ??h(v) (3.7)
The operator L
?
will be used in the following sections to characterize ? in a
compact manner.
3.2 Minimum Probability of Ruin
In this section we will present the veri?cation theorem which states the nec-
essary and su?cient conditions that ? must satisfy. First, note that when the
individual’s wealth is above c/r
d
, the probability of lifetime ruin is equal to
0; the individual can invest all of her wealth in the domestic (risk-free) bank
account and consume continuously at rate c with no possibility of running
out of money. To see this, consider the dynamics of the wealth process when
all of the individual’s wealth is invested in the domestic bank:
dV
t
= r
d
V
t
dt ?cdt
15
In this case, there is no stochastic integral involved; so it can be expressed
as an ordinary di?erential equation:
dV
dt
= r
d
V ?c
So for all V ? c/r
d
, we have that
dV
dt
? 0, i.e. there is no chance that the
wealth process will decrease in this situation, let alone reach zero.
Thus in addition to the stopping times ?
0
and ?
d
which we already de-
?ned, we also introduce the stopping time ?
c/r
d
= inf {t > 0 : V
t
? c/r
d
},
that is, the ?rst time that the individual’s wealth reaches c/r
d
(or more). If
we now de?ne the stopping time ? = ?
d
??
c/r
d
, it follows that we can express
? as follows:
? (v) = inf
?
P[ ?
0
< ? | V
0
= v ] (3.8)
We can now present the veri?cation theorem:
Theorem 3.2. Suppose h : R
+
? [0, 1] is a decreasing function, and ?
0
: R
+
?R
+
which satisfy the following conditions:
(i) h ? C
2
on [0, c/r
d
)
(ii) ?
0
? A
(iii) L
?
h(v) ? 0, for 0 ? ? ? v < c/r
d
(iv) L
?
0
(v)
h(v) = 0, for v ? (0, c/r
d
)
(v) h(0) = 1 and h(v) = 0 for v ? c/r
d
16
Then the minimum probability of lifetime ruin ? is given by:
? (v) = h(v) , v ? 0 (3.9)
And the optimal investment strategy ?
?
in the risky asset
˜
F is given by:
?
?
(v) = ?
0
(v) , v ? [0, c/r
d
] (3.10)
Proof. Suppose we have h which satis?es the properties stated above. Let
N be a Poisson process (independent of W) with rate parameter ?. The
stopping time ?
d
will be de?ned as the time of the ?rst jump of the process
N. Let ? be a function on the interval [0, c/r
d
] with 0 ? ?(v) ? v, and let
V
?
denote the wealth process under the investment strategy ?. We denote
?
s
= ?(V
?
s
). We will kill the wealth process at time ?
d
and assign W
?
d
= ?,
(the co?n state). Our convention will be that for any function f : R
+
?R
+
,
we let f(?) = 0. In particular, note that h(c/r
d
) = 0 and h(V
?
?
d
) = 0. Using
the wealth process dynamics as stated in 3.5 and It¯o’s formula as in 2.23, we
have the following:
h(V
?
t????
0
) ?h(v) =
_
t????
0
0
h
(V
?
s
)dV
?
s
+
1
2
_
t????
0
0
h
(V
?
s
)d[V
?
, V
?
]
s
+
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] (3.11)
17
=
_
t????
0
0
h
(V
?
s
){[r
d
V
?
s
+ (µ +r
f
?r
d
)?
s
?c]ds +??
s
dW
s
}
+
1
2
_
t????
0
0
h
(V
?
s
)(?
2
?
2
s
)ds +
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] (3.12)
Since the process jumps only at time ?
d
, then the jump at time s can be
expressed as h(V
?
s
) ?h(V
?
s
?
) = ?h(V
?
s
?
)?N
s
and so we write
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] = ?
_
t????
0
0
h(V
?
s
?)dN
s
(3.13)
In order to write the expression in a compact manner, we add and subtract
the term ?
_
t????
0
0
h(V
?
s
?
) in the right hand side:
h(V
?
t????
0
) = h(v) +
_
t????
0
0
{[r
d
V
?
s
+ (µ +r
f
?r
d
)?
s
?c]h
(V
?
s
) +
1
2
?
2
?
2
s
h
(V
?
s
)}ds
?
_
t????
0
0
?h(V
?
s
?)ds +
_
t????
0
0
??
s
h
(V
?
s
)dW
s
?
_
t????
0
0
h(V
?
s
?)dN
s
+?
_
t????
0
0
h(V
?
s
?)ds (3.14)
So the expression can be simpli?ed to:
h(V
?
t????
0
) = h(v) +
_
t????
0
0
L
?
h(V
?
s
)ds +
_
t????
0
0
??
s
h
(V
?
s
)dW
s
?
_
t????
0
0
h(V
?
s
?)d(N
s
??s) (3.15)
Taking the expectation of both sides, the third and fourth terms vanish (it
can be shown that the integrands satisfy su?cient conditions). So, following
18
from assumption (iii) in the theorem statement, we have:
E
v
[h(V
?
t????
0
)] = h(v) +E
v
__
t????
0
0
L
?
h(V
?
s
)ds
_
? h(v) (3.16)
Here, E
v
indicates that the expectation is conditional on V
0
= v. Therefore,
the process h(V
?
t????
0
), t ? 0, is a submartingale. Since h(0) = 1, h(V
?
?
0
??
) =
0, and h(V
?
c/r
d
), it follows (where 1 denotes the indicator function) that
h(V
?
?
0
??
) = 1
{?
?
0
<?}
. (3.17)
Now, taking expectations of both sides and applying the optional sampling
theorem gives
E
v
h(V
?
?
0
??
) = P
v
(?
?
0
< ?) ? h(v), (3.18)
since h(V
?
t????
0
) is a submartingale. Therefore
inf
?
P
v
(?
?
0
< ?) = ?(v) ? h(v). (3.19)
If we consider ?
0
as speci?ed in the theorem statement (namely, property
(iv), i.e. ?
0
is the minimizer of L
?
h), then it follows that h(V
?
0
t????
0
) is a
martingale. So we have that
E
v
h(V
?
0
?
0
??
) = P
v
(?
?
0
0
< ?) = h(v). (3.20)
We have therefore shown that the statements in 3.9 and 3.10 are true for
19
v ? [0, c/r
d
). Together with the assumption in (v) and the fact that
? (v) = inf
??A
P[ ?
0
< ? | V
0
= v ], the proof is complete.
The forms of the functions h and ? are discovered by imposing a few
additional properties that are not stated explicitly in the preceding theorem.
However, because that theorem asserts that the function h is unique on
the interval [0, c/r
d
), if we ?nd h and ? which satisfy the assumptions of
Theorem 3.2 and the additional assumptions, then the additional properties
are implicit.
We make the following additional hypotheses as well: In the constrained
case, we assume that ?
0
(the amount invested in the foreign bank
˜
F) is a
continuous function of v, and that there exists a wealth level v
l
such that
v ??
0
(v) = 0 for v < v
l
and v ??
0
(v) > 0 for v > v
l
. The idea here is that
when the individual has more wealth, it is wiser to invest a portion of it in a
risk-free asset. The subscript l denotes that the individual is, upon reaching
this level, “lending” some amount of money to the domestic bank.
We will consider the intervals [0, v
l
] and (v
l
, c/r
d
] separately. First we
look at the interval (v
l
, c/r
d
], and we assume that the borrowing constraint
is non-binding.
Proposition 3.3. Assume that 0 ? ?
0
(v) < v on the interval (v
l
, c/r
d
]. The
function h has the following form (with ? ? 1):
h(v) = ?
_
1 ?
r
d
c
v
_
d
, (3.21)
20
where
d =
1
2r
d
_
(r
d
+? +m) +
_
(r
d
+? +m)
2
?4r
d
?
_
> 1, (3.22)
and
m =
1
2
_
µ +r
f
?r
d
?
_
2
. (3.23)
The corresponding ?
0
on (v
l
, c/r
d
] is given by:
?
0
(v) =
µ +r
f
?r
d
?
2
1
d ?1
_
c
r
d
?v
_
. (3.24)
Proof. Items (iii), (iv), (v) of Theorem 3.2 require that we solve
?h(v) = (r
d
v ?c)h
(v) + min
?
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
(3.25)
with the boundary condition h(c/r
d
) = 0. We can show that we also have
the boundary condition h
(c/r
d
) = 0: Consider the solution ? of (3.25)
with ? = 0 (corresponding to the event that the individual never dies).
So h ? ? on some interval (c/r
d
? ?, c/r
d
], since the probability of ruin be-
fore death is necessarily less than the probability of ruin before in?nity (i.e.
P(?
0
< ?
d
) ? P(?
0
< ?)). So it is enough to show that ?
(c/r
d
) = 0, which
would imply h
(c/r
d
) = 0. Note that ? is a solution to the following (with
?(c/r
d
) = 0):
0 = (r
d
v ?c)?
(v) + min
?
_
(µ +r
f
?r
d
)??
(v) +
1
2
?
2
?
2
?
(v)
_
(3.26)
21
Pestien and Suddherth (1985) showed that the optimal investment strategy
?
?
maximizes (in our case) the quantity:
f(?) =
(µ +r
f
?r
d
)? ?(c ?r
d
v)
?
2
. (3.27)
By ordinary calculus, it is easily checked that the value of ? which maximizes
that expression is ?
?
= 2(c?r
d
v)/(µ+r
f
?r
d
). However, we also have (again
from ordinary calculus, this time applied to the minimization problem in
(3.26)) that
?
?
(v) = ?
µ +r
f
?r
d
?
2
?
(v)
?
(v)
(3.28)
Therefore, for v ? (c/r
d
??, c/r
d
], we have (for some k < 0)
?
(v) = k(c ?r
d
v)
m/r
d
(3.29)
So we showed that ?
(c/r
d
) = 0, which implies that h
(c/r
d
) = 0.
To be consistent with the hypothesis that the borrowing constraint is
non-binding on (v
l
, c/r
d
], it must be true that h is convex on (v
l
, c/r
d
]. To
see this, note that if h is not convex in some neighborhood of a point v
?
?
(v
l
, c/r
d
] (i.e. h
(v) < 0 in that neighborhood), then ?
0
is as large as possible
on that neighborhood, which contradicts the hypothesis that the borrowing
constraint is non-binding. So we have that h is convex on (v
l
, c/r
d
], and can
22
therefore consider its Legendre transform
˜
h:
˜
h(u) = min
v
[h(v) +vu] (3.30)
h can be recovered from
˜
h by
h(v) = max
u
[
˜
h(u) ?uv] (3.31)
From ordinary calculus, the value of v which minimizes the quantity in (3.30)
is v = (h
)
?1
(?u) =
˜
h
(u). Therefore the value of u which maximizes the
expression in (3.31) is u = ?h
(v). We can then make substitutions into
(3.25). Using v =
˜
h
(u), it follows that
h(v) =
˜
h(u) ?u
˜
h
(u), h
(v) = ?u, and h
(v) = ?
1
˜
h
(u)
. (3.32)
Additionally, as in (3.28), we use
? = ?
µ +r
f
?r
d
?
2
h
(v)
h
(v)
(3.33)
Making these substitutions in (3.25) gives
?h(v) = (r
d
v ?c)h
(v) ?
1
2
(µ +r
f
?r
d
)
2
?
2
(h
(v))
2
h
(v)
? ?[
˜
h(u) ?u
˜
h
(u)] = (r
d
˜
h
(u) ?c)(?u) +mu
2
˜
h
(u) (3.34)
Here, m is as given in (3.23). Simplifying further gives the following di?er-
23
ential equation:
?
˜
h(u) + (r
d
??)u
˜
h
(u) ?mu
2
˜
h
(u) = cu (3.35)
The general solution of this
˜
h(u) = D
1
u
B
1
+D
2
u
B
2
+
c
r
u (3.36)
where D
1
and D
2
are constants, and B
1
and B
2
are the roots of
?? ?(r
d
?? +m)B +mB
2
= 0, (3.37)
so
B
1
=
1
2m
_
(r
d
?? +m) +
_
(r
d
?? +m)
2
+ 4?m
_
> 1 (3.38)
B
2
=
1
2m
_
(r
d
?? +m) ?
_
(r
d
?? +m)
2
+ 4?m
_
< 0 (3.39)
Let u
c
= ?h
(c/r
d
) = 0, so
˜
h
(0) = c/r
d
. From the de?nition of
˜
h and
because h(c/r
d
) = 0, we have at u = u
c
= 0,
˜
h(0) = 0. (3.40)
From this it follows that D
2
= 0. We can then use (3.36) and (3.31) to
recover h:
h(v) = max
u
_
D
1
u
B
1
+
c
r
u ?vu
_
. (3.41)
24
The maximizing value of u here (by ordinary calculus) is:
u =
_
v ?c/r
d
D
1
B
1
_
1/(B
1
?1)
. (3.42)
Substituting this back into (3.41) gives:
h(v) = D
1
_
v ?c/r
d
D
1
B
1
_
B
1
B
1
?1
?(v ?c/r
d
)
_
v ?c/r
d
D
1
B
1
_ 1
B
1
?1
h(v) =
_
D
1
(D
1
B
1
)
B
1
B
1
?1
?
1
(D
1
B
1
)
1
B
1
?1
_
_
?
c
r
d
_
B
1
B
1
?1
_
1 ?
r
d
c
u
_
B
1
B
1
?1
. (3.43)
And we simplify this by denoting the leading constant quantity by ? and
noting that
B
1
B
1
?1
= d, so we obtain
h(v) = ?
_
1 ?
r
d
c
v
_
d
. (3.44)
Using this expression for h, the optimal investment strategy ?
0
is found by
minimizing (using ordinary calculus) the expression:
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
. (3.45)
And the value of ? which minimizes this expression is
?
0
(v) =
µ +r
f
?r
d
?
2
1
d ?1
_
c
r
d
?v
_
. (3.46)
25
Corollary 3.4. The lending level v
l
takes the following form:
v
l
=
x
1 +x
c
r
d
, (3.47)
where
x =
µ +r
f
?r
d
?
2
1
d ?1
. (3.48)
Proof. This follows from the assumption that ?
0
is continuous. Substituting
our value of x into (3.46), and setting ?
0
(v
l
) = v
l
, we have
x
_
c
r
d
?v
l
_
= v
l
(3.49)
which simpli?es to the expression in (3.47).
We therefore have an explicit expression for the lending level v
l
. Recall
that when wealth lies below this level, all of the wealth should be invested into
the foreign bank. The quantity v
l
varies nontrivially with changes in most of
the parameters, but there are a few things we can note about its behavior.
For instance, we can see that as c approaches zero, v
l
also approaches zero;
that is, if we have a low rate of consumption then we should invest most
of our money in the domestic bank. Indeed, this agrees with our intuition.
However, the behavior of v
l
with respect to the other parameters is more
di?cult to analyze.
Next we consider the interval [0, v
l
], on which ?
0
(v) = v.
Proposition 3.5. Under the assumption that ?
0
(v) = v on [0, v
l
], the func-
26
tion h solves the following:
?h = [(µ +r
f
)v ?c] h
+
1
2
?
2
v
2
h
(3.50)
with boundary conditions
h(0) = 1 and
h(v
l
)
h
(v
l
)
= ?
1
d
_
c
r
d
?v
l
_
. (3.51)
Proof. From part (iv) of Theorem 3.2, and with the substitution ?
0
(v) = v
on [0, v
l
], we have
L
?
0
(v)
h = 0
?[r
d
v + (µ +r
f
?r
d
) v ?c] h
+
1
2
?
2
v
2
h
??h = 0 (3.52)
So we have
?h = [(µ +r
f
)v ?c] h
+
1
2
?
2
v
2
h
. (3.53)
The boundary condition h(0) = 1 is directly from part (v) of Theorem 3.2.
The other boundary condition arises from the fact that h ? C
2
on the interval
[0, c/r
d
) (part (i) of Theorem 3.2). At v = v
l
, the boundary between the two
regions, h should satisfy:
h(v
l
) = ?
_
1 ?
r
d
c
v
l
_
d
(3.54)
? h
(v
l
) = ??d
r
d
c
_
1 ?
r
d
c
v
l
_
d?1
(3.55)
27
Combining these conditions gives
h(v
l
)
h
(v
l
)
= ?
1
d
_
c
r
d
?v
l
_
(3.56)
Using these boundary conditions, it is possible to solve the ordinary dif-
ferential equation (3.53) numerically. Then the continuity condition h(v
l
?) =
h(v
l
+) can be used to determine the unknown parameter ?. We now need
only to show that if h has the properties stated in Proposition 3.5, then
?
0
= v.
Proposition 3.6. Suppose h satis?es the equations (3.50) and (3.51) on
[0, v
l
]. Then
arg min
0???v
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
= v, v ? [0, v
l
]. (3.57)
Proof. De?ne a function f by
f(?) = (µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v) (3.58)
for v ? [0, v
l
]. In order to prove this proposition, it su?ces to show that
f
(v) ? 0 for v ? [0, v
l
]. That is,
f
(v) = (µ +r
f
?r
d
)h
(v) +?
2
vh
(v) ? 0 (3.59)
28
Solving for h
(v) in equation (3.50) and substituting into this inequality gives
(µ +r
f
?r
d
)h
(v) +
2
v
{?h(v) ?[(µ +r
f
)v ?c] h
(v)} ? 0
? [?(µ +r
f
+r
d
)v + 2c] h
(v) + 2?h(v) ? 0 (3.60)
Rearranging, this can be put in the following form:
h(v)
h
(v)
?
(µ +r
f
+r
d
)
2?
v ?
c
?
(3.61)
We de?ne functions y(v) and z(v) as follows:
y(v) =
h(v)
h
(v)
(3.62)
z(v) =
(µ +r
f
+r
d
)
2?
v ?
c
?
(3.63)
And we complete the proof by proving the following lemma (which asserts
that y ? z on [0, v
l
]).
Lemma 3.7. With y and z as given in (3.62) and (3.63), y > z on (0, v
l
)
and y = z at v = 0 and v = v
l
.
Proof. The equation in (3.53) can be rearranged as:
?
h(v)
h
(v)
= [(µ +r
f
)v ?c] +
1
2
?
2
v
2
h
(v) (3.64)
29
Note that
y
(v) =
h
(v)
2
?h(v)h
(v)
h
(v)
2
, (3.65)
so we can solve for the quantity
h
(v)
h
(v)
as well:
h
(v)
h
(v)
=
1 ?y
(v)
y(v)
(3.66)
Substituting these into (3.64) and rearranging yields the following:
?
2
v
2
(y
(v) ?1) = ?2?y(v)
2
+ 2[(µ +r
f
)v ?c]y(v) (3.67)
The function z(v) satis?es a similar ODE (this is easily veri?ed by substitu-
tion):
?
2
v
2
_
z
(v) ?
µ +r
f
+r
d
2?
_
= ?2?z(v)
2
+ 2
_
µ +r
f
+r
d
2
v ?c
_
z(v) (3.68)
We have that y(0) = z(0) = ?c/? (to see this, set v = 0 in (3.67)), and that
y(v
l
) = z(v
l
) = ?(1/d)(c/r
d
? v
l
). First we show that y
(v
l
) < z
(v
l
). If we
substitute y(v
l
) into (3.67), we have after simpli?cation
y
(v
l
) = 1 +
r
d
+m
?
?
r
d
?
d. (3.69)
Substituting the value of d in this expression, we can show that y
(v
l
) < z
(v
l
)
30
if and only if
?(µ +r
f
) +? +m <
_
(r
d
+? +m)
2
?4r?. (3.70)
And since µ +r
f
> r
d
, this is true if
?r
d
+? +m <
_
(r
d
+? +m)
2
?4r?. (3.71)
This inequality is true, and can be checked by squaring both sides. Therefore,
we have that y
(v
l
) < z
(v
l
). This means that y > z on the interval (v
l
??, v
l
),
for some ? > 0. The remainder of the proof will be done by contradiction:
Suppose that there exists ˜ v ? (0, v
l
) such that y(˜ v) = z(˜ v) and y > z on
(˜ v, v
l
). If we can show that no such ˜ v exists, the proof will be complete.
Since y(˜ v) = z(˜ v) and y > z on (˜ v, v
l
), we have y
(˜ v) ? z(˜ v). So by
substitution in (3.67) and (3.68), we have
1?
2?
?
2
˜ v
2
y(˜ v)
2
+
2[(µ +r
f
)˜ v ?c]
?
2
˜ v
2
y(˜ v)
?
(µ +r
f
+r
d
)
2?
?
2?
?
2
˜ v
2
z(˜ v)
2
+
2[
1
2
(µ +r
f
+r
d
)˜ v ?c]
?
2
˜ v
2
z(˜ v) (3.72)
Note that the middle terms cancel (as y(˜ v) = z(˜ v)), so substituting the value
of z(˜ v), what remains can be simpli?ed to
1 ?
µ +r
f
+r
d
2?
? ?
µ +r
f
?r
d
?
2
˜ v
_
µ +r
f
+r
d
2?
˜ v ?
c
?
_
(3.73)
The right hand side of this inequality is positive. We therefore have two
31
cases. In the case where
µ+r
f
+r
d
2?
? 1, we directly obtain our contradiction.
In the case when
µ+r
f
+r
d
2?
< 1, the inequality in (3.73) can be written as
˜ v ?
2c(µ +r
f
?r
d
)
?
2
[2? ?(µ +r
f
) ?r
d
] + [(µ +r
f
)
2
?r
2
d
]
(3.74)
If we can show that v
l
is less than that quantity, then ˜ v ? (0, v
l
) cannot
exist. It turns out that if we substitute in the value of v
l
we can show that
˜ v ?
2c(µ+r
f
?r
d
)
?
2
[2??(µ+r
f
)?r
d
]+[(µ+r
f
)
2
?r
2
d
]
is equivalent to the inequality in (3.70), which
was already shown to be true.
Therefore, we have shown that there cannot exist ˜ v ? (0, v
l
) such that
y(˜ v) = z(˜ v) and y > z on (˜ v, v
l
). So y > z on the whole interval (0, v
l
).
The results of this section are summarized in the following theorem.
Theorem 3.8. The constrained minimum probability of lifetime ruin
? ? C
1
(R
+
) ? C
2
(R
+
\ {c/r
d
}) is given by
?(v) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
h(v) if v ? [0, v
l
]
?
_
1 ?
r
d
c
v
_
d
, if v ? (v
l
, c/r
d
)
0 if v > c/r
d
(3.75)
where h solves the di?erential equation speci?ed in (3.50) and (3.51) and
where v
l
is as speci?ed in Corollary 3.4, and where
? = h(v
l
)
_
1 ?
r
d
c
v
l
_
?d
(3.76)
32
The optimal investment strategy ?
?
(v) is given by
?
?
(v) =
_
¸
¸
_
¸
¸
_
v if v ? [0, v
l
]
µ+r
f
?r
d
?
2
1
d?1
_
c
r
d
?v
_
if v ? (v
l
, c/r
d
]
(3.77)
33
4 Numerical Examples
Here we provide a few examples with numerical data to illustrate the results
of section 3.2. We assume the following parameter values:
• r
d
= 0.02; the domestic bank has an interest rate of 2% over in?ation.
• r
f
= 0.035; the foreign bank has an interest rate of 3.5% over in?ation.
• µ = 0.025; the drift of the exchange rate is 2.5%.
• ? = 0.20; the volatility of the exchange rate is 20%.
• c = 1; wealth is consumed at a rate of one unit per year.
• ? = 0.04; constant hazard rate of 4% such that the individual’s ex-
pected future lifetime is 25 years.
For these parameter choices, we have an approximate lending level of v
l
=
14.64. So the individual would invest all of her wealth in the foreign bank
when v ? 14.64, and some amount less than her total wealth when v > 14.64.
Moreover, the wealth level at which she invests only in the domestic bank
is c/r
d
= 50. Figure 4.1 shows the amount invested in the foreign bank for
wealth levels v ? [0, 50], computed using Theorem 3.8.
We can also express the optimal investment strategy in terms of the
fraction of total wealth invested. Figure 4.2 shows the fraction of total wealth
that the individual would invest in the foreign bank for v ? [0, 50].
The function ?(v) given in Theorem 3.8 can be evaluated numerically
for v ? 14.64 using a software ODE solver. After that solution is found,
34
the other piece of ?(v) can be evaluated. For this example, we have the
boundary conditions (as in (3.51)) of h(0) = 1 and
h(14.64)
h
(14.64)
= ?10.36. Solv-
ing numerically in Maple (using dsolve with numeric, method=bvp options)
gives a curve with h(14.64) ? 0.361. This value can then be used to solve for
? in Theorem 3.8, using (3.76). The resulting curve is shown in Figure 4.3.
Figure 4.1: Optimal investment strategy. Here v
l
= 14.64.
35
Figure 4.2: Percentage of total wealth invested.
Figure 4.3: Minimum probability of ruin. The level v
l
= 14.64 is indicated.
36
5 Summary and Conclusion
In this paper we consider the problem of minimizing the probability of lifetime
ruin of an individual investing in a market with foreign and domestic bank
accounts. Our model assumes that the investor is not allowed to borrow,
and that her consumption remains at a constant level. By extending the
work of Bayraktar and Young (2006), we ?nd expressions for the minimum
probability of ruin as well as the optimal investment strategy for any given
wealth level.
Moreover, part of our goal was to present the arguments leading to our
results in a very clear manner. To that end we included a reference of the
main concepts from probability theory and stochastic calculus which were
applied, and attempted to make the steps of each proof clear and justi?ed.
We ?nd that there exists a “lending level” of wealth at which the invest-
ment strategy changes. For wealth below that level, the individual invests all
of her wealth in the foreign bank account. For wealth above the lending level,
the individual instead is able to reduce her risk of ruin by investing a portion
of her wealth into the risk-free domestic bank. Naturally, an individual with
a su?ciently high amount of wealth will have zero risk of lifetime ruin as
long as her consumption is constant.
We do not address the case in which borrowing is allowed or in which
the consumption rate varies with total wealth. Bayraktar and Young (2006)
cover these cases, and their results naturally apply to our model as well. The
assumptions of this paper are in some ways simplistic, and the results could
37
be made more realistic by assuming random interest rates or other (possibly
random) consumption rates. Additionally, the assumption that only two
currencies are tradeable is itself a signi?cant simpli?cation.
Nevertheless, viewing these results in the context of foreign exchange
markets can give insight into the behavior of any investor seeking to minimize
the risk of bankruptcy.
38
6 Appendix: MATLAB and Maple Code
Maple code used to solve the ODE in (3.50) and (3.51):
> sol1 := dsolve(0.04*h(v)-(0.06*v-1)*(diff(h(v),v))
- 0.02*v^2*(diff(diff(h(v),v),v)) = 0,
h(0.01) = 1,
h(14.644660940672622)/(D(h))(14.644660940672622) = -10.355339059327376],
numeric, method = bvp, abserr = 0.001);
sol1(14.6446);
plots[odeplot](sol1, 0.001 .. 15, color = red);
MATLAB function for computing the lending level v
l
as in Corollary 3.4:
function[vl] = LendingLevel(rd,rf,mu,sigma,c,lambda)
m = 0.5*(mu+rf-rd)^2/sigma^2;
d = ( (rd+lambda+m) + sqrt((rd+lambda+m)^2-4*rd*lambda) )/(2*rd);
x = (mu+rf-rd)/(sigma^2*(d-1));
vl = (x*c)/((1+x)*rd);
MATLAB function for computing the function ?
?
(v) as in Theorem 3.8:
function[pi] = OptimalStrategyPlot(rd,rf,mu,sigma,c,lambda)
pi = zeros(1,251);
v = 0:c/(rd*250):c/rd;
m = 0.5*(mu+rf-rd)^2/sigma^2;
d = ( (rd+lambda+m) + sqrt((rd+lambda+m)^2-4*rd*lambda) )/(2*rd);
vl = LendingLevel(rd,rf,mu,sigma,c,lambda);
for i = 1:251
if v(i) < vl,
pi(i) = v(i);
else
pi(i) = (mu+rf-rd)/(sigma^2*(d-1)) * (c/rd - v(i));
end
end
39
MATLAB function used to compute the function ?(v) as in Theorem
3.8. Here, HVECEX was a global variable consisting of the data imported
from Maple regarding the ?rst half of the function (solved by the Maple code
shown above).
function[h] = h_example_plot()
global HVECEX
h_first = HVECEX;
lambda = 0.04; mu = 0.025; r_f = 0.035; r_d = 0.02; c = 1; sigma = 0.2;
m = 0.5*((mu+r_f-r_d)/sigma)^2;
d = (1/(2*r_d))*( (r_d+lambda+m) + sqrt((r_d+lambda+m)^2-4*r_d*lambda) );
vl = LendingLevel(r_d,r_f,mu,sigma,c,lambda);
v_first = (vl/100:vl/100:vl)’;
v_rest = (vl+vl/100:vl/100:50)’;
beta = h_first(length(h_first))*(1-r_d*vl/c)^(-d);
h_rest = beta*(1-r_d*v_rest/c).^d;
v = [ v_first ; v_rest ];
h = [ h_first ; h_rest ];
40
References
[1] Erhan Bayraktar and Virginia R. Young. Minimizing the Probability
of Lifetime Ruin under Borrowing Constraints. Insurance: Mathematics
and Economics, 41(1), 2006.
[2] Bj¨ ork, Tomas. Arbitrage Theory in Continuous Time. Oxford University
Press Inc, 2004.
[3] V. C. Pestien and W. D. Suddherth. Continuous-time red and black:
How to control a di?usion to a goal. Mathematics of Operations Research,
10(4):599–611, 1985.
[4] Philip E. Protter. Stochastic Integration and Di?erential Equations.
Springer, 2004.
[5] Steven E Shreve. Stochastic Calculus for Finance II. Springer, 2004.
[6] V. R. Young. Optimal investment strategy to minimize the probability
of lifetime ruin. North American Actuarial Journal, 8(4):105–126, 2004.
41
doc_157134530.pdf
In finance, an exchange rate (also known as a foreign-exchange rate, forex rate, FX rate or Agio) between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency.
ISG HS1 - 598
Minimizing the Probability of Ruin
in Exchange Rate Markets
A Professional Master’s Project Report
submitted to the Faculty
of the
WORCESTER POLYTECHNIC INSTITUTE
in partial fu?llment of the requirements for the
Degree of Master of Science
in
Financial Mathematics
by
Tyler A. Chase
April 2009
Professor Hasanjan Sayit, Project Advisor
This report represents the work of one or more WPI graduate students
submitted to the faculty as evidence of completion of a degree requirement.
WPI routinely publishes these reports on its web site without editorial or peer review.
Abstract
The goal of this paper is to extend the results of Bayraktar and Young (2006)
on minimizing an individual’s probability of lifetime ruin; i.e. the probability
that the individual goes bankrupt before dying. We consider a scenario in
which the individual is allowed to invest in both a domestic bank account and
a foreign bank account, with the exchange rate between the two currencies
being modeled by geometric Brownian motion. Additionally, we impose the
restriction that the individual is not allowed to borrow money, and assume
that the individual’s wealth is consumed at a constant rate. We derive for-
mulas for the minimum probability of ruin as well as the individual’s optimal
investment strategy. We also give a few numerical examples to illustrate
these results.
i
Contents
Abstract i
1 Introduction 1
2 Background 3
2.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Probability of Lifetime Ruin 11
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Minimum Probability of Ruin . . . . . . . . . . . . . . . . . . 15
4 Numerical Examples 34
5 Summary and Conclusion 37
6 Appendix: MATLAB and Maple Code 39
ii
1 Introduction
In the current state of the American economy, it is natural for individuals to
be concerned about their ?nancial well-being in the present as well as in the
future. In particular, some have reason to be concerned about the possibility
of bankruptcy during retirement. In a situation where the American dollar
is a bit weaker or more unstable, it is reasonable to suppose that some in-
dividuals may be interested in investing in a potentially more stable foreign
currency.
In this paper we begin to consider an extension of the work of Bayraktar
and Young (2006) in determining how an individual should invest her wealth
in order to minimize the probability that she ruins before death. We focus
on the scenario in which the individual’s rate of consumption is constant and
borrowing constraints are imposed. However, we look at a ?nancial market
model in which the individual has the option of investing some of her wealth
in a domestic bank account and some in a foreign bank account; the risk is
introduced by the random exchange rate between the two currencies.
The Foreign Exchange market is an interesting model to consider. For
some investors, the possibility of trading in a currency market can be more
appealing than trading in a stock exchange. The Foreign Exchange o?ers
high market liquidity, and has a high trading volume. Margins of pro?t are
lower than in other, possibly riskier markets, but there is still the potential
for signi?cant earnings. So while not every investor would be interested in
this market, its applications are relevant.
1
The most common criterion for optimization problems in ?nancial lit-
erature is the maximization of expected utility of consumption, and there
has been a substantial amount of work done on that subject. Bayraktar and
Young (2006) note that these methods generally depend on a subjective util-
ity function for consumption, whereas minimizing the probability of lifetime
ruin may be more appealing and comprehensible to individuals since that
criterion is more objective. And indeed, this technique has seen increased
application in recent years.
Our work closely follows that of Bayraktar and Young (2006), since our
market model is closely related. We consider only the “no-borrowing” case
with constant consumption; after that, it should not be di?cult to see how
the other cases would follow. Before presenting the main results, we review a
few of the de?nitions and theorems from probability and stochastic calculus,
that the reader may have a suitable reference. Later we give a few numerical
examples to demonstrate our results.
2
2 Background
2.1 Probability
To begin with, it would be helpful to establish the setting in which our work
takes place. Speci?cally, we will assume the existence of a continuous-time
?ltered probability space. We will give a brief de?nition of most of the
relevant fundamental concepts. While this is not strictly necessary, it helps
to assure that the reader is able to follow the reasoning presented in the
paper.
De?nition 2.1 (?-algebra). Let ? be a nonempty set, and let F be a collec-
tion of subsets of ?. F is a ?-algebra (also known as a ?-?eld) on ? if the
following conditions are satis?ed:
• ? ? F
• A ? F =? A
c
? F
• A
1
, A
2
, . . . ? F =?
?
i=1
A
i
? F
De?nition 2.2 (Probability measure). Let ? be a nonempty set, and F a
?-algebra of subsets of ?. A probability measure is a function P : F ? [0, 1]
such that
• P(?) = 1.
• If A
1
, A
2
, . . . ? F are disjoint, then P (
?
i=1
A
i
) =
?
i=1
P (A
i
).
3
De?nition 2.3 (Probability Space). A triple (?, F, P), consisting of a sam-
ple space ?, a ?-algebra F on ?, and a probability measure P on F, is called
a probability space.
De?nition 2.4 (Random Variable). Given a probability space (?, F, P), a
random variable is a function X : ? ? R with the property that for any
Borel set B ? B, the inverse image X
?1
(B) belongs to F.
De?nition 2.5 (Measurability). A random variable X is said to be measur-
able with respect to a ?-algebra G (or, X is G-measurable) if for any Borel
set B ? B the inverse image X
?1
(B) belongs to G.
These are the basic assumptions of most models in probability. To dis-
cuss results relying on stochastic calculus, we also would like to review the
concepts of ?ltrations and stopping times.
De?nition 2.6 (Filtration). Let (?, F, P) be a probability space, and let T be
a ?xed positive number. A continuous ?ltration is a collection of ?-algebras
with the following properties:
• ?t ? [0, T] , ?F
t
? F.
• s ? t =? F
s
? F
t
Moreover,
_
?, F, P, {F
t
}
0?t?T
_
is called a ?ltered probability space.
De?nition 2.7 (Stopping Time). A stopping time ? is a random variable
satisfying the following property: ?t ? [0, T], {? ? ? : ? (?) ? t} ? F
t
.
4
De?nition 2.8 (Stopped ?-algebra). Let
_
?, F, P, {F
t
}
0?t?T
_
be a ?ltered
probability space, and let ? be a stopping time. The stopped ?-algebra (stopped
at ?) is de?ned as:
F
?
= {A ? F : A ? {? ? t} ? F
t
}.
2.2 Stochastic Calculus
In addition to the material on probability theory, we also provide a reference
for fundamental de?nitions and theorems in stochastic calculus, as these are
necessary tools in the proofs of our results.
De?nition 2.9 (Stochastic Process). A continuous stochastic process is a
collection of random variables {X
t
, t ? [0, T]}. For each ? ? ?, X
t
(?) is a
deterministic function called the sample path, or trajectory.
De?nition 2.10 (Adaptedness). A stochastic process X
t
is said to be adapted
to a ?ltration {F
t
} if for every t, X
t
is F
t
-measurable.
De?nition 2.11 (C` adl`ag Process). A stochastic process X
t
is called c`adl`ag
if it has sample paths satisfying the following conditions almost surely:
• X
t
(?) is right-continuous, i.e. lim
t?a
+ X
t
(?) = X
a
(?) for all a.
• X
t
(?) has left-limits, i.e. lim
t?a
? X
t
(?) exists for all a.
The word “c` adl` ag” is a French acronym, standing for continue `a droite,
limit´ee `a gauche, literally “continuous on the left, limited on the right.”
5
De?nition 2.12 (Conditional Expectation). The conditional expectation of
a random variable X with respect to a ?-algebra G ? F, denoted E [X|G], is
itself a random variable with the following properties:
• E [X|G] is G-measurable.
• ?A ? G,
_
A
XdP =
_
A
E [X|G] dP
De?nition 2.13 (Martingale). Let
_
?, F, P, {F
t
}
0?t?T
_
be a ?ltered prob-
ability space. An adapted stochastic process M
t
is called a martingale with
respect to F
t
if it satis?es the following property:
E [M
t
|F
s
] = M
s
for all 0 ? s ? t (2.1)
De?nition 2.14 (Stopped Process). If X is a stochastic process and ? is a
stopping time, then we can de?ne the stopped process
X
t??
=
_
¸
_
¸
_
X
t
if t ? ?
X
?
if t > ?
(2.2)
In that case, the stochastic process is said to be “stopped” at time ?. The
stopped process is equal to the original process until time ?, and becomes
constant after that time, equal to the value of X
?
.
A similar notion is the “killed” process, which instead of taking on the
constant value X
?
at its killing time, takes a value ? (outside the range of
X) called the “co?n state”.
6
De?nition 2.15 (Brownian Motion). A standard Brownian motion (or Wiener
process) is a continuous stochastic process W
t
with independent increments
which are normally distributed: For all 0 = t
0
< t
1
< t
2
< . . . < t
m
, the
increments
W
t
1
?W
t
0
, W
t
2
?W
t
1
, . . . , W
t
m
?W
t
m?1
(2.3)
are independent and normally distributed with mean 0 and variance t
i
?t
i?1
.
De?nition 2.16 (It¯ o Integral). The It¯o integral
_
t
0
H
s
dW
s
of a c`adl`ag process
H is de?ned as:
_
t
0
H
s
dW
s
= lim
n??
n
i=1
H
t
i?1
_
W
t
i
?W
t
i?1
_
. (2.4)
where 0 = t
0
< t
1
< t
2
< . . . < t
n
is a partition of [0, t], growing ?ner as n
increases.
Theorem 2.17 (Properties of the It¯ o Integral). The It¯o integral I
t
=
_
t
0
H
s
dW
s
has the following properties:
• I
t
has continuous sample paths.
• I
t
is F
t
-adapted.
• I
t
is a martingale.
• I
t
has quadratic variation [I, I]
t
=
_
t
0
H
2
s
ds.
De?nition 2.18 (It¯ o Process). An It¯o Process is a stochastic process X of
7
the following form:
X
t
= X
0
+
_
t
0
µ
s
dW
s
+
_
t
0
?
s
ds (2.5)
where µ and ? are adapted stochastic processes, and X
0
is a nonrandom
initial value.
Alternatively, this can be written in the di?erential form:
dX
t
= µ
s
dW
s
+?
s
ds (2.6)
Also note that the quadratic variation of the It¯o process X is given by (in
integral and di?erential form):
[X, X]
t
=
_
t
0
µ
2
s
ds (2.7)
d [X, X]
t
= µ
2
t
dt (2.8)
Theorem 2.19 (It¯ o’s Formula). Let X
t
be an It¯o process and let f (t, x) be
a function for which the partial derivatives f
t
, f
x
, and f
xx
are de?ned and
continuous. Then
f (T, X
T
) = f (0, X
0
) +
_
T
0
f
t
(t, X
t
) dt +
_
T
0
f
x
(t, X
t
) dX
t
+
1
2
_
T
0
f
xx
(t, X
t
) d [X, X]
t
(2.9)
A Poisson process is a stochastic process which takes nonnegative integer
values characterized by a rate parameter ?. It is typically used to model the
8
number of events which occur in a given time interval.
De?nition 2.20 (Poisson Process). The Poisson process N
t
with rate pa-
rameter ? obeys a Poisson distribution with parameter ?t:
P (N
t
= k) =
(?t)
k
k!
e
??t
(2.10)
The Poisson process is an example of what is called a “pure jump” pro-
cess. It has stationary, independent increments. Moreover, the times between
successive jumps are independent and follow an exponential distribution with
parameter ? (i.e., an exponential distribution with mean 1/?).
Theorem 2.21 (Compensated Poisson Process). If N
t
is a Poisson process
with rate parameter ?, we de?ne the compensated Poisson process M
t
=
N
t
??t. M
t
is a martingale.
De?nition 2.22 (Jump Process). Let X
t
be an It¯o process, and let J
t
be a
“pure jump” process. That is, J
t
is an adapted, c`adl`ag process with ?nitely
many jumps on the interval (0, T], and is constant between jumps. We will
call a process of the following form a jump process:
Y
t
= X
t
+J
t
(2.11)
When discussing processes with jumps, It¯ o’s Formula takes a slightly
altered form.
Theorem 2.23 (It¯ o’s Formula for Jump Processes). Let Y
t
be a jump process,
and let f
be a function which is twice continuously di?erentiable. Then9
we have the following:
f (Y
t
) = f (Y
0
) +
_
t
0
f
(Y
s
) dY
c
s
+
1
2
_
t
0
f
(Y
s
) d [Y, Y ]
c
s
+
0<s?t
[f (Y
s
) ?f (Y
s
?)] (2.12)
Here, the quantity Y
c
denotes the continuous part of the jump process
Y . If Y is given in the form of (2.11), then Y
c
= X.
10
3 Probability of Lifetime Ruin
We will be considering the problem of minimizing an individual’s probability
of ruin under the condition that borrowing is forbidden. In section 3.1 we
outline the ?nancial market model used in the analysis. In 3.2 we present
and prove the main results.
3.1 Model
We will assume a model in which an individual has the option of investing
in two assets: A domestic bank account B, and a foreign bank account F.
Each of these banks will have its own ?xed interest rate, and the exchange
rate X between the two currencies will be modeled by a geometric Brownian
motion. So our assets have the following dynamics:
dB = r
d
Bdt
dF = r
f
F dt (3.1)
dX = µX dt +?X dW
where ? and ? are constants, and W is a standard Brownian motion. We
will prove a lemma which states that the above formulation is equivalent to
a situation in which the individual is allowed to invest in the domestic bank
account and in a domestic risky asset whose price is given by
˜
F = FX.
11
Lemma 3.1. The model in (3.1) is equivalent to the following model:
dB = r
d
Bdt
d
˜
F = (µ +r
f
)
˜
F dt +?
˜
F dW (3.2)
where
˜
F = FX.
Proof. The proof follows from the multidimensional version of It¯o’s lemma.
The two-dimensional case is as follows: If U and V are It¯o processes, and if
f (t, u, v) is a function which is twice continuously di?erentiable, then
df (t, U, V ) =f
t
(t, U, V ) dt +f
u
(t, U, V ) dU +f
v
(t, U, V ) dV
+
1
2
f
uu
(t, U, V ) dUdU +
1
2
f
vv
(t, U, V ) dV dV (3.3)
+f
uv
(t, U, V ) dUdV
where f
u
denotes the ?rst partial derivative of f with respect to u, and so
on. Using f (t, u, v) = uv, this reduces to
df (t, U, V ) = 0dt +V dU +UdV + 0dUdU + 0dV dV + 1dUdV
= V dU +UdV +dUdV
12
Substituting U = X and V = F, we obtain
d
˜
F = FdX +XdF +dXdF
= F (µXdt +?XdW) +X (r
f
Fdt) + (µXdt +?XdW) (r
f
Fdt)
=
˜
F (µdt +?dW) +
˜
F (r
f
dt) + 0
And this can be rearranged as
d
˜
F = (µ +r
f
)
˜
Fdt +?
˜
FdW (3.4)
So the possibility of investing in the domestic bank B and the foreign bank
F is equivalent to investing in the domestic bank and a domestic asset
˜
F
with dynamics as given in (3.4).
We will henceforth assume that the individual may act by investing a
portion of her wealth in
˜
F, with the remainder invested in B. In this for-
mulation,
˜
F can be interpreted as a risky asset while B is a risk-free asset;
we assume that µ + r
f
> r
d
(indeed, we are not interested in the problem
otherwise, since no investor should invest in a risky asset with lower expected
return than the risk-free asset).
We will also assume that the individual’s total wealth is continuously
consumed at a constant rate c.
Let V
t
denote the wealth of the individual at time t, and denote by ?
t
the amount she invests in the risky asset
˜
F. Then the amount invested in the
risk-free asset B is V
t
??
t
. Therefore the wealth process obeys the following
13
dynamics:
dV
t
=
?
t
˜
F
t
d
˜
F
t
+
(V
t
??
t
)
B
t
dB
t
?cdt
? dV
t
=
?
t
˜
F
t
_
(µ +r
f
)
˜
F
t
dt +?
˜
F
t
dW
t
_
+
(V
t
??
t
)
B
t
r
d
B
t
dt ?cdt
This can be simpli?ed as:
_
¸
_
¸
_
dV
t
= [r
d
V
t
+ (µ +r
f
?r
d
) ?
t
?c] dt +??
t
dW
t
V
0
= v
(3.5)
We now wish to de?ne what is meant by “lifetime ruin”. We let ?
0
denote
the ?rst time that V = 0, and let ?
d
denote the individual’s time of death.
Lifetime ruin is de?ned as the event in which the wealth process reaches zero
before the individual dies, i.e. the event {?
0
< ?
d
}. Here we will assume that
?
d
follows an exponential distribution with parameter ?, so that the expected
value of ?
d
is 1/? (later, we will model this using a Poisson process with rate
parameter ?, since the time between jumps in the Poisson process follows an
exponential distribution).
The minimum probability of ruin will be denoted by ? (v) (the argu-
ment v indicates that this probability is conditional on V
0
= v). So we are
minimizing the probability that ?
0
< ?
d
, with respect to the set of admissi-
ble trading strategies (denoted by A). For this paper, we also impose the
restriction that 0 ? ?
t
? V
t
(i.e. no borrowing or short-selling is possible).
14
Therefore the probability ? (v) is given by:
? (v) = inf
?
P[ ?
0
< ?
d
| V
0
= v ] (3.6)
For each real number ?, we can de?ne a second-order di?erential operator
L
?
which is associated with the minimization problem. For each open subset
G of R
+
and for each h ? C
2
(G), de?ne the function L
?
h : G ? R as
follows:
L
?
h(v) = [r
d
v + (µ +r
f
?r
d
) ? ?c] h
(v) +
1
2
?
2
?
2
h
(v) ??h(v) (3.7)
The operator L
?
will be used in the following sections to characterize ? in a
compact manner.
3.2 Minimum Probability of Ruin
In this section we will present the veri?cation theorem which states the nec-
essary and su?cient conditions that ? must satisfy. First, note that when the
individual’s wealth is above c/r
d
, the probability of lifetime ruin is equal to
0; the individual can invest all of her wealth in the domestic (risk-free) bank
account and consume continuously at rate c with no possibility of running
out of money. To see this, consider the dynamics of the wealth process when
all of the individual’s wealth is invested in the domestic bank:
dV
t
= r
d
V
t
dt ?cdt
15
In this case, there is no stochastic integral involved; so it can be expressed
as an ordinary di?erential equation:
dV
dt
= r
d
V ?c
So for all V ? c/r
d
, we have that
dV
dt
? 0, i.e. there is no chance that the
wealth process will decrease in this situation, let alone reach zero.
Thus in addition to the stopping times ?
0
and ?
d
which we already de-
?ned, we also introduce the stopping time ?
c/r
d
= inf {t > 0 : V
t
? c/r
d
},
that is, the ?rst time that the individual’s wealth reaches c/r
d
(or more). If
we now de?ne the stopping time ? = ?
d
??
c/r
d
, it follows that we can express
? as follows:
? (v) = inf
?
P[ ?
0
< ? | V
0
= v ] (3.8)
We can now present the veri?cation theorem:
Theorem 3.2. Suppose h : R
+
? [0, 1] is a decreasing function, and ?
0
: R
+
?R
+
which satisfy the following conditions:
(i) h ? C
2
on [0, c/r
d
)
(ii) ?
0
? A
(iii) L
?
h(v) ? 0, for 0 ? ? ? v < c/r
d
(iv) L
?
0
(v)
h(v) = 0, for v ? (0, c/r
d
)
(v) h(0) = 1 and h(v) = 0 for v ? c/r
d
16
Then the minimum probability of lifetime ruin ? is given by:
? (v) = h(v) , v ? 0 (3.9)
And the optimal investment strategy ?
?
in the risky asset
˜
F is given by:
?
?
(v) = ?
0
(v) , v ? [0, c/r
d
] (3.10)
Proof. Suppose we have h which satis?es the properties stated above. Let
N be a Poisson process (independent of W) with rate parameter ?. The
stopping time ?
d
will be de?ned as the time of the ?rst jump of the process
N. Let ? be a function on the interval [0, c/r
d
] with 0 ? ?(v) ? v, and let
V
?
denote the wealth process under the investment strategy ?. We denote
?
s
= ?(V
?
s
). We will kill the wealth process at time ?
d
and assign W
?
d
= ?,
(the co?n state). Our convention will be that for any function f : R
+
?R
+
,
we let f(?) = 0. In particular, note that h(c/r
d
) = 0 and h(V
?
?
d
) = 0. Using
the wealth process dynamics as stated in 3.5 and It¯o’s formula as in 2.23, we
have the following:
h(V
?
t????
0
) ?h(v) =
_
t????
0
0
h
(V
?
s
)dV
?
s
+
1
2
_
t????
0
0
h
(V
?
s
)d[V
?
, V
?
]
s
+
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] (3.11)
17
=
_
t????
0
0
h
(V
?
s
){[r
d
V
?
s
+ (µ +r
f
?r
d
)?
s
?c]ds +??
s
dW
s
}
+
1
2
_
t????
0
0
h
(V
?
s
)(?
2
?
2
s
)ds +
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] (3.12)
Since the process jumps only at time ?
d
, then the jump at time s can be
expressed as h(V
?
s
) ?h(V
?
s
?
) = ?h(V
?
s
?
)?N
s
and so we write
0<s?t????
0
[h(V
?
s
) ?h(V
?
s
?)] = ?
_
t????
0
0
h(V
?
s
?)dN
s
(3.13)
In order to write the expression in a compact manner, we add and subtract
the term ?
_
t????
0
0
h(V
?
s
?
) in the right hand side:
h(V
?
t????
0
) = h(v) +
_
t????
0
0
{[r
d
V
?
s
+ (µ +r
f
?r
d
)?
s
?c]h
(V
?
s
) +
1
2
?
2
?
2
s
h
(V
?
s
)}ds
?
_
t????
0
0
?h(V
?
s
?)ds +
_
t????
0
0
??
s
h
(V
?
s
)dW
s
?
_
t????
0
0
h(V
?
s
?)dN
s
+?
_
t????
0
0
h(V
?
s
?)ds (3.14)
So the expression can be simpli?ed to:
h(V
?
t????
0
) = h(v) +
_
t????
0
0
L
?
h(V
?
s
)ds +
_
t????
0
0
??
s
h
(V
?
s
)dW
s
?
_
t????
0
0
h(V
?
s
?)d(N
s
??s) (3.15)
Taking the expectation of both sides, the third and fourth terms vanish (it
can be shown that the integrands satisfy su?cient conditions). So, following
18
from assumption (iii) in the theorem statement, we have:
E
v
[h(V
?
t????
0
)] = h(v) +E
v
__
t????
0
0
L
?
h(V
?
s
)ds
_
? h(v) (3.16)
Here, E
v
indicates that the expectation is conditional on V
0
= v. Therefore,
the process h(V
?
t????
0
), t ? 0, is a submartingale. Since h(0) = 1, h(V
?
?
0
??
) =
0, and h(V
?
c/r
d
), it follows (where 1 denotes the indicator function) that
h(V
?
?
0
??
) = 1
{?
?
0
<?}
. (3.17)
Now, taking expectations of both sides and applying the optional sampling
theorem gives
E
v
h(V
?
?
0
??
) = P
v
(?
?
0
< ?) ? h(v), (3.18)
since h(V
?
t????
0
) is a submartingale. Therefore
inf
?
P
v
(?
?
0
< ?) = ?(v) ? h(v). (3.19)
If we consider ?
0
as speci?ed in the theorem statement (namely, property
(iv), i.e. ?
0
is the minimizer of L
?
h), then it follows that h(V
?
0
t????
0
) is a
martingale. So we have that
E
v
h(V
?
0
?
0
??
) = P
v
(?
?
0
0
< ?) = h(v). (3.20)
We have therefore shown that the statements in 3.9 and 3.10 are true for
19
v ? [0, c/r
d
). Together with the assumption in (v) and the fact that
? (v) = inf
??A
P[ ?
0
< ? | V
0
= v ], the proof is complete.
The forms of the functions h and ? are discovered by imposing a few
additional properties that are not stated explicitly in the preceding theorem.
However, because that theorem asserts that the function h is unique on
the interval [0, c/r
d
), if we ?nd h and ? which satisfy the assumptions of
Theorem 3.2 and the additional assumptions, then the additional properties
are implicit.
We make the following additional hypotheses as well: In the constrained
case, we assume that ?
0
(the amount invested in the foreign bank
˜
F) is a
continuous function of v, and that there exists a wealth level v
l
such that
v ??
0
(v) = 0 for v < v
l
and v ??
0
(v) > 0 for v > v
l
. The idea here is that
when the individual has more wealth, it is wiser to invest a portion of it in a
risk-free asset. The subscript l denotes that the individual is, upon reaching
this level, “lending” some amount of money to the domestic bank.
We will consider the intervals [0, v
l
] and (v
l
, c/r
d
] separately. First we
look at the interval (v
l
, c/r
d
], and we assume that the borrowing constraint
is non-binding.
Proposition 3.3. Assume that 0 ? ?
0
(v) < v on the interval (v
l
, c/r
d
]. The
function h has the following form (with ? ? 1):
h(v) = ?
_
1 ?
r
d
c
v
_
d
, (3.21)
20
where
d =
1
2r
d
_
(r
d
+? +m) +
_
(r
d
+? +m)
2
?4r
d
?
_
> 1, (3.22)
and
m =
1
2
_
µ +r
f
?r
d
?
_
2
. (3.23)
The corresponding ?
0
on (v
l
, c/r
d
] is given by:
?
0
(v) =
µ +r
f
?r
d
?
2
1
d ?1
_
c
r
d
?v
_
. (3.24)
Proof. Items (iii), (iv), (v) of Theorem 3.2 require that we solve
?h(v) = (r
d
v ?c)h
(v) + min
?
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
(3.25)
with the boundary condition h(c/r
d
) = 0. We can show that we also have
the boundary condition h
(c/r
d
) = 0: Consider the solution ? of (3.25)
with ? = 0 (corresponding to the event that the individual never dies).
So h ? ? on some interval (c/r
d
? ?, c/r
d
], since the probability of ruin be-
fore death is necessarily less than the probability of ruin before in?nity (i.e.
P(?
0
< ?
d
) ? P(?
0
< ?)). So it is enough to show that ?
(c/r
d
) = 0, which
would imply h
(c/r
d
) = 0. Note that ? is a solution to the following (with
?(c/r
d
) = 0):
0 = (r
d
v ?c)?
(v) + min
?
_
(µ +r
f
?r
d
)??
(v) +
1
2
?
2
?
2
?
(v)
_
(3.26)
21
Pestien and Suddherth (1985) showed that the optimal investment strategy
?
?
maximizes (in our case) the quantity:
f(?) =
(µ +r
f
?r
d
)? ?(c ?r
d
v)
?
2
. (3.27)
By ordinary calculus, it is easily checked that the value of ? which maximizes
that expression is ?
?
= 2(c?r
d
v)/(µ+r
f
?r
d
). However, we also have (again
from ordinary calculus, this time applied to the minimization problem in
(3.26)) that
?
?
(v) = ?
µ +r
f
?r
d
?
2
?
(v)
?
(v)
(3.28)
Therefore, for v ? (c/r
d
??, c/r
d
], we have (for some k < 0)
?
(v) = k(c ?r
d
v)
m/r
d
(3.29)
So we showed that ?
(c/r
d
) = 0, which implies that h
(c/r
d
) = 0.
To be consistent with the hypothesis that the borrowing constraint is
non-binding on (v
l
, c/r
d
], it must be true that h is convex on (v
l
, c/r
d
]. To
see this, note that if h is not convex in some neighborhood of a point v
?
?
(v
l
, c/r
d
] (i.e. h
(v) < 0 in that neighborhood), then ?
0
is as large as possible
on that neighborhood, which contradicts the hypothesis that the borrowing
constraint is non-binding. So we have that h is convex on (v
l
, c/r
d
], and can
22
therefore consider its Legendre transform
˜
h:
˜
h(u) = min
v
[h(v) +vu] (3.30)
h can be recovered from
˜
h by
h(v) = max
u
[
˜
h(u) ?uv] (3.31)
From ordinary calculus, the value of v which minimizes the quantity in (3.30)
is v = (h
)
?1
(?u) =
˜
h
(u). Therefore the value of u which maximizes the
expression in (3.31) is u = ?h
(v). We can then make substitutions into
(3.25). Using v =
˜
h
(u), it follows that
h(v) =
˜
h(u) ?u
˜
h
(u), h
(v) = ?u, and h
(v) = ?
1
˜
h
(u)
. (3.32)
Additionally, as in (3.28), we use
? = ?
µ +r
f
?r
d
?
2
h
(v)
h
(v)
(3.33)
Making these substitutions in (3.25) gives
?h(v) = (r
d
v ?c)h
(v) ?
1
2
(µ +r
f
?r
d
)
2
?
2
(h
(v))
2
h
(v)
? ?[
˜
h(u) ?u
˜
h
(u)] = (r
d
˜
h
(u) ?c)(?u) +mu
2
˜
h
(u) (3.34)
Here, m is as given in (3.23). Simplifying further gives the following di?er-
23
ential equation:
?
˜
h(u) + (r
d
??)u
˜
h
(u) ?mu
2
˜
h
(u) = cu (3.35)
The general solution of this
˜
h(u) = D
1
u
B
1
+D
2
u
B
2
+
c
r
u (3.36)
where D
1
and D
2
are constants, and B
1
and B
2
are the roots of
?? ?(r
d
?? +m)B +mB
2
= 0, (3.37)
so
B
1
=
1
2m
_
(r
d
?? +m) +
_
(r
d
?? +m)
2
+ 4?m
_
> 1 (3.38)
B
2
=
1
2m
_
(r
d
?? +m) ?
_
(r
d
?? +m)
2
+ 4?m
_
< 0 (3.39)
Let u
c
= ?h
(c/r
d
) = 0, so
˜
h
(0) = c/r
d
. From the de?nition of
˜
h and
because h(c/r
d
) = 0, we have at u = u
c
= 0,
˜
h(0) = 0. (3.40)
From this it follows that D
2
= 0. We can then use (3.36) and (3.31) to
recover h:
h(v) = max
u
_
D
1
u
B
1
+
c
r
u ?vu
_
. (3.41)
24
The maximizing value of u here (by ordinary calculus) is:
u =
_
v ?c/r
d
D
1
B
1
_
1/(B
1
?1)
. (3.42)
Substituting this back into (3.41) gives:
h(v) = D
1
_
v ?c/r
d
D
1
B
1
_
B
1
B
1
?1
?(v ?c/r
d
)
_
v ?c/r
d
D
1
B
1
_ 1
B
1
?1
h(v) =
_
D
1
(D
1
B
1
)
B
1
B
1
?1
?
1
(D
1
B
1
)
1
B
1
?1
_
_
?
c
r
d
_
B
1
B
1
?1
_
1 ?
r
d
c
u
_
B
1
B
1
?1
. (3.43)
And we simplify this by denoting the leading constant quantity by ? and
noting that
B
1
B
1
?1
= d, so we obtain
h(v) = ?
_
1 ?
r
d
c
v
_
d
. (3.44)
Using this expression for h, the optimal investment strategy ?
0
is found by
minimizing (using ordinary calculus) the expression:
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
. (3.45)
And the value of ? which minimizes this expression is
?
0
(v) =
µ +r
f
?r
d
?
2
1
d ?1
_
c
r
d
?v
_
. (3.46)
25
Corollary 3.4. The lending level v
l
takes the following form:
v
l
=
x
1 +x
c
r
d
, (3.47)
where
x =
µ +r
f
?r
d
?
2
1
d ?1
. (3.48)
Proof. This follows from the assumption that ?
0
is continuous. Substituting
our value of x into (3.46), and setting ?
0
(v
l
) = v
l
, we have
x
_
c
r
d
?v
l
_
= v
l
(3.49)
which simpli?es to the expression in (3.47).
We therefore have an explicit expression for the lending level v
l
. Recall
that when wealth lies below this level, all of the wealth should be invested into
the foreign bank. The quantity v
l
varies nontrivially with changes in most of
the parameters, but there are a few things we can note about its behavior.
For instance, we can see that as c approaches zero, v
l
also approaches zero;
that is, if we have a low rate of consumption then we should invest most
of our money in the domestic bank. Indeed, this agrees with our intuition.
However, the behavior of v
l
with respect to the other parameters is more
di?cult to analyze.
Next we consider the interval [0, v
l
], on which ?
0
(v) = v.
Proposition 3.5. Under the assumption that ?
0
(v) = v on [0, v
l
], the func-
26
tion h solves the following:
?h = [(µ +r
f
)v ?c] h
+
1
2
?
2
v
2
h
(3.50)
with boundary conditions
h(0) = 1 and
h(v
l
)
h
(v
l
)
= ?
1
d
_
c
r
d
?v
l
_
. (3.51)
Proof. From part (iv) of Theorem 3.2, and with the substitution ?
0
(v) = v
on [0, v
l
], we have
L
?
0
(v)
h = 0
?[r
d
v + (µ +r
f
?r
d
) v ?c] h
+
1
2
?
2
v
2
h
??h = 0 (3.52)
So we have
?h = [(µ +r
f
)v ?c] h
+
1
2
?
2
v
2
h
. (3.53)
The boundary condition h(0) = 1 is directly from part (v) of Theorem 3.2.
The other boundary condition arises from the fact that h ? C
2
on the interval
[0, c/r
d
) (part (i) of Theorem 3.2). At v = v
l
, the boundary between the two
regions, h should satisfy:
h(v
l
) = ?
_
1 ?
r
d
c
v
l
_
d
(3.54)
? h
(v
l
) = ??d
r
d
c
_
1 ?
r
d
c
v
l
_
d?1
(3.55)
27
Combining these conditions gives
h(v
l
)
h
(v
l
)
= ?
1
d
_
c
r
d
?v
l
_
(3.56)
Using these boundary conditions, it is possible to solve the ordinary dif-
ferential equation (3.53) numerically. Then the continuity condition h(v
l
?) =
h(v
l
+) can be used to determine the unknown parameter ?. We now need
only to show that if h has the properties stated in Proposition 3.5, then
?
0
= v.
Proposition 3.6. Suppose h satis?es the equations (3.50) and (3.51) on
[0, v
l
]. Then
arg min
0???v
_
(µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v)
_
= v, v ? [0, v
l
]. (3.57)
Proof. De?ne a function f by
f(?) = (µ +r
f
?r
d
)?h
(v) +
1
2
?
2
?
2
h
(v) (3.58)
for v ? [0, v
l
]. In order to prove this proposition, it su?ces to show that
f
(v) ? 0 for v ? [0, v
l
]. That is,
f
(v) = (µ +r
f
?r
d
)h
(v) +?
2
vh
(v) ? 0 (3.59)
28
Solving for h
(v) in equation (3.50) and substituting into this inequality gives
(µ +r
f
?r
d
)h
(v) +
2
v
{?h(v) ?[(µ +r
f
)v ?c] h
(v)} ? 0
? [?(µ +r
f
+r
d
)v + 2c] h
(v) + 2?h(v) ? 0 (3.60)
Rearranging, this can be put in the following form:
h(v)
h
(v)
?
(µ +r
f
+r
d
)
2?
v ?
c
?
(3.61)
We de?ne functions y(v) and z(v) as follows:
y(v) =
h(v)
h
(v)
(3.62)
z(v) =
(µ +r
f
+r
d
)
2?
v ?
c
?
(3.63)
And we complete the proof by proving the following lemma (which asserts
that y ? z on [0, v
l
]).
Lemma 3.7. With y and z as given in (3.62) and (3.63), y > z on (0, v
l
)
and y = z at v = 0 and v = v
l
.
Proof. The equation in (3.53) can be rearranged as:
?
h(v)
h
(v)
= [(µ +r
f
)v ?c] +
1
2
?
2
v
2
h
(v) (3.64)
29
Note that
y
(v) =
h
(v)
2
?h(v)h
(v)
h
(v)
2
, (3.65)
so we can solve for the quantity
h
(v)
h
(v)
as well:
h
(v)
h
(v)
=
1 ?y
(v)
y(v)
(3.66)
Substituting these into (3.64) and rearranging yields the following:
?
2
v
2
(y
(v) ?1) = ?2?y(v)
2
+ 2[(µ +r
f
)v ?c]y(v) (3.67)
The function z(v) satis?es a similar ODE (this is easily veri?ed by substitu-
tion):
?
2
v
2
_
z
(v) ?
µ +r
f
+r
d
2?
_
= ?2?z(v)
2
+ 2
_
µ +r
f
+r
d
2
v ?c
_
z(v) (3.68)
We have that y(0) = z(0) = ?c/? (to see this, set v = 0 in (3.67)), and that
y(v
l
) = z(v
l
) = ?(1/d)(c/r
d
? v
l
). First we show that y
(v
l
) < z
(v
l
). If we
substitute y(v
l
) into (3.67), we have after simpli?cation
y
(v
l
) = 1 +
r
d
+m
?
?
r
d
?
d. (3.69)
Substituting the value of d in this expression, we can show that y
(v
l
) < z
(v
l
)
30
if and only if
?(µ +r
f
) +? +m <
_
(r
d
+? +m)
2
?4r?. (3.70)
And since µ +r
f
> r
d
, this is true if
?r
d
+? +m <
_
(r
d
+? +m)
2
?4r?. (3.71)
This inequality is true, and can be checked by squaring both sides. Therefore,
we have that y
(v
l
) < z
(v
l
). This means that y > z on the interval (v
l
??, v
l
),
for some ? > 0. The remainder of the proof will be done by contradiction:
Suppose that there exists ˜ v ? (0, v
l
) such that y(˜ v) = z(˜ v) and y > z on
(˜ v, v
l
). If we can show that no such ˜ v exists, the proof will be complete.
Since y(˜ v) = z(˜ v) and y > z on (˜ v, v
l
), we have y
(˜ v) ? z(˜ v). So by
substitution in (3.67) and (3.68), we have
1?
2?
?
2
˜ v
2
y(˜ v)
2
+
2[(µ +r
f
)˜ v ?c]
?
2
˜ v
2
y(˜ v)
?
(µ +r
f
+r
d
)
2?
?
2?
?
2
˜ v
2
z(˜ v)
2
+
2[
1
2
(µ +r
f
+r
d
)˜ v ?c]
?
2
˜ v
2
z(˜ v) (3.72)
Note that the middle terms cancel (as y(˜ v) = z(˜ v)), so substituting the value
of z(˜ v), what remains can be simpli?ed to
1 ?
µ +r
f
+r
d
2?
? ?
µ +r
f
?r
d
?
2
˜ v
_
µ +r
f
+r
d
2?
˜ v ?
c
?
_
(3.73)
The right hand side of this inequality is positive. We therefore have two
31
cases. In the case where
µ+r
f
+r
d
2?
? 1, we directly obtain our contradiction.
In the case when
µ+r
f
+r
d
2?
< 1, the inequality in (3.73) can be written as
˜ v ?
2c(µ +r
f
?r
d
)
?
2
[2? ?(µ +r
f
) ?r
d
] + [(µ +r
f
)
2
?r
2
d
]
(3.74)
If we can show that v
l
is less than that quantity, then ˜ v ? (0, v
l
) cannot
exist. It turns out that if we substitute in the value of v
l
we can show that
˜ v ?
2c(µ+r
f
?r
d
)
?
2
[2??(µ+r
f
)?r
d
]+[(µ+r
f
)
2
?r
2
d
]
is equivalent to the inequality in (3.70), which
was already shown to be true.
Therefore, we have shown that there cannot exist ˜ v ? (0, v
l
) such that
y(˜ v) = z(˜ v) and y > z on (˜ v, v
l
). So y > z on the whole interval (0, v
l
).
The results of this section are summarized in the following theorem.
Theorem 3.8. The constrained minimum probability of lifetime ruin
? ? C
1
(R
+
) ? C
2
(R
+
\ {c/r
d
}) is given by
?(v) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
h(v) if v ? [0, v
l
]
?
_
1 ?
r
d
c
v
_
d
, if v ? (v
l
, c/r
d
)
0 if v > c/r
d
(3.75)
where h solves the di?erential equation speci?ed in (3.50) and (3.51) and
where v
l
is as speci?ed in Corollary 3.4, and where
? = h(v
l
)
_
1 ?
r
d
c
v
l
_
?d
(3.76)
32
The optimal investment strategy ?
?
(v) is given by
?
?
(v) =
_
¸
¸
_
¸
¸
_
v if v ? [0, v
l
]
µ+r
f
?r
d
?
2
1
d?1
_
c
r
d
?v
_
if v ? (v
l
, c/r
d
]
(3.77)
33
4 Numerical Examples
Here we provide a few examples with numerical data to illustrate the results
of section 3.2. We assume the following parameter values:
• r
d
= 0.02; the domestic bank has an interest rate of 2% over in?ation.
• r
f
= 0.035; the foreign bank has an interest rate of 3.5% over in?ation.
• µ = 0.025; the drift of the exchange rate is 2.5%.
• ? = 0.20; the volatility of the exchange rate is 20%.
• c = 1; wealth is consumed at a rate of one unit per year.
• ? = 0.04; constant hazard rate of 4% such that the individual’s ex-
pected future lifetime is 25 years.
For these parameter choices, we have an approximate lending level of v
l
=
14.64. So the individual would invest all of her wealth in the foreign bank
when v ? 14.64, and some amount less than her total wealth when v > 14.64.
Moreover, the wealth level at which she invests only in the domestic bank
is c/r
d
= 50. Figure 4.1 shows the amount invested in the foreign bank for
wealth levels v ? [0, 50], computed using Theorem 3.8.
We can also express the optimal investment strategy in terms of the
fraction of total wealth invested. Figure 4.2 shows the fraction of total wealth
that the individual would invest in the foreign bank for v ? [0, 50].
The function ?(v) given in Theorem 3.8 can be evaluated numerically
for v ? 14.64 using a software ODE solver. After that solution is found,
34
the other piece of ?(v) can be evaluated. For this example, we have the
boundary conditions (as in (3.51)) of h(0) = 1 and
h(14.64)
h
(14.64)
= ?10.36. Solv-
ing numerically in Maple (using dsolve with numeric, method=bvp options)
gives a curve with h(14.64) ? 0.361. This value can then be used to solve for
? in Theorem 3.8, using (3.76). The resulting curve is shown in Figure 4.3.
Figure 4.1: Optimal investment strategy. Here v
l
= 14.64.
35
Figure 4.2: Percentage of total wealth invested.
Figure 4.3: Minimum probability of ruin. The level v
l
= 14.64 is indicated.
36
5 Summary and Conclusion
In this paper we consider the problem of minimizing the probability of lifetime
ruin of an individual investing in a market with foreign and domestic bank
accounts. Our model assumes that the investor is not allowed to borrow,
and that her consumption remains at a constant level. By extending the
work of Bayraktar and Young (2006), we ?nd expressions for the minimum
probability of ruin as well as the optimal investment strategy for any given
wealth level.
Moreover, part of our goal was to present the arguments leading to our
results in a very clear manner. To that end we included a reference of the
main concepts from probability theory and stochastic calculus which were
applied, and attempted to make the steps of each proof clear and justi?ed.
We ?nd that there exists a “lending level” of wealth at which the invest-
ment strategy changes. For wealth below that level, the individual invests all
of her wealth in the foreign bank account. For wealth above the lending level,
the individual instead is able to reduce her risk of ruin by investing a portion
of her wealth into the risk-free domestic bank. Naturally, an individual with
a su?ciently high amount of wealth will have zero risk of lifetime ruin as
long as her consumption is constant.
We do not address the case in which borrowing is allowed or in which
the consumption rate varies with total wealth. Bayraktar and Young (2006)
cover these cases, and their results naturally apply to our model as well. The
assumptions of this paper are in some ways simplistic, and the results could
37
be made more realistic by assuming random interest rates or other (possibly
random) consumption rates. Additionally, the assumption that only two
currencies are tradeable is itself a signi?cant simpli?cation.
Nevertheless, viewing these results in the context of foreign exchange
markets can give insight into the behavior of any investor seeking to minimize
the risk of bankruptcy.
38
6 Appendix: MATLAB and Maple Code
Maple code used to solve the ODE in (3.50) and (3.51):
> sol1 := dsolve(0.04*h(v)-(0.06*v-1)*(diff(h(v),v))
- 0.02*v^2*(diff(diff(h(v),v),v)) = 0,
h(0.01) = 1,
h(14.644660940672622)/(D(h))(14.644660940672622) = -10.355339059327376],
numeric, method = bvp, abserr = 0.001);
sol1(14.6446);
plots[odeplot](sol1, 0.001 .. 15, color = red);
MATLAB function for computing the lending level v
l
as in Corollary 3.4:
function[vl] = LendingLevel(rd,rf,mu,sigma,c,lambda)
m = 0.5*(mu+rf-rd)^2/sigma^2;
d = ( (rd+lambda+m) + sqrt((rd+lambda+m)^2-4*rd*lambda) )/(2*rd);
x = (mu+rf-rd)/(sigma^2*(d-1));
vl = (x*c)/((1+x)*rd);
MATLAB function for computing the function ?
?
(v) as in Theorem 3.8:
function[pi] = OptimalStrategyPlot(rd,rf,mu,sigma,c,lambda)
pi = zeros(1,251);
v = 0:c/(rd*250):c/rd;
m = 0.5*(mu+rf-rd)^2/sigma^2;
d = ( (rd+lambda+m) + sqrt((rd+lambda+m)^2-4*rd*lambda) )/(2*rd);
vl = LendingLevel(rd,rf,mu,sigma,c,lambda);
for i = 1:251
if v(i) < vl,
pi(i) = v(i);
else
pi(i) = (mu+rf-rd)/(sigma^2*(d-1)) * (c/rd - v(i));
end
end
39
MATLAB function used to compute the function ?(v) as in Theorem
3.8. Here, HVECEX was a global variable consisting of the data imported
from Maple regarding the ?rst half of the function (solved by the Maple code
shown above).
function[h] = h_example_plot()
global HVECEX
h_first = HVECEX;
lambda = 0.04; mu = 0.025; r_f = 0.035; r_d = 0.02; c = 1; sigma = 0.2;
m = 0.5*((mu+r_f-r_d)/sigma)^2;
d = (1/(2*r_d))*( (r_d+lambda+m) + sqrt((r_d+lambda+m)^2-4*r_d*lambda) );
vl = LendingLevel(r_d,r_f,mu,sigma,c,lambda);
v_first = (vl/100:vl/100:vl)’;
v_rest = (vl+vl/100:vl/100:50)’;
beta = h_first(length(h_first))*(1-r_d*vl/c)^(-d);
h_rest = beta*(1-r_d*v_rest/c).^d;
v = [ v_first ; v_rest ];
h = [ h_first ; h_rest ];
40
References
[1] Erhan Bayraktar and Virginia R. Young. Minimizing the Probability
of Lifetime Ruin under Borrowing Constraints. Insurance: Mathematics
and Economics, 41(1), 2006.
[2] Bj¨ ork, Tomas. Arbitrage Theory in Continuous Time. Oxford University
Press Inc, 2004.
[3] V. C. Pestien and W. D. Suddherth. Continuous-time red and black:
How to control a di?usion to a goal. Mathematics of Operations Research,
10(4):599–611, 1985.
[4] Philip E. Protter. Stochastic Integration and Di?erential Equations.
Springer, 2004.
[5] Steven E Shreve. Stochastic Calculus for Finance II. Springer, 2004.
[6] V. R. Young. Optimal investment strategy to minimize the probability
of lifetime ruin. North American Actuarial Journal, 8(4):105–126, 2004.
41
doc_157134530.pdf