Page 1 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 11
December 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 221
Determination of Optimal Salary via Defined Benefit
Pension Plan with Early Retirement
Adaji1, M. O; Onah2, E. S; Kimbir3, A. R. & Aboiyar4, T.
1Department of Mathematics/Statistics, Benue Polytechnic, Ugbokolo
2,3,4Department of Mathematics/Statistics/Computer Science,
University of Agriculture, Makurdi Nigeria
Abstract
The paper seeks to apply the mathematical model
formulated by Adaji, M. O. et al (2015) to
determine the financial value of the retirement
benefit Sf by applying Smooth Pasting condition.
We examined the smooth pasting condition
(tangency condition) along the optimal point at
the boundary for an American put by considering
the gradient, ∂V
∂S more closely and found that as
long as V(S) coincides with the straight line, K −
S gradient equals −1. We used Microsoft excel
spread sheet facility to perform the individual
computation for last 25 years of service for
university Senior Lecturers. The optimal salary
Sf was determined by drawing a vertical line from
the tangent perpendicularly to the horizontal axis
(salary axis). We recommend the appliction of this
model to individuals as well as cooperate
organisations so as maximise the expected
retirement benefit. Early retirement alternate
should be encouraged so that the teeming
population of unemployed youths can take up the
vacancies created as a result of early retirement.
Keyword: Smooth Pasting
Introduction
The smooth pasting property (condition),
states that the value function must be
continuously differentiable everywhere, and
yields conditions, which uniquely determine
the optimal stopping region. Art of smooth
pasting includes a heuristic justification for
the differentiability of value functions at
optimal stopping thresholds. In pure stopping
problems, “smoothness” requires (and means)
that the value function is once differentiable,
and is known as the smooth pasting condition.
Optimal stopping problems are linked to free
boundary problems. This connection was
discovered by McKean (1965) and it was
formulated as a free boundary problem that
can be solved, an extra condition is needed.
The principle of smooth pasting provides this
condition. It was first adopted by
Mikhalevich (1958), and was studied in
greater depth by Shiryaev (2006).
It was Bensoussan (1984), and later Karatzas
(1988), that first used no-arbitrage methods to
show that the price of the American put is the
solution to an optimal stopping problem. This
work followed that of McKean (1965), who
was the first to derive a free boundary
problem for the ‘discounted’ American call
with gain function
Φ(S) = e
−rτ(S − K)
+
Page 2 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 11
December 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 222
2.0 Assumptions of the Model
a) The model satisfies smooth pasting condition
b) A member of the plan would retire when he/she maximizes the benefits of retirement among
all possible dates (stopping times) to retire.
c) Optimal stopping problem with a value function V(St) = sup
τ≤T
ESe
−τrVτ
(K − St
) satisfies
geometric Brownian motion, dSt = μStdt + σStdWt
d) The infinitesimal generator of the (strong) Markov process S is given by
LSV = rS ∂
∂S +
σ
2
2
S
2
∂
2
∂S
2
.
e) Standard Markovian arguments suggest that V from assumption ( g) solves the following free
boundary problem of parabolic type
LSV = rV
2.1 Parameters and Variables of the Model
The following parameters (functions) and variables are used in this research work:
V = V(S,t ) = V(St
) is the financial value of retirement benefit in the time interval 0 < t ≤ T;
S = St = salary at time, t;
Sf = Optimal salary;
K= strike salary;
S − K = Payoff for the call option (American call option for a fixed K and any given salary, S)
representing an employer’s option
K − S = Payoff for the put option (American put option for a fixed K and any given salary, S)
representing an employee’s option
(S − K)
+ = max
S
(S − K, 0) assumed to occur at the optimal boundary (call option)
(K − S)
+ = min
S
(K − S, 0) assumed to occur at the optimal boundary (put option)
V(Sf)= Optimal financial value of retirement benefit (is also the same as the optimal retirement
benefits) with respect to salary
t = Time (in year) spent with the pension plan
r =the salary growth rate or Accrual rate
μ =(r − δ) is the expected return of the salary (asset)
δ =annual dividend yield δ ≥ 0 of the asset (salary) (when δ = 0, then μ = r)
σ = The volatility of the salary (also the standard deviation)
T =Worker's expected retirement time (maturity or expiry time) in years
τ= stopping time
τf = optimal stopping time
Page 3 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 11
December 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 223
C, D = continuation set and stopping set respectively
Wt =geometric Brownian process, (Disturbance factor)
k1 , k2= arbitrary constants,
w+ ,w−= respective positive and negative roots of an auxiliary equation
Rd = d − dimensionalEuclidean space
LS = infinitesimal operator of S
Vτ = value function at stopping time, τ,
Φτ = gain function at stopping time, τ,
ES =expectation with respect to S
(Ω,F, P) =probability space
Ft =filtration
Definition 1. Let Ω be some space of functions from [0, ∞] into R. The shift operator θt
: Ω → Ω
defined by
(θt
(ω)) (s) = ω(t + s)
for ωεΩ (Typically, we regard ωεΩ as a sample path of some stochastic process.) Suppose that S =
(St)t≥0 is a stochastic process on the probability space Ω,F, P the following useful results are
given without proof.
Remark: The Shift Operator
The shift operator is useful in defining the (strong) Markov property.
The Measure PS
Let W = (Wt)t≥0 be a standard Brownian motion under the measure P. Thus each Wt
is a random
variable defined on a probability space (Ω,F, P), and W0 = 0 under P. Now define St = S + Wt
,
for all 0 ≤ t < ∞. Then St
is a random variable on the same probability space. Moreover, we see
that S0 = S under P.
Definition 2. If a process S = (St)t≥0 is equipped with the filtration((Ft
)t≥0, with F =
σ(⋃t≥0 Ft
), then S has the (strong) Markov property if any of the following three equivalent
conditions hold:
ES
(f(Xτ+h
)⁄Ft
) = Ex
(f(Sτ+h
)| Sτ) (1)
ES
(Sτ+h
)|Fτ
) = ESτ
(f(Sh)) (2)
Ex(Y ∘ θτ
|Fτ
) = ESτ
(Y) (3)
for all S, all stopping times τ, all h > 0, any bounded Borel-measurable function f, and any
(bounded) F-measurable random variable Y.
