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Journal of Mathematical Finance, 2019, 9, 494-521
http://www.scirp.org/journal/jmf
ISSN Online: 2162-2442
ISSN Print: 2162-2434
DOI:
10.4236/jmf.2019.93026 Aug. 22, 2019 494 Journal of Mathematical Finance
Fast Fourier Transform Based Computation of
American Options under Economic Recession
Induced Volatility Uncertainty
Philip Ajibola Bankole, Olabisi O. Ugbebor
Department of Mathematics, University of Ibadan, Ibadan, Nigeria
Abstract
The menace of Economic recession to uncertainty in the payoff of invest-
ments and standard of living cannot be over emphasized. This paper presents
fast Fourier transform method for the valuation of American style options
under the exposure of Economic recession. A multi-factor affine Exponential
jump model with Recession induced Stochastic volatility and Intensity, which
is a
partial Integro-Differential Equation (
PIDE) is presented. We show how
to determine the characteristic function of the model via generating function.
A close form characteristic formula for a financial claim satisfying the PIDE
in pricing both European style and American style options in Fourier based
transform was done. A numerical based Fourier transform algorithm FFT for
European call option valuation was extended to the model under study. The
algorithm was further extended to American call options valuation by adding
premium price to the European call options price. Numerical result was pre-
sented to reflect the effect of economic recession induced volatility on options
prices and that of the usual volatility. The result shows some significant vicis-
situdes in the options values in the two states of the Economy. The result
output indicated that the model is effective and reliable compared to other
existing models. The fast Fourier transform (FFT) approach gave better op-
tion value and compared to both Black-Scholes Merton (BSM) and American
Option solver as shown in the table under numerical result section. We used
Nigerian Flourmill Stock (NFS) prices for data calibration and reported the
stock performance during the first Nigerian recession and recovery year in
the Appendix section.
Keywords
Economic Recession, Volatility Change, Characteristic Function in Affine
Form, Partial Integro-Differential Equation PIDE, American Dividend Paying
Options, Fourier Transform, Fast Fourier Transform (FFT)
How
to
cite
this
paper:
Bankole, P.A. and
Ugbebor
, O.O. (2019) Fast Fourier Trans-
form
Based Computation of American
Options
under Economic Recession In-
duced
Volatility Uncertainty.
Journal
of
Mathematical
Finance
,
9
, 494-521.
https://doi.org/10.4236/jmf.2019.93026
Received:
May 16, 2019
Accepted:
August 19, 2019
Published:
August 22, 2019
Copyright
© 2019 by author(s) and
Scientific
Research Publishing Inc.
This
work is licensed under the Creative
Commons
Attribution International
License
(CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 495 Journal of Mathematical Finance
1. Introduction
Financial mathematics encompasses many relevant areas in which option valua-
tion is one of them and connects many different fields of study in mathematics.
It relies on the application of various mathematical concepts and tools for sur-
vival. Such concepts are Probability theory, optimization theory, numerical
analysis, partial differential equations, Ordinary Differential Equations, integral
representations, transformation and many more. Among various transformation
techniques obtainable in Mathematics such as Hankel transform, Mellin trans-
form, Hilbert transform, Laplace transform, z-transform, and Fourier transform,
this paper applies fast Fourier transform based computation of Options focusing
on uncertainty in the options value induced by Economic Recession.
Instability of Economy has created a wide disparity between Economic activi-
ties and the values of Assets found in the economy. It is very obvious that Assets’
values are been affected by the state of the Economy. The effect of Economic re-
cession to the falling of the payoff of investments and standard of living is
enormous and should not be taken for granted. This is evident as the data from
Nigeria Stock Exchange (NSE) indices prior to recession outbreak, during reces-
sion and after recession revealed huge lower performance of Nigeria stocks dur-
ing the Nigeria economy recession of 2016. The National Bureau of Economic
Research (NBER) has been recognised in the United States as an official body
saddled with the responsibility of providing an accurate information on eco-
nomic recession dates and business cycle publishing since 1929. The definition
of recession according to NBER cited in [1] is a significant decline in economic
activity spread across the economy, lasting more than a few months, normally
visible in production, employment, real income, and other indicators. Some
other definitions of economic recession obtainable in financial press emphasised
that recession begins with two consecutive quarters of decline in Gross Domestic
Product (GDP). NBER ponders on GDP as the single best measure of total eco-
nomic activity and reflects on the GDP definition to be too narrow in measuring
economic activity and to reliably date economic recessions. There exists scenario
whereby recession may not include two consecutive quarters of negative growth
such as United States recession of 2001 but according to Thomas Hsu [1], de-
clines in GDP are closely correlated to recession periods. Nigerian economic re-
cession outbreak in year 2016 was based on decline in GDP and rise in some
other macroeconomic indicators such as high inflation rate and unemployment
rate after two consecutive quarters. Some measures were invented by axiomatic
methods: for example, as cited in [2]. Probability Measure was invented by A.N.
Kolmogoroff in 1933, Possibility Measure by L.A. Zadeh in 1978 and Uncertain
Measure (B. Liu) in 2007 (see [2] [3] [4] [5] [6]). Probability theory has been
seen as the vehicle for dealing with uncertainty in finance and insurance risks.
Probability theory as a mathematical theory is useful in describing and analysing
situations where randomness or uncertainty are present [7]. Definition of
un-
certainty
had been given by different authors in various scenarios. According to
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 496 Journal of Mathematical Finance
[8] (Dungan
et al.
, 2002), Uncertainty is a multi-faceted characterization about
data or predictions made from data that may include several concepts including
error, accuracy, validity, quality, noise and confidence and reliability. Uncer-
tainty connotes difficulty in predicting the outcome of an event due to inexact
knowledge of information concerning the outcome of the event in a particular
time. Uncertainty appears to endogenously increase during recessions, as lower
economic growth induces greater micro and macro uncertainty [9]. It was
stressed in [10] that investment growth experiences downward trend during the
period of recession.
The following authors [11]-[21] have considered uncertainty with respect to
models. The aspect of uncertainty in terms of economic recession was not dis-
cussed in their formulations. It was recognized that prices of insurance and fi-
nancial risks are not solely determined by mathematics or economic theories but
are also impacted by activities taking place in financial markets. During the pe-
riod of economic crisis, prices of assets tend to change stochastically in the fi-
nancial market (especially while dealing with stocks) since there is much ten-
dency for fluctuations of asset prices due to fluctuation in the underlying term
structures typically, the volatility variation. The study of volatility of stock prices
is very essential due to its effect on the prices of stocks in the financial market.
Pfante, O. and Bertschinger, N. (2018), [21] consider the uncertainty of volatility
estimates from Heston Greeks. It was stressed in [22] that volatility has been the
cause for several Statistical properties of observed stock prices processes. It was
further emphasized in [22] that volatility clustering is commonly accompanied
by other large fluctuations and similar for small changes. As volatility is not ob-
served it has to be estimated from market prices,
i.e.
, as the implied volatility
from option prices. There is strong tendency that predictions of the volatility of
asset prices induced by economic recession factor among other market risk fac-
tors may not really be accurate. This becomes a challenge for investors that do
not like taking risk, that is, the risk averters. There are some existing methods of
calculating the volatility of asset prices in financial market especially while deal-
ing with instantaneous prices (
i.e.
the real time prices) of assets in the financial
market. Among the methods known to us are 1) Standard deviation approach, 2)
using Historical value of the volatility and 3) implied volatility. The Standard
deviation approach tells us how tightly the stock price is grouped around the
mean or moving average. When the prices are spread apart, the implication is
that one has a relatively large standard deviation but if the prices are relatively
closed to each other or bunched together, this connotes that the standard devia-
tion is negligible.
The causal effect of volatility on stock prices is discussed briefly as follows.
The stock market prices rise when volatility decreases and increase in volatility
causes fall in stock market value (prices). Increase in volatility leads to increase
in market risk but decrease in returns of the market. In the case of options on
stocks market, the causal effect of volatility changes on options depends on the
type. For example, increase in volatility leads to increase in call options value but
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 497 Journal of Mathematical Finance
for put options, increase in volatility leads to decline in the payoff of the put op-
tions vice versa. An intuition behind the introduction of recession induced vola-
tility uncertainty is revealed by huge volatility fluctuations during the period of
Economic recession compared to the period of normalcy (recession-free). This
in turn affects investors’ prediction in the market. Since Economic recession in-
duces a high level of uncertainty on investors activities to include decision mak-
ing and the payoffs of stocks in general, we then proposes giving a close atten-
tion to volatility changes in relation to economic recession and financial models
for options valuation. Nigeria economic recession outbreak in 2016 and its ef-
fects on the payoffs uncertainty of Nigeria Stocks Exchange (NSE) among other
investments is among the motivating factors for proposing economic recession
induced volatility formulation in Options pricing. A good knowledge of the be-
havior or the level of uncertainty in economic recession induced volatility by in-
vestor’s will help in decision making during recession period.
The rest part of the paper is organized as follows: Preliminaries on financial
modeling is discussed in Section 2 in addition to some other subsections to in-
clude Fourier transform, uncertainty and uncertain measure. Section 3 deals
with accounting for jumps in the asset price linked with recession, Section 4
shows the model formulation. Numerical Fourier based transform of options is
presented in Section 5 while Section 6 is conclusion.
2. Preliminaries on Financial Modeling
2.1. Some Literature Review on Uncertainty
The theory of uncertainty in financial market could be traced back to the re-
search work of [2] [3] [4]. Without uncertainty, the probabilities of risky events
are known and frictionless markets can precisely price contracts contingent on
risky events broadly. The volatility of the stock market or GDP is often used as a
measure of uncertainty because when a data series becomes more volatile it is
harder to forecast. Other common measures of uncertainty include forecaster
disagreement, mentions of “uncertainty” in news, and the dispersion of produc-
tivity shocks to firms [9]. It was further stressed by Nicholas Bloom [9] that the
volatility of stock markets, bond markets, exchange rates, and GDP growth all
rise suddenly during economy recessions. In Philip
et al
. [10], it was stressed that
investment under the exposure of economic recession tends to have negative
growth. This in turn poses a lot of challenges on investors in a financial or mon-
ey market as investors’ decision in the market also depends on the daily infor-
mation on the state of the economy.
In the history of options pricing in financial market, the famous Black -
Scholes model [23] with the assumption of constant rate of return and volatility
have been criticized by many researchers due to the fact that it does not reflect
the stochastic nature of financial markets wholly. As a result of this deficiency in
the B-S model, some other realistic models have been formulated and come to
stay in the sense that the models shows better random movement of financial
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 498 Journal of Mathematical Finance
market. Example of such models are: Double exponential jump model, Regime
switching model, Stochastic volatility models of Heston [24], etc. In [25], it was
asserted that regime-switching behaviour captures changing preferences and be-
liefs of investors concerning asset prices as the state of financial market
changes. Regime-Switching formulation in valuation of financial market was
known to be introduced by J. Hamilton [26]. Since then, there have been some
development on considering his suggestion. A closed form formula for the
valuation of European call option in a two-state economy under the assump-
tion of no-arbitrage was presented in [27] by Guo, X. European and Ameri-
can-style derivatives pricing was presented in [28] by Bollen using lat-
tice-based approach.
Fourier transform approach of pricing options especially European-style op-
tions have been looked into by various authors. The one with great popularity
known to us among others was that of Carr and Madan (1999) [29] in pricing
European call option. Some other authors have also shown other mathematical
formulation of Fourier transform of European options such as in [30]. As far as
we know, the issue of economy recession factor in option pricing has not been
incorporated into fast Fourier Transform of both European and American-style
options. Very few authors have worked on Fourier Transform of American-style
options compare to European style options. The notable one known to us as at
the time of this research was the work of Oleksandr, Z. in (2010) [31].
Geske-Johnson scheme with Richardson Extrapolation was adopted to numeri-
cally extend fast Fourier Transform algorithm to American-style security, which
features a continum of potential exercise times up to expiration.
Our contribution to the existing literature on options valuation is highlighted
in sequel:
1) We incorporate economic recession induces volatility uncertainty into ex-
ponential jump model with stochastic volatility and intensity. The economic re-
cession induce volatility is assumed to be an uncertain variable.
2) Derivation of Characteristic function of the affine model with recession
uncertainty effect is presented.
3) The Fourier transform of the affine model is performed.
4) The Characteristic function of the affine model is extended to fast Fourier
transform algorithm of Carr & Madam [29] in pricing European options. It was
extended to pricing American style options by adding
time premium
such that
the limit value tends to zero at the expiry time.
5) Volatility Surface of the affine model based on the recession induced un-
certainty is presented.
2.2. Uncertainty
Definition 1: [5] [6]. An uncertain variable
ξ
is a
measurable
function from
an uncertainty space to the set of real numbers.
Definition 2: O.O. Ugbebor [6]. Let
Γ
be a non-empty set. A collection
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10.4236/jmf.2019.93026 499 Journal of Mathematical Finance
consisting of subsets of
Γ
is called an algebra over
Γ
if the following three
conditions hold:
Γ∈
;
if
∧∈
, then
c
∧∈
; and
if
12
, ,,
n
∧∧ ∧∈
, then
1
n
i
i=
∧∈
Under the notion of countable union, if
12
, ,,
n
∧∧ ∧∈
such that
1
n
i
i=
∧∈
the collection
becomes a
σ
- algebra over
Γ
.
Example 1: Suppose
∧
is a proper subset of
Γ
. Then the set
{ }
,, ,
c
∅∧∧ Γ
is a
σ
-algebra over
Γ
.
Remark 1: The collection
Γ
is closed under countable union, countable in-
tersection, difference and limit.
Definition 3: [5] [6] Let
Γ≠∅
be a non-empty set and suppose
is a
σ
-algebra over
Γ
. Then
( )
Γ,
is called a
measurable space
and any member
in
is a measurable set.
Definition 4: [6] A function
( )
:Γ,R
ξ
= →
is said to be
measurable
if
( ) ( )
{ }
1
|
ξ γ ξγ
−
= ∈Γ ∈ ∈
(1)
where
( )
Γ,
is a measurable space and
( )
R
is a Borel set of real numbers.
2.3. Uncertain Measure
B. Liu, [4] [5] highlighted the following three axioms for an uncertain process:
Axiom 1: (Normality Axiom);
{ }
Γ1=
for the universal set
Γ
.
Axiom 2: (Duality Axiom):
( )
()
1
c
∧+ ∧ =
for any even
∧
.
Axiom 3: (Subadditivity Axiom);
{ }
{ }
1
1ii
i
i
∞∞
=
=
∧≤ ∧
∑
for every
countable sequence of events
12
,,∧∧
.
Remark 2:
For some application of B. Liu (2007)’s uncertainty theory, the reader should
see the research work of the following authors in [12] [13] [14] as cited earlier.
2.4. Fourier Transform
Fourier transform in mathematical finance was firstly used to determine the dis-
tribution of an underlying asset price under the stochastic volatility model by the
inversion method, Stein
et al.
(1991) [18]. The following definitions and condi-
tions are necessary for the understanding of Fourier transform technique.
Definition 5:
Let
( )
fx
be a Lebesgue-measurable function of
xR∈
, then
the
2
L
-norm of f is defined by
( )
()
1
22
df fx x
∞
−∞
=∫
(2)
where
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 500 Journal of Mathematical Finance
{ }
2
:.L ff= <∞
(3)
Equivalently,
( )
fx
is a piecewise integrable real function over the entire real
line satisfying the condition
( )
d.fx x
∞
−∞
<∞
∫
(4)
For the Fourier transform and inverse Fourier transform of a function to exist,
it is necessary that the function is absolutely integrable over
R
with finite value
as defined in (3) above. An absolutely integrable functions on a given interval
( )
,ab
are said to be on the space of
()
1
,L ab
. The notion of Fourier transform
can be extended to the square integrable functions.
Definition 6:
A square integrable functions is defined by
( )
2d.
fx x
∞
−∞ <∞
∫
(5)
The space of square integrable functions is represented by
( )
2
LR
over the
real line or simply an interval
( )
,ab
.
Definition 7:
Let
:fR R→
be a real-value function for
xR∈
, then the
Fourier transform of f is defined as
()
( )
( ) ( )
ˆ
; e d, ,
ix
fxw fw fx x R
ω
ω
∞−
−∞
= = ∈
∫
(6)
where
1
i= −
and
ω
is a parameter.
We can recover
( )
fx
from
( )
f
ω
by the inverse Fourier transform.
Definition 8:
The inverse Fourier transform of the real valued function
( )
fx
is defined by
( )
( )
( ) ( )
1
1
ˆ ˆ
; e d,
2π
ix
f x fx f x R
ω
ω ωω
∞
−
−∞
= = ∈
∫
(7)
which belongs to either
1
L
or
2
L
- spaces.
3. Accounting for Jumps in the Asset Prices Linked with
Recession
The major market parameters which most financial modeler will wish to con-
sider while formulating a financial model relies on price, interest rate, dividend
rate, volatility and time. In an economy threatened by recession, asset price
tends to experience lots of fluctuation. Asset price especially stocks can never be
stable. It is very necessary to consider interest rate and volatility to be stochastic
in nature. Even if one considers interest rate to be constant, in reality volatility
cannot be constant. The Stock volatility is seen as a measure of the uncertainty
on the payoff or returns of the stock which investors look up to for decision
making and taking. During economic recession, the rate of volatility variation is
considered higher than the state of normalcy of an economy. As a result an
economy recession factor is invoked in this paper as we shall see later.
Jumps in the Underlying Stock Price
Consider the dynamics of a stock price
( )
St
given by
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10.4236/jmf.2019.93026 501 Journal of Mathematical Finance
( )
( ) ( )
( )
( ) ( )
( )
( )
dd d e 1d
s
St r q tm t t W t Nt
St
ν
λσ
= −− + + −
(8)
where
r
is the interest rate,
q
is the dividend rate,
( )
Nt
is a Poisson process
with stochastic intensity
( )
t
λ
,
m
is an average jump amplitude given as
( )
ˆ
e1
P
mE
ν
= −
where jump size
ν
is a random variable. The equation consists
of
diffusion process
and
jump process
. There are two sources of fluctuations in
the stock price which we classified as “
changes in economic state
” and “
changes
due to supply and demand factors
”. The inclusion of jump in the model is to ca-
ter for the arrival of useful information into the market that will have an abnor-
mal consequence on the stock price. Among the factors that are responsible for
jumps in the stock price highlighted by Matthew, S.M in [32], we added eco-
nomic recession factor. The factors are grouped into:
Firm specific jumps: Caused by news inflow to the market on individual
firm’s profit/loss report.
Industry/sector specific jumps: Caused by news that can affect specific
company or industry such as news on sudden declaration of
Holiday
.
Market specific jumps: This jumps are caused by news such as oil prices,
interest rates, credit spreads etc. that affect the market.
Economic recession jumps: This is caused by the news inflow to the market
on the state of the economy to include deep economy crisis with recession
probability indicators.
In our model formulation of stock price to capture economy recession in-
duced volatility uncertainty on stock in the next section, we further grouped the
above highlighted factors “(a)-(c)” as
other sources
with volatility
*
t
σ
while the
factor (d) Economic Recession induced volatility source as
rec
t
σ
.
Poisson process
( )
Nt
has probability density function given as
( )
( )
e , 0, 1, 2,
!
t
t
t
PN
ν
λ
λ
νν
ν
−
= = =
(9)
where
λ
is the intensity. In line with Matthew [24], if information on eco-
nomic recession or other source of panic enters the financial market (stock)
causing an instantaneous jump in the stock price such that the price jump
change from
( ) ( )
t
St St
ν
→
where
t
ν
is taking as the absolute magnitude
jump. Then the relative change in price is given by
( )
( )
( )
()
( )
()
( )
( )
1
d1.
t
t
t
St
St St St
St St St
ν
νν
−
−
= = = −
(10)
4. The Model Formulation
Let
( )
ξγ
be an economy recession induced parameter variation define on an
uncertain space
( )
Γ,,
, with
{ }
12
,
γ γγ
=
such that
{ } { }
12
,.
γα γ β
= =
Then
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( )
1
2
0, if
1, if
γγ
ξγ γγ
=
==
(11)
is an uncertain variable. We have that
{ } ( )
{ }
{ }
1
0 |0
ξ γξγ γ α
= = = = =
and
{ } ( )
{ }
{ }
2
1 |1
ξ γξγ γ β
= = = = =
such that
{ } { }
12
1
γ γ αβ
+ ≡+=
.
Example 2: Let the above economic recession induce parameter variation
( )
ξγ
defined on an uncertain space
( )
Γ,,
, with
{ }
12
,
γ γγ
=
be defined
such that
{ } { }
12
0.32, 0.68
γγ
= =
satisfying Equation (11), then
{ } ( )
{ }
{ }
1
0 | 0 0.32
ξ γξγ γ
= = = = =
and
{ } ( )
{ }
{ }
2
1 | 1 0.68
ξ γξγ γ
= = = = =
Let
{}{ }
()
ˆ
Ω,, ,
mket eco
tt
⊆
be a filtered probability space where
mket
t
and
eco
t
describes the filtration in the market and the economy where market
activities takes place, respectively. The
mket
t
and
eco
t
are the market infor-
mation and economy information available up to a time
(
]
0,tT∈
respectively.
Definition 9: [33] An uncertain random variable is a measurable function
p
R
ξ
∈
(resp.
pm
R
×
) from an uncertainty probability space
()
,,PΓ×Ω ⊗ ×
to the set in
p
R
(resp.
pm
R
×
), for any Borel set
p
AR∈
(resp,
pm
R
×
) the given set
{ } ( ) ( )
{ }
, :,AA
ξ γω ξ γω
∈ = ∈Γ×Ω ∈ ∈ ∈ ⊗
.
The expected value of the uncertain random variable
ξ
is defined as
[ ] [ ]
{ } ( ) { } ( )
00
dd dd
p
E EE
r rP r rP
ξξ
ξωξω
∞∞
ΩΩ
=
=≥ −≤
∫∫ ∫∫
(12)
where
p
E
and
E
represent the expected values under the uncertainty space
and the probability space, respectively.
The above definition connects both the notion of probability and uncertainty
such that the random variable is defined from a probability space to uncertainty
space. It makes sense to introduce the notion of uncertainty to financial models
especially while pricing during economy recession or strong financial crisis in
the market. The works of the authors cited in this paper to have contributed to
parameter uncertainty in financial models did not consider uncertainty with re-
spect to economic recession. We extended the notion of uncertainty to the term
structure of stochastic volatility to include economy recession.
Let an asset
X
be described as a two-state regime switching process which is
free to jump between the states defined as
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1, if the economy is in a state of Expansion;
2, if the economy is in a state of Recession.
t
X
=
(13)
Suppose further that there is room for transition between the two states which
evolves as a Poisson process
( )
( )
*
exp , , 1, 2
lm lm
Prob t t t l m
λ
>= − =
(14)
where
lm
λ
is the rate of transition from state
l
to state
m
and the times spent in
state
l
before transiting to state
m
is
*
lm
t
.
Assumption 1: Assume there are two sources of volatility on the asset,
t
X
, i.e
volatility
t
rec
X
σ
and
s
othersources
X
σ
representing market volatility triggered by
economy recession and other sources (usual) respectively such that
s
market rec usual
X ts
σ σσ
= +
, but
rec usual
ts
σσ
≠
(15)
Assumption 2: It is assumed that an economy cannot be in the two states at
the same time. Therefore, when pricing in the financial (stock) market, the
choice of the volatility parameter depends on the state of the economy. For ex-
ample, if the stock is under the exposure of economy recession, then market vo-
latility of the stock
market rec othersources
ts
σ σσ
= +
since recession will have an influ-
ence on the market but if there is no recession, then market volatility
*
market
ts
σσ
=
usual volatility, here
ts
=
, since
0
rec
s
σ
=
.
Suppose further that the asset
X
price is defined on a filtered probability space
( )
ˆ
, ,,
mket rec
tt
Ω
such that the market filtration is generated by the combina-
tion of Wiener process and jump process at a given time,
[ ]
0,tT∈
and
ˆ
P
is
taken as a risk-neutral probability measure. The dynamics of the underlying
stock price
( )
St
(not necessarily a stock) is given by
() ( )
( )
( ) ( ) ( )
()
() ( )( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
()
()() ( )
0
*
0
0
d d d e1 d , 0 0
d d d ,0 0
d d d ,0 0
s
rec
St r q tmSt t t W t St Nt S S
t t t tWt
t t t t Wt
ν
σ σσ
λ λλ
λσ
σ κβ β σ ξσ σ σ
λ κθλ ξ λ λ λ
= −− + + − = >
= +− + =>
=−+ =>
(16)
where
r
is the interest rate,
q
is the dividend rate,
( )
Nt
is a Poisson process
with stochastic intensity
( )
t
λ
,
m
is an average jump amplitude given as
( )
ˆ
e1
P
mE
ν
= −
where jump size
ν
is a random variable. The parameters
0
σ
ξ
>
and
0
λ
ξ
>
are constants with mean - reverting rates
σ
κ
and
λ
κ
which are also taken to be positive constants. The stock market stochastic vola-
tility
( )
*
*
, if the economy is in recession;
, if economy is recession free.
rec
t
σσ
σσ
+
=
(17)
where
rec
σ
is economy recession induced volatility in the stock market due to
the information flow from the Recessed Economy into the stock market while
*
σ
is the volatility in the market from other sources. Furthermore, the con-
stants
*
β
and
rec
β
are respectively regarded as the usual long-term volatility,
long-term volatility influenced by recession during the period of economy reces-
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 504 Journal of Mathematical Finance
sion and
θ
is the intensity constant. The
rec
β
as specified above is to account
for the possibility of economy recession source of long-term volatility in addi-
tion to the usual long-term volatility that may arise during the life-span of the
quoted option on the stock. Also,
( )
s
Wt
and
( )
Wt
σ
are Wiener processes but
( )
Wt
λ
evolve independently of the correlated Wiener process
( )
s
Wt
and
( )
Wt
σ
in the above model. The correlation of
()
s
Wt
and
( )
Wt
σ
is such that
( ) ( )
,d
s
WtW t t
σ
ρ
=
with
ρ
is less than 1. As we take
m
to be an average
jump amplitude since the stock price is expected to jump either upward or
downward. Then setting
( )
1 e d1
11
ud
pq
mf
ν
νν
ξξ
∞
−∞
= + −≡ −
−+
∫
(18)
where
p
is the probability of upward move jump and
q
, the downward move
jump such that
1pq+=
. In [16], the mean positive jump and negative jumps
were given as
1
u
ξ
and
1
d
ξ
, respectively. An analysis on double exponential
jump model with stochastic volatility and stochastic intensity was done by Jiex-
iang Huang et al in [34] and solved. The major distinction between our formula-
tion here and that of [34] is the incorporation of economic recession induced
uncertainty into the term structure of the volatility of the model. Consequently,
we propose the model in Equation (16) to be referred to as an “
Uncertain Affine
Exponential jump model with Recession induced Stochastic volatility and sto-
chastic Intensity
”.
Remarks 3:
The stochastic volatility
( )
t
x
t
σ
of the stock has been formulated to capture
recession induced volatility.
4.1. Determination of Characteristics Function
The notion of characteristics function is indispensable in the study of random
variables in the context of jump diffusion processes. It is a viable tool in Fourier
transformation of options pricing. It is worth noting that the distribution func-
tion of jump diffusion processes in closed form may not be readily available or
known but the characteristics function is explicitly known. However, the cha-
racteristics function of some certain stochastic processes or model may not be
readily available in closed form but it can be determined.
Definition 10: The characteristics function of a random variable
X
is a func-
tion
( )
( )
( )
e e d, ,
iX ix
XX
E fx x R
ωω
ϕω ω
∞
−∞
= = ∈
∫
(19)
where
( )
X
fx
is the probability density function of the random variable
X
.
This definition almost coincide with the definition of inverse Fourier Trans-
form in (6) with the exception of
1
2π
. For a Poisson process say
t
N
with
parameter
λ
, the characteristics function is given by
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 505 Journal of Mathematical Finance
()
()
()
{}
e exp e 1
t
iN i
N
Et
ωω
ϕω λ
≡= −
(20)
There is relationship between Moment-generating function and characteris-
tics function. Moment generating function say,
M
of a real valued random varia-
ble is very useful in the study of probability theory as an alternative specification
of the probability distribution of the random variable
( )
Xt
.
Definition 11: Moment-generating function of a random variable,
X
, is de-
fined as
( )
( )
e, ,
tX
x
Mt E tR= ∈
(21)
if the expectation exists.
Example 3: For a Poisson
()
Pois
λ
distribution, the moment-generating
function
( )
( )
e1
e
t
x
Mt
λ
−
=
and its characteristic function
( )
( )
e1
e
it
t
λ
ϕ
−
=
.
For a continuous probability density function, the moment-generating func-
tion is
() ( )
ed
tx
x
M t fx x
∞
−∞
=∫
(22)
where
( )
fx
is the probability distribution function.
To this end, let
( ) ( )
: lnxS
ττ
=
with
Tt
τ
= −
or
*
tt
τ
= −
as the case may
be where
( )
x
τ
is the log stock price in the model (16) and
τ
is the time dif-
ference between the maturity time and the present time or the time difference
between the optimal time and the present time for exercising the option as the
case may be. The latter is for Early exercise of American option peradventure
one is able to determine the optimal exercise time for the option. In order to de-
rive the characteristics function for the proposed model in 16, we need to con-
sider Feynman-Kac Formula in what follows.
4.2. The Feynman-Kac Formula Revisited
The formula states that a probabilistic expectation wrt some Ito-diffusion
processes can be obtained as a solution of a related partial differential equation.
For an example, for a 1-dimensional stochastic process
( )
( )
,,
d
tt
Xt X X
′
=
which is a solution of stochastic differential equation
( ) ( )
d ,d ,d
ii
tit itt
X tX t tX W
µσ
= +
where
, 1, ,
i
t
Wi d=
are Wieners process with correlation
d ,d d
ij
t t ij
WW t
ρ
=
.
Let
( )
( )
1
,,
d
x xx=
be some payoff.
Then the function
( ) ( )
,: |
Tt
gtx E X X x= =
(23)
is a solution of the PDE
( ) ()()
2
1 ,1
1
, ,, 0
2
dd
ij i j
i ij
i ij
gg g
tx tx tx
t x xx
µ ρσ σ
= =
∂∂ ∂
++ =
∂ ∂ ∂∂
∑∑
(24)
with final condition
( ) ( )
,gtx x=
. For 1-dimensional stochastic process
Xt
which is a solution of the SDE
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 506 Journal of Mathematical Finance
( ) ( )
d ,d,d.
t t tt
X tX t tX W
σ
µ
=
(25)
By Feynman-Kac formula, for any function say
( ) ( )
( )
, ,|
tt
gtX E gtX X t
=
(26)
is the solution to the PDE
( ) ( )
2
2
2
1
, , 0.
2
tt
gg g
tX tX
tx x
µσ
∂∂ ∂
++ =
∂∂ ∂
(27)
The Equation (26) is useful as it can be extended to
characteristics function
as
()( )
, ,, e .
iX
gX X
τ
ω
ττ
τ ϕω τ
= =
(28)
Similarly,
( ) ( )
, ,, e e | .
tT
iX iX
tt t
g X Xt E X
ωω
τ ϕω
= = =
(29)
Remarks 4:
1)
( )
,,X
τ
ϕω τ
is the characteristics function of
X
τ
.
2) The moment-generating function
( ) ( )
xx
Mi
ωϕ ω
= −
.
4.3. Solution to the Proposed Uncertain Affine Exponential Jump
Model with Recession Induced Stochastic Volatility and
Intensity
Consider the Equation (16) rewritten in the form
( ) ( )
( )
() ( )
( )
( ) ( )
( ) ( )
()
( ) ( ) ( )
() ()
( )
( ) ( ) ( )
0
*
0
0
d ln d d e 1 d , 0 0
d d d ,0 0
d d d ,0 0
s
rec
St r q tm t t W t Nt S S
t t t tWt
t t t t Wt
ν
σ σσ
λ λλ
λσ
σ κβ β σ ξσ σ σ
λ κθλ ξ λ λ λ
= −− + + − = >
= +− + =>
=−+ =>
(30)
where
( )
d ,d d
s
Wt W t
σ
ρ
=
and
() () () ()
d ,d 0 d ,d
s
Wt Wt Wt Wt
λ σλ
= =
and the volatility is taken as
( )
*rec
t
σ σσ
= +
as defined in Equation (17). The
PIDE for the moment-generating function
( )
M
ω
of
: lnxS
ττ
=
for the model
above is given by
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
*2
2
11
22
1
2
1
2
d0
x xx
rec
x
M r q mM M
MM
M tM M
Mx M f
τ
σ σ σ σσ
σ λ λ λ λλ
στ λτ σ
κ β β στ ξστ
ρξσ κ θ λ ξ λ
λ ν ν νν
∞
−∞
+ −− − +
+ +− +
+ +− +
+ +− =
∫
(31)
where
e
x
M
τ
ω
τ
=
is the final condition and
( )
e d1mf
ν
νν
∞
−∞
= −
∫
.
The solution to the PIDE (30) is speculated to be of the form
( )
( ) ( ) ( ) ( )
,, ,
,, ,, e .
rq x A B C
Mx
τ ττ
τω ω ωτ ωτ σ ωτ λ
τ ττ
ω σ λτ
− ++ + +
=
(32)
Substituting Equation (32) into (31) gives
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 507 Journal of Mathematical Finance
( ) ( ) ( )
( )
( )
()
( )
( ) ( )
( )
( )
( ) ( )
2
* 22
22
,, ,
1e d1
2
1
,,
2
1
, ,,
2
rec
AB C
mf
BB
B CC
ττ
νω
ττ
σ τ στ
σ τ λ τ λτ
ωτ ωτ σ ωτ λ
ττ τ
σω ω λ ω ν ν
κ β β σ ωτ ξσ ωτ
ρξσω ωτ κ θ λ ωτ ξ λ ωτ
∞
−∞
∂∂ ∂
−+ +
∂∂ ∂
= −− − −
+ +− +
+ +− +
∫
(33)
Rearranging the equation in terms of the state variables volatility and intensity
results to
( ) ( ) ( )
( ) ( ) ( ) ( )
()
( )
( )
( ) ( )
22 2
22
*
1 11
, ,,
2 22
,1, , e d1
2
,, ,0
rec
BB BB
CC Cm f
ABC
σ σσ τ
νω
λλ τ
σλ
ξ ωτ ρξ ω ωτ κ ωτ ω ω σ
τ
ωτ ξ ωτ κ ωτ ω ν ν λ
τ
ωτ κ β β ωτ κθ ωτ
τ
∞
−∞
∂
+ + − −+
∂
∂
+ + − +− −
∂
∂
+ ++ + =
∂
∫
(34)
where
()
( ) ( )
jump
e1 d
f
νω
ννϕ ω
∞
−∞
−=
∫
(35)
is the
moment-generating function
of the jump size distribution and
m
is an av-
erage jump amplitude given earlier by
( )
e d 1.
mf
ν
νν
∞
−∞
= −
∫
(36)
Equating the coefficients of the term structures (the stochastic volatility and
the intensity) in Equation (35) to zero, the following ordinary differential equa-
tions were obtained:
( )
( )
( ) ( )
( )
*
,,,
rec
ABC
σλ
ωτ κ β β ωτ κθ ωτ
τ
∂=−+ +
∂
(37)
( ) ( )
( )
( )
( )
22 2
,11
,,
22
BBB
σ σσ
ωτ ξ ωτ κ ρξ ω ωτ ω ω
τ
∂=− +− + −
∂
(38)
( ) ( ) ( ) ( )
()
22
,1, , e d1
2
CC Cm f
νω
λλ
ωτ ξ ωτ κ ωτ ω ν ν
τ
∞
−∞
∂=− + −− −
∂
∫
(39)
The above systems of solutions were solved as follows.
Starting with the Equation (38) first, we can see that Equation (38) is a Ricatti
differential equation. Setting
( ) ( )
( )
2
,2
D
BD
σ
τ
ωτ ξτ
′
=
(40)
in Equation (38) and simplify further, one will have a second order differential
equation given by
( )
( )
( )
2
22
2
10.
4
DD
D
σσ σ
κ ρξ ω ξ ω ω τ
τ
τ
∂∂
+− + − =
∂
∂
(41)
The general solution of the Equation (41) is given by
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 508 Journal of Mathematical Finance
( )
12
11
exp exp
22
Dk k
ττ
τψ ψ
−+
−+
=
(42)
where
( ) ( )
( )
22
.
σσ σσ σ
ψ κ ρξ ω κ ρξ ω ξ ω ω
±
=− +− −−
(43)
At
0
τ
=
, the following boundary conditions hold:
( )
( )
12
21
0
11
0 0.
22
D kk
D kk
ψψ
+−
= +
′=−=
(44)
The sum
( )
( )
22
22
σσ σ
ψ ψ κ ρξ ω ξ ω ω ς
+−
+ − − −≡=
with
( )
( )
22
σσ σ
ς κ ρξ ω ξ ω ω
=− −−
and the product
( )
2
.
σ
ψψ ξ ωω
−+
=−−⋅
The constants
1
k
and
2
k
has values
( )
0
2
D
ψ
ς
+
and
( )
0
2
D
ψ
ς
−
respectively
using the boundary conditions in Equation (47) and initial conditions
( ) ( ) ( )
,0 ,0 ,0 0ABC
ωωω
= = =
Substituting for
( )
D
τ
′
and
( )
D
τ
in Equation (40)
( ) ( )
( )
2
,2
D
BD
σ
τ
ωτ ξτ
′
=
and after simplification gives
( )
( )
11
22
2
11
22
ee
,
ee
B
ττ
ττ
ψψ
ψψ
ωτ ω ω
ψψ
+−
+−
−−
−−
− +
−
=−−
+
(45)
( )
( )
2
1e
,.
e
B
τ
τ
ς
ς
ωτ ω ω ψψ
−
−
−+
−
=−− +
(46)
In line with Artur [35]. Letting
( ) ( ) ( )
,,,AEF
ωτ ωτ ωτ
≡+
.
We rewrite Equation (37) as
( ) ( )
,,
AE F
ωτ ωτ
ττ τ
∂∂ ∂
= +
∂∂ ∂
(47)
such that
( ) ( ) ( )
*
,, ,
rec
EBB
σσ
ωτ κ β ωτ κ β ωτ
τ
∂= +
∂
(48)
( ) ( )
, ,.
FC
λ
ωτ κθ ωτ
τ
∂=
∂
(49)
Integrating the Equation (48) firstly, the following equations emerged.
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 509 Journal of Mathematical Finance
( ) ( ) (
)
*
00
, ,d ,d
rec
E B ss B ss
ττ
σσ
ωτ κβ ω κβ ω
= +
∫∫
(50)
( ) () (
)
*
00
, ,d ,d
rec
E B ss B ss
ττ
σσ
ωτ κβ ω κβ ω
= +
∫∫
(51)
( ) ( )
( )
( )
( )
*
22
00
22
, dd
rec
Ds Ds
E ss
Ds Ds
ττ
σσ
κβ κβ
ωτ ξξ
′′
−−
= +
∫∫
(52)
( )
( )
( )
*
20
2
, ln
rec
s
s
E Ds
τ
σ
κβ β
ωτ ξ
=
=
−+
=
(53)
( )
( )
( )
( )
*
2
2
, ln .
0
rec
D
ED
σ
κβ β τ
ωτ ξ
−+
=
(54)
( )
( )
11
*22
2
2ee
, ln .
2
rec
E
ττ
ψψ
σ
κβ β ψψ
ωτ ς
ξ
+−
−−
−+
−+ +
=
(55)
The result is finally written as
( )
( )
*
2
2e
, 2ln .
2
rec
E
τ
ς
σ
τ
σ
κβ β ψψ
ωτ ψ ς
ξ
−
−+
+
−+
+
= +
(56)
Similarly, integrating the ODE in the Equation (49), one will have
( )
2
e
, 2 ln .
2
F
τ
λτ
λ
κθ χχ
ωτ ψ
ξ
−
−+
+
−+
= +
(57)
Writing an explicit solution for
( )
,A
ωτ
as the sum of the solutions
( )
,E
ωτ
and
( )
,
F
ωτ
in (56) and (57) respectively yield
( )
( )
*
2
2
e
, 2 ln 2
e
2 ln .
2
rec
A
τ
τ
ς
σ
τ
σ
λτ
λ
κβ β ψψ
ωτ ψ ς
ξ
κθ χχ
ψ
ξ
−
−+
+
−
−+
+
+
+
=−+
+
++
(58)
Analogously,
( )
1e
,2
Λe
C
τ
τ
ς
ωτ ψψ
−
−
−+
−
=
++
(59)
where
( ) ( ) ( )
()
e d1 e d1ff
νω ν
ω νν ω νν
∞∞
−∞ −∞
Λ = −− −
∫∫
(60)
( )
22
2Λ
λλ
κ ξω
= −
(61)
λ
χκ
±
=
(62)
( ) ( )
( )
22
σσ σσ σ
ψ κ ρξ ω κ ρξ ω ξ ω ω
±
=− +− −−
(63)
but
( )
( )
22.
σσ σ
ς κ ρξ ω ξ ω ω
=− −−
The summary of the solution to the model (16) in which we have incorporated
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 510 Journal of Mathematical Finance
economic recession induced volatility parameter on the stock market price is
given by the characteristic function approach (the core for Fourier transform) in
sequel as
( )
( ) ( ) ( ) ( )
,, ,
,, ,, e .
rq x A B C
Mx
τ ττ
τω ω ωτ ωτ σ ωτ λ
τ ττ
ω σ λτ
− ++ + +
=
(64)
where
( )
( )
*
2
2
e
, 2 ln 2
e
2 ln 2
rec
A
τ
τ
ς
σ
τ
σ
λτ
λ
κβ β ψψ
ωτ ψ ς
ξ
κθ χχ
ψ
ξ
−
−+
+
−
−+
+
+
+
=−+
+
++
( )
( )
2
1e
,e
B
τ
τ
ς
ς
ωτ ω ω ψψ
−
−
−+
−
=−− +
( )
1e
,2Λe
C
τ
τ
ς
ωτ ψψ
−
−
−+
−
=
++
( ) ( ) ( )
()
Λe d1 e d1ff
νω ν
ω νν ω νν
∞∞
−∞ −∞
= −− −
∫∫
()
22
2Λ
λλ
κ ξω
= −
λ
χκ
±
=
( ) ( )
( )
22
σσ σσ σ
ψ κ ρξ ω κ ρξ ω ξ ω ω
±
=− +− −−
( )
( )
22
.
σσ σ
ς κ ρξ ω ξ ω ω
=− −−
For a financial claim
( )
,,,gx
σ λτ
satisfying the Partial Integro-Differential
Equation (PIDE) given in Equation (30) such that the payoff function say
( )
e,
x
fK
is satisfied by the claim,
i.e.
( )
( )
, , ,0 e ,
x
gx f K
σλ
=
. The
Fourier
transform
of
( )
gx
simply put defined on the PIDE in the light of the defini-
tion defined in Equation (5) is given as
( ) ( )
( )
( )
ˆ; ed
iwx
gw gx w gx x
+∞ −
−∞
= =
∫
(65)
and the corresponding
inverse Fourier transform
using definition in (6) is given
by
( ) ( )
( )
( )
1
1
ˆ ˆ
; ed
2π
iwx
gx gw x gw w
+∞
−
−∞
= =
∫
(66)
In light of Theorem 3.1 given by Artur Sepp [35], and Theorem 3.2 given by
Lewis (2001) [36] on page 11, we gave another modified version as follow.
Theorem 5: (Characteristic formula)
Let an asset price
lnxS
ττ
=
possesses an affine analytic characteristic func-
tion
( )
w
τ
ϕ
at time
T
τ
≤
. Define a regularity strip
( )
{ }
::Reg S w w
αβ
= <<I
where
( )
wI
is the imaginary part of
w
lying between
α
and
β
in the re-
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 511 Journal of Mathematical Finance
gularity strip. Suppose further that
( )
( )
e
wx
gx
−I
is defined on space of
( )
1
LR
such that
()()
ˆ,
g
gw w S
∈I
where
g
S
is the payoff strip and
( )
ˆ
gw
obeys the
Fourier transform given in Equation (65). Then the option value is given by
( )
( )
( )
()( )
()
( )
eˆd
2π
rT t iw
T
iw
g xt wg w w
ϕ
−− +∞
−∞
= −
∫I
I
(67)
where
g
w Reg S S∈
.
Proof:
In a risk-neutral world
Q
, we expressed
( )
( )
( )
( )
( )
e |, .
rT t
QT
g xt E g xt t T
−−
= <
(68)
( )
()
()
()
()
e
rT t
Q
gxt IE gxT
−−
=
(69)
()
()
() ( )
( )
( )
()
1ˆ
e ed
2π
iw
r T t iwx T
Qiw
g x t IE g w w
+∞
−− −
−∞
=
∫
I
I
(70)
( )
( )
( ) ( ) ( )
( )
( )
( )
( )
eˆ
ee
2π
rT t iw iwx T iwx T
Q
iw
g xt E gw
−− +∞ −−
−∞
=
∫
I
I
(71)
( )
()
()( )
( ) ( )
()
()
eˆ
ed
2π
rT t iw iwx T
T
iw
g xt wg w w
ϕ
−− +∞ −
−∞
= −
∫
I
I
(72)
Corollary 6 : (Characteristic formula for early exercise Option)
For an American option, consider a range of time
tT
τ
≤≤
, where
t
is the
initial (starting) time,
τ
is an early exercise time and
T
is the expiry time.
Suppose one is able to determine an optimal payoff time
*
T
τ
<
in a stopping
region, then the early exercise payoff of the claim
( )
( )
gx
τ
is given as
( )
( )
( )
( ) ( )
( )
( )
eˆ
ed
2π
riw iwx
iw
gx wgw w
ττ
τ
τϕ
−+∞ −
−∞
= −
∫
I
I
(73)
whenever
*
ττ
=
.
The proof follows immediately from the above theorem in addition to opti-
mality condition attached to early exercising of American options whenever the
early exercise time
*
ττ
=
.
The justification of the corollary rely on the fact that an American option can
be exercised at any time up to the expiry time. By symmetric property, we as-
sume that there exists
( )
w
τ
ϕ
−
since
g
w Reg S S∈
, and by extension the en-
tire integrand exists whenever
g
w Reg S S∈
.
5. Numerical Fourier Based Transform of
A good number of references such as Carr & Madan (1999) [29] among others
has agreed that if the corresponding characteristic function of a risk-neutral
density is given or derived, the Fourier transform representation can be written
analytically.
Let the characteristics function
( )
,,
xiw X
τ
ϕτ
of the moment generating
function
( )
,,
x
M wX
τ
τ
derived above holds for the affine model. In general, we
rewrite
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 512 Journal of Mathematical Finance
( )
( )
( )
( ) ( )
exp exp dw E iwS iwS q S S
τ τ ττ τ τ
ϕ
∞
−∞
= =
∫
(74)
where
τ
:= Optimal time in the time horizon
{ }
012
0
n
ttt t T
=<<< < =
.
In the case of European call
( )
call
T
Ek
maturing at time
tT=
with exercis-
ing price
ek
K=
defined on an underlying stock
S
. Suppose further that
( )
T
qs
represent the probability distribution function pdf of
ln
TT
sS=
. Then the fair
price
()
call
T
Ek
is expressed as the present value of the expected payoff defined
by
( )
( )
()
e e e d.
call rT s k
TT
k
E k qss
∞
−
= −
∫
(75)
With reference to Carr & Madan (1999) [29], we rewrite modified call to ac-
commodate damping factor say
R
α
+
∈
so that
( ) ( )
e
k call
TT
ck E k
α
= ×
.
Define
( )
u
τ
ψ
as the Fourier transformation representation of
( )
T
ck
such
that
( ) ( )
ed
iuk
TT
k ckk
ψ
∞
−∞
=∫
. (76)
Then the call option price function is written as
()( ) ()
exp ed
2π
call iuk
TT
k
E k uu
αψ
∞−
−∞
−
=
∫
(77)
( ) ( ) ( )
0
exp ed
π
call iuk
TT
k
E k uu
αψ
∞−
−
=
∫
(78)
( )
u
τ
ψ
in (78) is the Fourier transform of the call price given by
() ( ) ( )
exp du iuk c k k
ττ
ψ
∞
−∞
=
∫
(79)
( ) ( ) ( ) ( )
( )
()
exp exp exp e e d d
sk
T
k
u iuk rT k q s s k
τ
ψα
∞∞
−∞
= ×− −
∫∫
(80)
( ) ( ) ( )
( )
( )
1
exp e e d d
siu k
s k iuk
T
u rT q s k s
α
α
τ
ψ
∞++
++
−∞ −∞
=−−
∫∫
(81)
( ) ( ) ( )
( ) ( )
( )
1
exp e e d d
ss iu k iu k
T
u rT q s k s
αα
τ
ψ
∞+ + ++
−∞ −∞
=−−
∫∫
(82)
Integrating the term in the second integral wrt
k
gives
( ) ( ) ( )
( ) ( )
11
ee
exp d
1
iu s iu s
T
u rT q s s
iu iu
αα
τ
ψαα
++ ++
∞
−∞
=−−
+ ++
∫
(83)
Further simplification yields
( ) ( ) ( )
( )
( )
22
1
exp 21
ui
u rT ui u
τ
τ
ϕα
ψαα α
−+
= −
+− + +
(84)
By rationalising the base yields
( ) ( ) ( )
( )
( )
( )
( )
( ) ( )
22
22 22
1 21
exp
21 21
aa
bb
u i ui u
u rT
uiu uiu
τ
τ
ϕ α αα α
ψ
αα α αα α
− + × +− − +
= −
+− + + +− − +
(85)
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10.4236/jmf.2019.93026 513 Journal of Mathematical Finance
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
22
22 22
1 21
exp
21 21
ab ab
u i ui u
u rT
uiu uiu
τ
τ
ϕ α αα α
ψ
αα α αα α
+−
− + × +− − +
= −
+− + + +− − +
(86)
From difference of two’s squares, the denominator reduces to
() ( )( )
( )
( )
( )
()
( )
( )
22
22
22
2 22 2
21
exp 1
21
ab
ui u
u rT u i
ui u
ττ
αα α
ψ ϕα αα α
+− − +
= − × −+
+− − +
(87)
Since
2
1i= −
, then negative sign in the denominator changed to positive,
Hence we have
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
22
22
22
22 2
21
exp 1
21
ab
ui u
u rT u i
uu
ττ
αα α
ψ ϕα αα α
+
+− − +
= − −+ ×
+− + +
(88)
a
and
b
as used above are for ease simplification purpose and henceforth sup-
pressed.
( ) ( )
( )
( )
( )
( )
( )
( )
22
4 3 2 2 24 2 2
e1
21
2 2 4 41
rT
u ui
ui u
uu u
ττ
ψ ϕα
αα α
ααα αα αα
−
= −+
+− − +
×
+ +− + + + ++
(89)
Finally,
( ) ( )
(
)
( )
( )
( )
22
4 3 2 2 24
21
e1
221
rT
ui u
u ui
uu
ττ
αα α
ψ ϕα α αα αα
−
+− − +
= −+ ×
+ ++ ++ +
(90)
Substituting Equation (90) into (79), we now have a complete analytical for-
mula for the option price as
( ) ( ) ( )
()
( )
( )
( )
0
22
4 3 2 2 24
exp ee 1
π
21 d.
221
call iuk rT
T
k
Ek u i
ui u u
uu
τ
αϕα
αα α
α αα αα
∞−−
−
= −+
+− − +
×
+ ++ ++ +
∫
(91)
Equivalently stated as
( ) ( )
( )
( )
( )
( )
( )
( )
0
22
4 3 2 2 24
exp e1
π
21 d.
221
rT iuk
call
T
k
Ek u i
ui u u
uu
τ
αϕα
αα α
α αα αα
∞−+
−
= −+
+− − +
×
+ ++ ++ +
∫
(92)
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 514 Journal of Mathematical Finance
Extending this formulation to American option, we express the value of
American call price as the sum of European call Fourier prices and Early exercise
premium price as
( )
( )
, where log .
call
Tt e
tT
A E kP k K
≤
=+=
(93)
t
P
represent the early premium price of the American call option such that
lim 0
tTt
P
→
=
, since at the maturity time, continuation region varnishes and there
is no waiting time any longer. In other words, once the option reaches the ma-
turity date, if it has not been exercised at an early time
tT<
, then the option
must be exercised at
tT
=
. The implication is that the American option prices
coincides with that of European option prices at the maturity date,
T
.
Hence,
( ) ( )
( )
( )
( )
( )
( )
( )
( )
0
22
4 3 2 2 24
exp e
π
1 21 d.
221
rT iuk
t
t
k
Ak
u i ui u uP
uu
τ
α
ϕ α αα α
α αα αα
∞−+
−
=
− + +− − +
×+
+ ++ ++ +
∫
(94)
To extend FFT algorithm which is a numerical technique of Discrete Fourier
Transform (DFT) to American options valuation for the models under study
with economic recession induced uncertainty, we define Fast Fourier Transform
(FFT) equation as
()
( )( )
2π
111
1
e ,1 .
Ni jk
N
j
j
Hk y k N
−− −−
=
= ≤≤
∑
(95)
Applying FFT algorithm to (94), we have
( )
( )
( )
( )
( )
1
1
exp e 1.
π,
j
Niu u
u Tj t
j
k
Ak u P u N
ϖζ
τ
αψη
− −+ −
=
−
≈ + ≤≤
∑
(96)
where
1uN≤≤
,
( )
1
u
ku
ϖζ
=−+ −
and
2N
ϖζ
=
.
ζ
is the size of consistent spacing existing between
N
values of log strikes
K
.
Setting
( )
1
j
uj
η
= −
, and substituting into Equation (96) yields
( )
( )
( )( )
( )
11
1
exp ee ,1.
π
jj
Niujuiu
u Tj t
j
k
Ak u P u N
ζη ϖ
τ
αψη
− −−
=
−
≈ + ≤≤
∑
(97)
The size of the consistent spacing
ζ
between the
N
values of
k
has the fol-
lowing relation
iff
2π2π.
NN
ζ ζη
η
= ⇔=
(98)
The relation is observable if we make comparison between Equations (95) and
(97).
The smaller the values assumed by
η
, the better the fineness of the integra-
tion grid and vice versa.
Numerical Experiment
Consider an American Stock with initial price
0
100S=
, Strike Price
80K=
,
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10.4236/jmf.2019.93026 515 Journal of Mathematical Finance
Risk free interest rate
0.04r=
, Dividend rate
0.002q=
, Time to Maturity
1T=
year, Volatilities
{ }
*
0.1, 0.00,0.025
rec
σσ
= =
, Integrability Parameter
0.25
α
=
, Fineness of integration grid point
12
2N=
. The call option value on
the underlying American stock is reported below.
Table 1 above shows the payoff of a dividend paying American style call op-
tions during economic recession period and recession free period in Nigeria
which the proposed volatility change formulation presented in this paper
stressed. The option prices obtained via Fast Fourier Transform (FFT) approach
yields better results compared to the values obtained by BSM and American call
options solver prices. This confirmed numerical performance of Fast Fourier
Transform (FFT) technique among other methods of pricing options such as
Analytical payoff via BSM and American call option solver.
Figure 1 shows the true representation of the call options value obtained dur-
ing recession free period in Table 1 above while Figure 2 shows the payoff of the
Figure 1. FFT call options payoffs comparison among other methods without recession
induced volatility.
Table 1. Numerical value for a dividend paying American call options.
Volatility
{ }
,
rec
σσ
∗
Dividend
(
q
)
BSM
Price
American
Option
Pricing
Solver
FFT
Price
Exercising
Time
(yr.)
0.1, 0.000 0.002 20.249557 20.249556 20.398278
1
12
0.1, 0.025 0.002 20.249557 20.249556 20.398278
1
12
0.1, 0.000 0.002 20.746027 20.746026 20.894006 0.25
0.1, 0.025
0.002
20.746157
20.746127
20.894135
0.25
0.1, 0.000 0.002 21.484649 21.484550 21.631524 0.5
0.1, 0.025 0.002 21.491432 21.490722 21.638307 0.5
0.1, 0.000 0.002 22.949394 22.948914 23.094093 1
0.1, 0.025
0.002
23.009897
23.009970
23.154597
1
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 516 Journal of Mathematical Finance
Figure 2. FFT call options payoffs comparison among other methods with recession in-
duced volatility.
American style call options during recession period. The inference we can draw
from both the Table 1 and the two Figure 1 and Figure 2 is that during a very
short maturity period of a month in the options life span, the options value in
both economic states (recession and recession-free) coincides while as the ma-
turity period increases beyond a month, the upward volatility change induced by
recession causes slight increase in the call options prices. If further experiment is
carried out on put options, reverse will be the case in terms of volatility change
effect on the put options payoff.
6. Conclusions
The notion of economic recession and its effect on volatility uncertainty on the
payoff of European and American Options based on some certain assumptions
was presented in this paper. Economic recession is becoming a global issue and
not too far to be recognized as a re-occuring incident. In the history of economic
recession, US is at the forefront as the continent has experienced several eco-
nomic recessions. An intuition behind the introduction of economic recession
induced volatility uncertainty in this research could be traced to Nigeria eco-
nomic recession outbreak in 2016. This has an enormous effect on investors, fi-
nancial institutions and every other economic activities in the country and by
extension affected some aspect of foreign transactions (exchange), import and
export etc. by the individual within and outside the country. In relation to the
options (or stock) market, volatility has been a reference point to the level of
fluctuation in the market price of the underlying asset. It is seen as a metric for
the speed and amount of changes stochastically in the underlying asset’s prices.
An investor’s acquaintance with volatility gives a better comprehension of why
option prices behave in certain ways and this will guide them in decision mak-
P. A. Bankole, O. O. Ugbebor
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10.4236/jmf.2019.93026 517 Journal of Mathematical Finance
ing.
The model we investigated is proposed to be referred to as
Uncertain Affine
Exponential jump model with Recession induced Stochastic volatility and Inten-
sity
. Fourier transform of the model was implemented having derived an affine
close form of the characteristic function. A numerical based Fourier transform
algorithm called
Fast Fourier transform
(FFT) which was a variant of Carr &
Madan [29] FFT algorithm for European call option valuation was adopted to
determine the European call Fourier prices. The algorithm was further extended
to American call options valuation by adding premium price to the European
call options prices.
We also reported Nigerian Flourmill stock performance, during recession and
recovery year in this study. The Stock volatility is seen as a measure of the un-
certainty on the payoff or returns of the stock which may require an estimation.
The two common methods in practice for an estimation of volatility of stocks or
assets generally are known to be an estimation of historical volatility or implied
volatility on that stock. Estimation of Recession induced volatility may be diffi-
cult to be determined accurately. Nevertheless, the fact we try to establish here is
that the stocks price experiences a high level of uncertainty during the period of
recession. The volatility of stocks tends to increase during recession compared to
the period of normalcy. One major point is that volatility is never constant in an
ideal real life situation. The Flourmill stock prices data used for calibration pur-
pose revealed that during Nigerian recession, the stock prices became more vola-
tile compared to other periods. According to the Assumption 1 above in this
paper, we suggest the use of historical data prior to recession period to deter-
mine the level of uncertainty posed by Economic recession which we refer to as
Economic recession induced volatility
. We hope the figures we generated using
MATLAB was presented in the Appendix section. Figures A1-A5 showed more
details in terms of volatility change effect on stock prices induced by economic
recession.
Acknowledgements
The authors would like to thank the editors and the anonymous reviewers for
their useful comments and suggestions.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa-
per.
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P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 520 Journal of Mathematical Finance
Appendix
The following Figures A1-A5 were generated using Nigerian Flour mill stock
prices of year 2016-2017 through MATLAB.
Figure A1. Nigerian flourmill stock prices movement during recession 2016 and recovery year 2017.
Figure A2. Comparison of stock prices variance during Nigeria economic recession.
P. A. Bankole, O. O. Ugbebor
DOI:
10.4236/jmf.2019.93026 521 Journal of Mathematical Finance
Figure A3. Bar charts of flourmill stock prices movement during Nigeria economic re-
cession and recovery year.
Figure A4. Flourmill stock price metrics during economic recession period.
Figure A5. Rotated side view of flourmill stock prices metrics during Nigerian economic
recession 2016.
