Journal of Mathematical Finance, 2013, 3, 465-475
Published Online November 2013 (http://www.scirp.org/journal/jmf)
http://dx.doi.org/10.4236/jmf.2013.34049
Open Access JMF
Assessing the Risks of Trading Strategies Using
Acceptability Indices
Masimba E. Sonono1, Hopolang P. Mashele2
1Unit for Business Mathematics and Informatics, North-West University, Potchefstroom, South Africa
2Centre for Business Mathematics and Informatics, North-West University, Potchefstroom, South Africa
Email: 23756144@nwu.ac.za, phillip.mashele@nwu.ac.za
Received August 4, 2013; revised September 30, 2013; accepted October 11, 2013
Copyright © 2013 Masimba E. Sonono, Hopolang P. Mashele. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
The paper looks at the quantification of risks of trading strategies in incomplete markets. We realized that the no-arbi-
trage price intervals are unacceptably large. From a risk management point of view, we are concerned with finding
prices that are acceptable to the market. The acceptability of the prices is assessed by risk measures. Plausible risk
measures give price bounds that are suitable for use as bid-ask prices. Furthermore, the risk measures should be able to
compensate for the unhedgeable risk to an extent. Conic finance provides plausible bid-ask prices that are determined
by the probability distribution of the cash flows only. We apply the theory to obtain bid-ask prices in the assessment of
the risks of trading strategies. We analyze two popular trading strategies—bull call the spread strategy and bear call
spread strategy. Comparison of risk profiles for the strategies is done between the Variance Gamma Scalable Self De-
composable model and the Black-Scholes model. The findings indicate that using bid-ask prices compensates for the
unhedgeable risk and reduces the spread between bid-ask prices.
Keywords: Conic Finance; Coherent Risk Measure; Acceptability Indices; Incomplete Markets; Bid-Ask Prices;
Continuous Time Models
1. Introduction
The paper focuses on the quantification of risks of trad-
ing strategies, particularly when the market is incomplete.
The incompleteness of the market gives rise to many
martingales, each of which produces a no-arbitrage price.
Thus there is no exact replication so as to obtain a unique
price. Furthermore, the no-arbitrage price intervals are
unacceptably large. From a risk management point of
view, we are concerned with finding the prices which are
acceptable. The acceptability of these prices is assessed
by risk measures.
In the financial literature, two major classes of risk
measures have gained ground in assessing the risks of
financial positions. Foremost, we have coherent meas-
ures introduced by [1]. Since then, the theory of coherent
risk measures has been applied to several problems in
finance. Secondly, there is the grounding work of [2], in
which they proposed a new class of performance meas-
ures known as acceptability indices. The acceptability
indices can be considered as an extension of coherent
risk measures. Under the acceptability framework, a fi-
nancial position is acceptable if its distribution function
withstands high levels of stress, or in other words, a
stressed sampling of the financial position has a positive
expectation. In this paper, our contribution is assessing
the risk profiles of trading strategies using the acceptabil-
ity framework.
The rest of the paper is organized as follows: Section 2
looks at the problem of pricing in incomplete markets.
Section 3 gives an overview of risk measures and pre-
sents new acceptability indices based on the family of
distortion functions. Section 4 presents a brief detail on
conic finance and provides closed form expressions for
the bid-ask prices. Section 5 presents the models that are
used in this work. Section 6 presents numerical tests on
assessing the risks of two trading strategies. Section 7 is
the conclusion.
2. Problem of Pricing in Incomplete Markets
We start by motivating the problem through explaining
the mathematical structure of good deal bounds by [3],
also found in [4]. The good deal bounds determine the
M. E. SONONO, H. P. MASHELE
466
range of values of a risky position payoff. Let be the
set of replicable payoffs, be the market price to
replicate a payoff Y, and
R

Y
R
A
be an acceptance set of
payoffs that are acceptable to the situation. The lower
good deal bound for a payoff
X
is:
 
sup .
YR
bXYY XA
 
(1)
This payoff might be interpreted as a bid price. Equa-
tion (1) tells us that if
X
can be bought for less than
, then there is a that can be bought for

bX Y
Y
such that a cost . The upper good deal
bound, which might be interpreted as the ask price for
 
0bX Y
X
, is given by:
 
inf .
YR
aXbxYYXA
  (2)
Equation (2) tells us that selling
X
or buying
X
yields the same effect. The interpretation of
bX is
the cost that renders
X
to be acceptable. As [5] pro-
pose: any valuation principle that gives price bounds
induces a risk measure and vice versa. The accept-
ance set
A
must include the set of riskless payoffs,
0ZZ
, which is the acceptance set that generates
no-arbitrage bounds. The set
A
does not intersect with
the set

0ZZ of pure losses. The acceptance set
A
must be consistent with market prices, , or arbi-
trage occurs.
Now, an incomplete market is one in which there are
many martingale measures . The price bounds in
Equations (1) and (2) form an interval of arbitrage-free
prices for
Q
X
:
 
inf, sup
QQ
QQ
I
EX EX
(3)
where is a set of equivalent martingale measures.
The problem with the interval of the arbitrage-free prices
for
X
is that it is usually too wide for the no-arbitrage
bounds to serve as useful bid-ask prices.
In practice, derivatives traders are aware of the incom-
pleteness of the markets and after making trades on cer-
tain positions, they are not able to hedge away all the risk.
Instead, they must bear the risk associated with the trade.
To cover their business expenses and to earn compensa-
tion for bearing the risk they are not able to hedge, trad-
ers establish bid-ask intervals around the expected dis-
counted payoff.
Now, in constructing the bid-ask prices, the difficulty
posed by incomplete markets is very significant because
of adverse selection. For instance, if the ask price is too
high, few potential investors will be willing to pay so
much and the result is foregone profits. If the ask price is
too low, the resulting trade is bad for a trader and entails
likely losses. So, to ensure that trades made at bid and
ask prices are beneficial, it helps to use methods that
produce bounds for the prices that are suitable for use as
bid-ask prices and are adequate to minimize unhedgeable
risk to an extent. In the process, we will be able to quan-
tify risk since any valuation method that yields price
bounds also induces a risk measure [5].
3. Risk Performance Measures
In this section, we give a brief overview of the risk
measures. In general, a risk measure, :X
, is a
functional that assigns a numerical value to a random
variable representing an uncertain payoff.
3.1. Coherent Risk Measure
Definition Coherent Risk Measure
A risk measure
is coherent if it satisfies the follow-
ing axioms:
Translation Invariance:

XrX
 

,
for all,X
.
Monotonicity:

X
Y

if
X
Y a.s.
Positive Homogeneity:

X
X
 
,
for 0
.
Subadditivity:

X
YX

 Y,
for all ,XY
Relevance:
0X
if 0X and 0X
.
The last property is included although it is not a de-
terminant of coherency. Translation invariance axiom
implies that by adding a fixed amount
to the initial
position and investing it in a reference instrument, the risk
X
decreases by
. The monotonicity axiom pos-
tulates that if
XY
for every state of nature
,
is more risk because it has higher risk potential. Y
The positive homogeneity axiom implies that risk
linearly increases with size of the position, that is to say
that the size of the risk of a position should scale with the
size of the position. This is just a natural requirement,
though this condition may not be satisfied in the real
world since markets may be illiquid. The subadditivity
axiom implies that the risk of a portfolio is always less
than the sum of the risks of its subparts. This axiom en-
sures that diversification decreases the risk.
According to the basic representation theorem proved
by [1] for a finite
, any coherent risk measure admits a
representation of the form:
inf Q
Q
X
EX
 , (4)
with a certain set of probability measures with respect
to . A cash flow
P
X
is acceptable if it has negative risk,
that is
X
0
.
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M. E. SONONO, H. P. MASHELE 467
3.2. Acceptability Indices
Cherny and Madan, defined a subclass of risk measures
called acceptability indices, defined formally as:
3.3. Definition. Index of Acceptability
The acceptability index is as a mapping
from the set
of bounded random variables to the extended half-line
0, . The index satisfies the following four properties:
Monotonicity
If dominates Y
X
, that is
X
Y, then
 
X
Y

.
Scale invariance

X
stays the same when
X
is scaled by a posi-
tive number, that is for .

cX
X
0c
Quasi-concavity
If
X
Y
and
YY
, then


1
X
YY
 
 ,
for any
0,1
.
Fatou Property (Convergence)
Let

n
X
be a sequence of random variable.
1
n and Xn
X
converges in probability to a random
variable X. If

n
X
x
, then

X
x
.
The acceptability indices are constructed by replacing
the cumulative distribution function of
X
,
X
F
x
, by a
risk adjusted distribution, XX

F
x. The corre-
sponding risk measure is the negative expectation of the
zero cost cash flow under the distorted distribution func-
tion:
 

d,
X
XxFx

 
(5)
where
is a family of concave distortion functions on
[0,1] increasing pointwise in the stress level parameter
.
A higher value of
results in severe distortion of the
distribution function of
X
. Then, the acceptability index,

X
, is the largest stress level
such that the expec-
tation of
X
remains positive under the distortion or in
other words the distorted cash flow remains acceptable:
 

sup:0 .XX

  (6)
Cherny and Madan introduced four acceptability indi-
ces based on the family of distortion functions which are
namely: AIMIN, AIMAX, AIMINMAX, AIMAXMIN.
AIMIN is the largest number
x
such that the expec-
tation of the minimum of 1
x
draws from cash flow
distribution is still positive. Let
11
min, ,
law
x
YXX
,
where
1
,,
1
x
X
X
are independent draws from
X
.
The concave distortion function is given by:


1
11,, 0,1
x
xyyxy
  (7)
AIMAX constructs a distribution from which one
draws numerous times and takes the maximum to get
the cash flow distribution being evaluated. Let,

11
max, ,
law
x
YY
X
1
,
where 1
,,
x
YY
are independent draws of . The
concave distortion function is given by:
Y


1
1,,0,
x
xyyx y
 1
1
(8)
AIMAXMIN is constructed by first using the MIN-
VAR and then followed by the MAXVAR to create
worst case scenarios.
Let

11 1
max, ,min, ,
law
xx
YYX X


,,
,
where 11
x
X
X
are independent draws of
X
and
11
,,
x
YY
are independent draws of . Combining
the MINVAR and MAXVAR, we have the distortion
function:
Y
 


1
11,11 ,0,1
xx
xyyxy
 (9)
AIMAXMIN is constructed by first using the
MAXVAR and then followed by the MINVAR to
create worst case scenarios. Let
11
min, ,,
law
x
YZZ
law
11
max, ,,
x
Z
ZX
,,
where 11
x
Z
Z
are independent draws of
Z
.
Combining the MINVAR and MAXVAR, we have the
distortion function:


1
1
1
11 ,0,,
x
x
xyyxy




1
(10)
The acceptability indices are more plausible in assess-
ing the risks of financial positions. The acceptability in-
dices have been used heavily in the theory of conic fi-
nance, which we review next.
4. Conic Finance Theory
We look at the principles of conic finance as set out in
[6]. The market serves a passive counterparty accepting
the opposite side of zero cost trades proposed by market
participants. The departure of conic finance from the
traditional one price economy is that trade now depends
on the direction of trade, with the market buying at bid
price and selling at ask price. Cash flows to trade are
modeled as bounded random variables on a fixed prob-
ability space
,,P for a base probability measure
selected by the economy.
Now, for a risk with a cash flow outcome denoted by
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M. E. SONONO, H. P. MASHELE
468
the random variable
X
with a distribution
F
x
Q
at a
fixed period, we develop bid-ask prices at which the cash
flow is bought and sold such that the net cash flow is an
acceptable risk. The set of acceptable risks is defined by
a convex cone of random variables that contains the
non-negative cash flows. [1] showed that any acceptable
set (cone) of acceptable risks, there exists a convex
set of probability measures , equiva-
lent to , with the property that if and only if:
Q
X
P
0, all.
Q
EX Q (11)
The acceptability of a cash flow can then be com-
pletely determined by its distribution function. Accept-
ability of cash flows is linked to positive expectation via
concave distortion. So for some concave distribution
function , the cash flow distribution
function

u

0u
 
1
F
xPXx
is acceptable if:


dxFx
 
0. (12)
[6] show that the bid price,
bx, for the cash flow
X
is given by:
 


d0inf
Q
Q
bxxFxE X

 
,
(13)
and the ask price is given by:
 


d10sup.
Q
Q
axxF xE X

 
(14)
The bid and ask prices for call and put options can be
obtained by using closed formulas which are obtained on
integration by parts. Let be the random variable at
time of an underlying asset. The call option
and put option , where
S
T
CSK
PKS

K
is the strike price. The following are the closed bid and
ask prices expressions:
 

1
S
K
aCFx x
 
d,
d,
.
(14)
 

1
S
K
bCFxx

(15)
 

0d,
K
S
aPFx x

(16)
 


011 d
K
S
bPFx x

(17)
s
F
is the distribution function of and is important
because the bid and ask prices are determined completely
by this distribution.
S
5. Continuous Time Models for Option
Pricing
This section looks at the models that are used for option
pricing. It is acknowledged that the relatively most liquid
traded assets with market information are quoted vanilla
options. In practice, trades mark to market their models
to quoted vanilla options before they can price non-
quoted options. As a result, this has led to demands for
models that are capable of synthesizing the surface of
vanilla options. It is well known that the geometric
Brownian model is not capable of synthesizing the sur-
face of vanilla options, although it remains a standard
quoting model in the markets. Improvements on this
model are offered by Lévy processes, which were found
to be successful in synthesizing across strikes for a given
maturity. The following is a brief overview of the mod-
els.
5.1. Black-Scholes Model
The log-normal process models continuously compound-
ed returns using the general Brownian motion so that:
,

X
ttWt

 (18)
where
Wt is a standard Weiner process,
is the
instantaneous drift and
is the instantaneous volatility
of returns. The stochastic differential equation of the
stock price is:

dddSt SttWt

, (19)
where
is the growth rate of the stock and is related to
as follows 2
12

 . The stochastic differential
equation can be solved to give the following dynamics of
the stock price:

2
1
0exp 2
St StWt
 





. (20)
The characteristic function for the logarithm of the
stock price is:



ln 22
11
eexpln0
22
iuS t
EiuSt
 
2
ut






 


(21)
5.2. Variance Gamma Model
[7] define a Variance Gamma process,

,, ,Xt

, as
a time changed Brownian motion as follows:


,, ,,
X
ttW
 
 t (22)
where
t
is a Gamma process with parameters
and , that is,
a
b

~tGammaatb

,ab
, where the gamma
probability density function is given by:


1
,,e,0.
aa bx
bx
fxab x
a
(23)
and
are respectively the instantaneous drift and
volatility and
Wt is a standard Brownian motion.
The Variance Gamma process uses a gamma process to
time change a Brownian motion. The density function of
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M. E. SONONO, H. P. MASHELE 469
a Variance Gamma process is known in closed form and
requires the computation of the modified Bessel function
of the second kind which can be time consuming. Thus
we resort to using the characteristic function, which is
found by the conditioning on the jump

t
as in many
Lévy processes and is given by:


22
t
u
1
1.
2
Xt uiu







(24)
The dynamics of the stock price are given by:


0exp, ,,,StSt Xt

 (25)
where
is the instantaneous expected return of the
stock evaluated at calendar time and
is a compensa-
tor term The characteristic function for the logarithm of
stock price is:


 



ln
eexpln0
iuS t
Xt
EiuSt





 .u
(26)
The compensator term can be found from the charac-
teristic function and is given by:



1nl Xt i
t

.
5.3. Variance Gamma Scalable Self
Decomposable (VGSSD) Model
Sato process model was first introduced by [8]. The Sato
process was shown to be effective in synthesizing many
options on numerous underliers at the same time. The
idea behind the model was to construct stochastic proc-
esses that had inhomogeneous independent increments
from Lévy processes with homogeneous independent
increments such that the higher moments are constant
over the time horizon.
The starting point for the construction of the Sato
model is the self-decomposable law. Loosely speaking,
the self-decomposable law describes random variables
that decompose into the sum of a scaled down version of
themselves and an independent residual term. The scal-
ing property means the distribution of increments of var-
ious time scales can be obtained from that of other time
scale by rescaling the random variable at that time scale.
Thus the distribution at larger time scales are derived
from those at smaller time scales, which are easier to
estimate as the data are sufficient. [9] proposed that the
self-decomposable law is associated with the unit time
distribution of self-similar additive process whose in-
crements are independent, but not necessarily stationary.
It is known that stock prices are moved by many
pieces of information. If the pieces of information are
considered as a sequence of independent random vari-
ables , then the price changes are con-
sequences of the impacts from all i
:1,2,
i
Zi
Z
. Now, let
0
ni
i
SZ
denote their sum. Suppose that there exist
centering constants n and scaling constants n
b such
that the distribution of nn n
converges to the dis-
tribution of the random variable
cbS c
X
, which belongs to a
family law “class L”. Then the random variable
X
is
said to have the class L property. So, the price change
over the time horizon is the outcome of many independ-
ent random variables which can be approximated as a
random variable
X
that has the law of “class L”. [10]
define the self-decomposable law as follows.
5.4. Definition Self Decomposable Law
A random variable
X
is self-decomposable if for all
0,1c,
,
law c
X
cX X (27)
where c
X
is a random variable independent of
X
.
The self-decomposable random variable
X
can be
decomposed into a partial of itself and another inde-
pendent random variable. [10] also shows that one may
associate with such a self-decomposable law at unit time
a process with independent but inhomogeneous incre-
ments by defining the marginal law of the process at time
points upon scaling the law at unit time. Therefore we
have that:
t
Xt ,tXt
0.
(28)
Thus we can study the price changes easily using
self-decomposable laws, which are easier to handle than
class L.
Self-decomposable laws are an important sub-class of
the class of infinitely divisible laws [11]. The character-
istic function of the self-decomposable laws has the form
(see [10])

22 e
iux
R
u1
xp11 ,
x
gx
Ei iuxx
x
1
2
ru
ee
iux d


 


(29)
where ,r
are constants, ,
20


2
R
gx
x
1dxx

,
and
g
x is an increasing function when 0x
and
decreasing function when . An infinitely divisible
law is self-decomposable if the corresponding Lévy den-
sity has the form
0x
g
x
x
,
where
g
x is increasing for negative
x
and de-
creasing for positive
x
.
The dynamics of the stock price is defined as:
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M. E. SONONO, H. P. MASHELE
470
 
0exp ,StSrt Xtt

(30)
where is a compensator term. The Sato process
used in this work is the one constructed from the vari-
ance gamma process and is known as the Variance
Gamma Scalable Self Decomposable (VGSSD) process.
The variance gamma process is defined by time changing
an arithmetic Brownian motion with drift

t
and volatil-
ity
by an independent gamma process with unit
mean rate and variance rate
. Let
;
Gt
be the
gamma process, then the variance gamma process is
written as:


,,,;; ,
VG
XtGtWGt
 

(
31)
where is an independent standard Brownian mo-
tion.

Wt
The gamma process is an increasing pure jump Lévy
process with independent identically distributed incre-
ments over regular non overlapping intervals of length
that are gamma distributed with density h
h
f
g
where:

1e,0
hg
hh
v
g
fg g
h





.
(32)
The VGSSD is constructed from the variance gamma
process by defining the scaled stochastic process
X
t
such that it is equal in law to where

1
VG
tX
1
VG
X
is a variance gamma random variable at unit time. It fol-
lows that the characteristic function of
X
t is given
by [7]



22
1 1
1.
2
VG
Xt X
uiutut






(33)
Since the VGSSD is a scaled stochastic process, its
higher moments remain constant over time.
6. Numerical Tests
Next, focus is shifted to analyzing the risk profiles of
option investing strategies. We examine two option
strategies which are namely bull call spread strategy and
bear call spread strategy. We determine the maximum
risk, maximum reward and breakeven price for each of
the strategies. Comparison of risk profiles is done be-
tween the VGSSD model and the Black-Scholes model.
The Black-Scholes model is considered here since it is
the one that is mostly used by industrial practioners. So,
the Black-Scholes is a proxy for market prices. The the-
ory of conic finance provides bid-ask prices, which de-
pend on the risk appetite of investors. For evaluation of
bid-ask prices, we use acceptability indices based on the
MAXMINVAR. The options used in the strategies are of
European type and are applied to Single Stocks Futures
(SSF) options offered in the South African financial
markets.
A bull call spread is a bullish strategy formed by buy-
ing an “in-the-money call option” (lower strike) and sell-
ing “out-of-the-money” (higher strike). Both call options
must be on the same underlying and expiration date. The
strategy’s net effect is to bring down the cost and break-
even (long call strike price + net debt) on a buy call (long
call) strategy.
A bear call spread is a bearish strategy formed by buy-
ing an “out-of-the-money” call option (higher strike) and
selling an “in-the-money” call option (lower strike). Both
call options must be on the same underlying security and
expiration date. The strategy's concept is to protect the
downside of the sold call option by buying a call option
of higher strike price. Then, the investor receives a net
credit since the call option which has been bought has a
higher strike price than the sold option. The breakeven
will be the sum of the strike price of the short call option
plus the premium received.
For numerical illustration purposes, we used names of
two large South African banks—ABSA and Standard
Bank. Note that, the illustrations do not pertain to any
real positions on the banks. The bid-ask prices were
computed at various theoretical prices of the underlying
on the expiration date. The 3-month JIBAR is used as a
proxy for the risk-free interest rate. To realize model
calibration, we need market prices. Simulated data set of
bid-ask options at different strikes maturing on the same
date were generated using the models introduced in the
previous Section.
The illustrations that follow merely suggest what an
investor can do given the different risk appetites on an
investor. The illustrations are implemented at stress (risk)
level of 0.01, 0.05 and 0.10.
6.1. Bull Call Spread Risk Profile
6.1.1. Scenario
An investor owns 100 shares in ABSA Bank (ASAQ),
which in early July are trading at a Single Stock Future
(SSF) fair value of R140. The investor believes the mar-
ket will be bullish in the coming 6 months and decided to
create a bull call spread. So the investor buys a DEC
ASAQ 140 call option and sells a DEC ASAQ call op-
tion with a higher strike price, so as to create the bull call
spread strategy. The concern for the investor is on the
appropriate higher strike which can create an attractive
strategy.
1) At different stress (risk) levels, the investor deter-
mines the bid-ask prices for the range of strike prices.
2) The investor analyzes the risk profiles at each strike
price choice so as to create an appropriate trade.
3) The investor finally assesses the performance of the
Open Access JMF
M. E. SONONO, H. P. MASHELE 471
strategy, given a range of possible values of the underly-
ing at expiration for the appropriate strike price from step
2 at a stress level of 0.01.
In addition, the investor gathers the following infor-
mation:
3month JIBAR rate5% .01
Time toexpiration6r 12 y
Dividend yield0%(assumpti. on)
In order to create the strategy appropriately, the inves-
tor implemented the following steps.
1) Bid-Ask Prices at Different Stress Levels

γ
The calibrated parameters used for this strategy are
0.240
in the Black-Scholes model and
0.226,0.131, 0.08, 0.480


in the VGSSD model. An attractive bull call spread is
created when an investor buys a lower strike call and
sells a higher strike call. In the scenario presented above,
the investor has the choices shown in Table 1. Table 2
shows the bid-ask prices for the options using both the
Black-Scholes model and VGSSD model.
2) Risk Profile Analysis
Next, we look at the risk profiles for each of the
choices using bid-ask prices provided in step 1 at a
stress level of 0.01. Under the Black-Scholes model, an
attractive strategy can be created by choice 4) as shown
in Table 3. The reason is that the risk and breakeven
point is lower whilst maximum reward and maximum
Return on Investment (ROI) are high enough to be at-
tractive.
Also under the VGSSD model, an attractive strategy
can be created using choice 4) as shown in Table 4. The
reason being again that the risk and breakeven is lower
whilst the maximum reward and maximum Return on
Investment (ROI) are high enough to be attractive.
3) Scenario Analysis at the Expiration Date
After choosing an attractive choice from step 2, we
now look at the profit/loss of the strategy at expiration
for a range of prices for the underlying. We compare the
profit/loss under the two models—Black-Scholes model
and VGSSD model. Table 5 shows the profit/loss of the
strategy under the two models. Figure 1 shows the plot
of the profit/loss of the strategy for a range of prices of
the underlying at expiration. From Figure 1, it can be
observed that the breakeven point is lower using the
VGSSD model than the Black-Scholes model. A lower
breakeven point is ideal for a strategy which intends to
reduce risk.
6.1.2. C omm ent on the Str ategy
The spread was observed to be lower in the VGSSD
model than in the Black-Scholes model. Reduced spread
can minimize the unhedgeable risk, which can be a major
Table 1. Bull call spread investor choices.
Step 1 Long Call Buy R140 Strike Call
Step 2 1) Short Call Sell R141 Strike Call
Or 2) Short Call Sell R142 Strike Call
Or 3) Short Call Sell R143 Strike Call
Or 4) Short Call Sell R144 Strike Call
Or 5) Short Call Sell R145 Strike Call
boost for option trading strategies. As a result, the cost of
trade is lowered as the sold options can offset the cost of
the bought option. In conclusion, the strategy becomes
less risky in terms of lower risk and lower breakeven
point but offers limited potential reward, which can still
be highly attractive.
6.2. Bear Call Spread Risk Profile
6.2.1. Scenario
In early July an investor believes the SSF fair price of
Standard Bank (SBKQ) is going to fall from the current
levels of R120 to around R117.50. The investor wants to
create an attractive bear call spread. So the investor
writes a SEP SBKQ 119 call option and buys a higher
SEP SBKQ strike call, so as to create a bear call strategy.
A little bit of concern to the investor is on the appropriate
higher strike to choose so as to create an attractive strat-
egy.
1) Now at different stress (risk) levels, the investor
determines the bid-ask prices for the range of higher
strike prices.
2) The investor analyzes the risk profiles at each of
strike price choices so as to create an appropriate trade.
3) Finally, the investor accesses the performance of the
strategy for the appropriate strike price in step 2 at a
stress level of 0.01 given a range of possible values of
the underlying at expiration.
1) Bid-Ask Prices at Different Stress
γ Levels
The calibrated parameters used for this strategy are
0.306
in the Black-Scholes model and
0.285,0.070, 0.060, 0.510


in the VGSSD model. An attractive bear call spread is
created when an investor sells a lower strike call and
buys a higher strike call. In the scenario presented here,
the investor has the choices shown in Table 6. Table 7
shows the bid-ask prices for the options using both the
Black-Scholes model and VGSSD model.
2) Risk Profile Analysis
We now look at the risk profiles for each of the
choices using bid-ask prices provided in step 1 at a stress
level of 0.01. In Table 8 a potential strategy can be cre-
ated using a strike which provides reduced risk and a
lower breakeven point. In addition, the gain on this strat-
egy is the net credit received upon entering the trade. As
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M. E. SONONO, H. P. MASHELE
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472
Table 2. Bull call spread bid-ask prices at different stress le vels.
Black-Scholes Model VGSSD Model
Stress
Level S K Bid Ask Spread S K Bid Ask Spread
0.01 140 140 11.05 11.83 0.78 140 140 11.04 11.77 0.73
141 10.56 11.32 0.76 141 10.54 11.25 0.71
142 10.07 10.81 0.74 142 10.04 10.73 0.69
143 9.62 10.33 0.71 143 9.59 10.26 0.67
144 9.16 9.85 0.69 144 9.15 9.79 0.64
145 8.71 9.37 0.66 145 8.70 9.32 0.62
0.05 140 140 10.50 14.59 4.09 140 140 10.49 14.28 3.79
141 10.03 14.00 3.79 141 10.01 13.68 3.67
142 9.56 13.40 3.84 142 9.53 13.09 3.56
143 9.09 12.80 3.71 143 9.07 12.52 3.45
144 8.67 12.28 3.61 144 8.66 12.01 3.35
145 8.24 11.72 3.48 145 8.23 11.47 3.24
0.10 140 140 9.85 18.45 8.60 140 140 9.81 17.67 7.86
141 9.38 17.72 8.34 141 9.35 16.99 7.64
142 8.93 17.03 8.10 142 8.92 16.35 7.43
143 8.49 16.33 7.84 143 8.46 15.67 7.21
144 8.07 15.67 7.60 144 8.05 15.05 7.00
145 7.68 15.05 7.37 145 7.67 14.46 6.79
Table 3. Bull call spread risk profile using Black-Scholes mode l.
Step 1 Long Call
Buy R140 Strike Call@R11.77
Step 2 Short Call Risk Reward Breakeven Max ROI
1) Sell R141 Strike Call@R10.54 R1.23 -R0.23 R141.23 18.70%
2) Sell R142 Strike Call@R10.04 R1.73 R0.27 R141.73 15.61%
3) Sell R143 Strike Call@R9.59 R2.18 R0.82 R142.18 37.61%
4) Sell R144 Strike Call@R9.15 R2.62 R1.38 R142.62 52.67%
5) Sell R145 Strike Call@8.70 R3.07 R1.93 R143.07 62.87%
Table 4. Bull call spread risk profile using VGSSD model.
Step 1 Long Call
Buy R140 Strike Call@R11.83
Step 2 Short Call Risk Reward Breakeven Max ROI
1) Sell R141 Strike Call@R10.56 R1.27 R0.27 R141.27 21.26%
2) Sell R142 Strike Call@R10.07 R1.76 R0.24 R141.76 13.64%
3) Sell R143 Strike Call@R9.62 R2.21 R0.79 R142.21 35.75%
4) Sell R144 Strike Call@R9.16 R2.67 R1.33 R142.67 49.81%
5) Sell R145 Strike Call@R8.71 R3.12 R1.88 R143.12 60.26%
a result choice 4) is attractive to create the strategy since
the net credit is fairly high, and the breakeven point as
well as the risk are reduced.
In Table 9 a potential strategy again can be created
using a strike which provides reduced risk and lower
breakeven. Also, the gain on this strategy is the net credit
received upon entering the trade. As a result choice 4) is
attractive to create the strategy since the net credit is
fairly high, and the breakeven point is reduced and the
risk is fairly low.
3) Scenario Analysis at the Expiration Date
Next, we look at the profit/loss of the strategy at expi-
ration for a range of prices for the underlying using the
choice selected in Step 2. We compare the profit/loss under
the two models—Black-Scholes model and VGSSD model.
Table 10 shows the profit/loss of the strategy under the
M. E. SONONO, H. P. MASHELE 473
Table 5. Bull call spread profit/loss under Black-Scholes and VGSSD models.
Black-Scholes Model VGSSD Model
ASAQ@expiry Profit/Loss ASAQ@expiry Profit/Loss
135 2.67 135 2.62
136 2.67 136 2.62
137 2.67 137 2.62
138 2.67 138 2.62
139 2.67 139 2.62
140 2.67 140 2.62
141 1.67 141 1.62
142 0.67 142 0.62
142.67 0.00 142.62 0.00
143 0.33 143 0.38
144 1.33 144 1.38
145 1.33 145 1.38
146 1.33 146 1.38
147 1.33 147 1.38
148 1.33 148 1.38
149 1.33 149 1.38
150 1.33 150 1.38
Table 6. Bear call spread investor choices.
Step 1 Short Call Sell R119 Strike Call
Step 2 1) Long Call Buy R120 Strike Call
Or 2) Long Call Buy R121 Strike Call
Or 3) Long Call Buy R122 Strike Call
Or 4) Long Call Buy R123 Strike Call
Or 5) Long Call Buy R124 Strike Call
Table 7. Bear call spread bid-ask prices at different stress levels.
Black-Scholes Model VGSSD Model
Stress
Level S K Bid Ask Spread S K Bid Ask Spread
0.01 120 119 6.91 7.45 0.54 120 119 6.87 7.40 0.73
120 6.40 6.90 0.50 120 6.37 6.86 0.71
121 5.92 6.40 0.48 121 5.88 6.35 0.69
122 5.47 5.93 0.46 122 5.43 5.89 0.67
123 5.02 5.45 0.43 123 5.01 5.43 0.64
124 4.62 5.03 0.41 124 4.61 5.02 0.41
0.05 120 119 6.57 9.42 2.85 120 119 6.55 9.34 2.79
120 6.06 8.76 2.70 120 6.04 8.67 2.63
121 5.61 8.16 2.55 121 5.59 8.09 2.50
122 5.18 7.61 2.43 122 5.17 7.53 2.36
123 4.75 7.03 2.28 123 4.73 6.95 2.22
124 4.35 6.50 2.15 124 4.33 6.42 2.09
0.10 120 119 6.16 12.25 6.09 120 119 6.14 12.05 5.91
120 5.67 11.44 5.77 120 5.66 11.28 5.62
121 5.25 10.74 5.49 121 5.23 10.56 5.33
122 4.81 10.00 5.19 122 4.80 9.85 5.05
123 4.42 9.33 4.91 123 4.41 9.19 4.78
124 3.71 8.08 4.37 124 3.69 7.94 4.25
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474
Table 8. Bear call spread risk profile using Black-Scholes model.
Step 1 Short Call
Sell R119 Strike Call@R6.91
Step 2 Long Call Risk Reward Breakeven Max ROI
1) Buy R120 Strike Call@R6.90 R0.99 R0.01 R119.01 1.01%
2) Buy R121 Strike Call@R6.40 R1.49 R0.51 R119.51 34.23%
3) Buy R122 Strike Call@R5.93 R2.02 R0.98 R119.98 48.51%
4) Buy R123 Strike Call@R5.45 R2.54 R1.46 R120.46 57.48%
5) Buy R124 Strike Call@5.03 R3.12 R1.88 R120.88 60.26%
Table 9. Bear call spread risk profile using VGSSD model.
Step 1 Short Call
Sell R119 Strike Call@R6.87
Step 2 Long Call Risk Reward BreakevenMax
ROI
1) Buy R120 Strike
Call@R6.86 R0.99 R0.01 R119.011.01%
2) Buy R121 Strike
Call@R6.35 R1.48 R0.52 R119.5235.14%
3) Buy R122 Strike
Call@R5.89 R2.02 R0.98 R119.9848.51%
4) Buy R123 Strike
Call@R5.43 R2.56 R1.44 R120.4456.25%
5) Buy R124 Strike
Call@5.02 R3.15 R1.85 R120.8558.73%
Figure 1. Plot of bull call spread profit/loss.
two models. Figure 2 shows the plot of the profit/loss of
the strategy for a range of prices of the underlying at ex-
piration. The breakeven point is lower in the VGSSD
model than the Black-Scholes model, which is ideal in
creating a strategy with reduced risk.
Table 10. Bull call spread profit/loss under the Black-Scho-
les and VGSSD models.
Black-Scholes Model VGSSD Model
SBKQ@expiryProfit/LossSBKQ@expiry Profit/Loss
125 2.54 125 -2.56
124 2.54 124 -2.56
123 2.54 123 -2.56
122 1.54 122 -1.56
121 0.54 121 -0.56
120.46 0.00 120.44 0.00
120 0.46 120 0.44
119 1.46 119 1.44
118 1.46 118 1.44
117 1.46 117 1.44
116 1.46 116 1.44
115 1.46 115 1.44
Figure 2. Plot of bear call spread profit/loss.
6.2.2. C omm ent on the Str ategy
Under this strategy, the spread is reduced under the
VGSSD model than the Black-Scholes model. The lower
spread implies reduced cost of risk and lower breakeven
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M. E. SONONO, H. P. MASHELE 475
point. However, in this strategy reducing the risk impacts
on the potential reward under the VGSSD model as
compared to the Black-Scholes model.
7. Conclusions
In this paper, we have looked at the quantification of
risks of trading strategies in incomplete markets. We
established that the no-arbitrage price intervals are unac-
ceptably large. We need intervals with prices which are
acceptable to the market. The acceptability of the prices
is assessed by risk measures. Ideal risk measures are
those that produce price bounds that are suitable for use
as bid-ask prices and are able to compensate for un-
hedgeable risk. Plausible risk measures we look at are
coherent risk measure and acceptability indices. Accept-
ability indices are heavily used in the theory of conic
finance, which we used to assess the risk of trading
strategies. Conic finance provides plausible bid-ask pric-
es which are determined only by the probability distribu-
tion of the cash flow.
We assess the risks of financial positions using two
strategies-bull call spread and bear call spread. Com-
parison of the risk profiles for the trading strategies is
done between the VGSSD model and the Black-Scholes
model. The findings showed that the spread was reduced,
especially using the VGSSD model as compared to the
Black-Scholes model. In addition, the findings showed
that the bid-ask price intervals are able to compensate for
the unhedgeable risk. Ultimately, reward from the strate-
gies had a potential of increasing.
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