Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model

Abstract

We provide ready-to-use formulas for European options prices, risk sensitivities, and P&L calculations under Lévy-stable models with maximal negative asymmetry. Particular cases, efficiency testing, and some qualitative features of the model are also discussed.

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risks
Article
Simple Formulas for Pricing and Hedging European
Options in the Finite Moment Log-Stable Model
Jean-Philippe Aguilar 1,* and Jan Korbel 2,3,4
1BRED Banque Populaire, Modeling Department, 18 quai de la Râpée, 75012 Paris, France
2Section for the Science of Complex Systems, Center for Medical Statistics, Informatics, and Intelligent
Systems (CeMSIIS), Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria;
jan.korbel@meduniwien.ac.at
3Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria
4Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University,
11519 Prague, Czech Republic
*Correspondence: jean-philippe.aguilar@bred.fr; Tel.: +33-1-4004-7429
Received: 27 February 2019; Accepted: 1 April 2019; Published: 3 April 2019


Abstract:
We provide ready-to-use formulas for European options prices, risk sensitivities, and P&L
calculations under Lévy-stable models with maximal negative asymmetry. Particular cases, efficiency
testing, and some qualitative features of the model are also discussed.
Keywords: stable distributions; Lévy process; option pricing; risk sensitivities; P&L explain
1. Introduction
The pricing of financial derivatives, such as options, is an important yet difficult task in
mathematical finance, in particular when one wishes to implement a model capturing realistic
market patterns. Probably the most popular option-pricing model is the one introduced in
Black and Scholes (1973)
; in this model, the instantaneous variations of an underlying asset are
modeled by the geometric Brownian motion which, from the mathematical point of view, is described
by the diffusion (or heat) equation. The Black-Scholes model has become popular among practitioners
notably because of its simplicity, and because it admits a closed formula for the option price. However,
the model fails in abnormal periods, typically during financial crises or periods of instability (Acharya
and Richardson 2009); moreover, it does not reproduce observable features such as the shape of the
volatility smile for short maturity, or the maturity pattern of the volatility smirk on equity index
options markets (see Cont and Tankov 2004;Zhang and Xiang 2008). The main reason for this is
that the Black-Scholes model is based on oversimplified assumptions, and, as a Gaussian model,
underestimates the probability of large price jumps in real markets.
It is, therefore, necessary to introduce more appropriate models, with the capability to capture
the complex behavior of financial markets. Many generalizations of the Black-Scholes model based
on different approaches have been introduced. Let us mention, among the others, models based on
stochastic volatility (Heston 1993), jump processes (Cont and Tankov 2004), regime switching models
(Duan et al. 2002) or multifractals (Calvet and Fisher 2008). These generalizations are coming from
very different fields, from econometric models to Econophysics and complex dynamical systems.
Particularly interesting are the generalizations based on fractional calculus. In this class of
models, the underlying diffusion equation is extended to derivatives of non-natural order. These
fractional derivative operators can be defined in many different ways, see e.g., Podlubny (1998) for
a general overview. The main advantage of models based on such generalized diffusion equation
lies in their ability to describe complex dynamics involving presence of large jumps, memory effects
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or risk redistribution (Kleinert and Korbel 2016). This class of fractional models include fractional
Brownian motion pricing model (Necula 2008), mixed models (Sun 2013), models with time-fractional
derivatives (Kleinert and Korbel 2016), or models with fractional diffusion of varying order (Korbel
and Luchko 2016).
The first option-pricing model connected to generalized diffusion equation, introduced in Carr and
Wu (2003) and called Finite Moment Log-Stable (FMLS) option-pricing model, makes the assumption
that the instantaneous log returns of the underlying price are driven by a specific class of Lévy process;
it is linked to fractional calculus because the model can equivalently be described by replacing the
space derivative operator in the diffusion equation by the so-called Riesz-Feller fractional derivative.
Historically, it was introduced by Carr and Wu to reproduce the maturity pattern of the implied
volatility maturity smirk (the phenomenon that, for a given maturity, implied volatility are higher
for out-of-the-money puts than for out-of-the-money calls); it is widely observed that the smirk (as a
function of moneyness) does not flatten out as maturity increases, which is in contradiction with the
Gaussian hypothesis: if the risk-neutral density were converging to the normal distribution, then the
smirk would flatten for longer maturities. Carr and Wu deliberately violate the Gaussian hypothesis by
assuming that the log returns of the market price are driven by a Lévy process and, when furthermore
assuming that its distribution is strongly asymmetric (a fat left tail and a thin right tail), then the
model generates the expected behavior for the implied volatility smirk. The tail index value (also
known as stability parameter)
α∈(
1, 2
]
of the Lévy process controls the negative slope of the smirk,
which is flat when
α=
2 (Gaussian case) and becomes steeper when
α
decreases, thus generating any
observable slope in equity index options markets. Let us note that the maximal asymmetry assumption
is plausible because large drops are more commonly observed in financial markets than large rises,
and, moreover, ensures the option prices and all its moments to remain finite (which gave the name to
the model).
The FMLS model is efficient when compared to other well-known models not only to generate
complex volatility phenomena, but also to provide conservative valuations in the context of portfolio
risk management (as demonstrated in Robinson 2015) and this makes it a good candidate for pricing
and hedging. Unfortunately, the resulting option prices cannot be expressed in terms of elementary
functions, which makes the model hard to use for practitioners. The main aim of this paper is therefore
to provide a simple mathematical representation of the European option prices and related quantities
(risk sensitivities, expected profit & loss) driven by the FMLS model, which can be easily used by
any trader. Let us note that the use of fractional models would typically require the practitioner
to implement advanced mathematical techniques such as integral transforms, complex analysis, or
numerical methods; however, it has recently been shown that, for a wide class of space-time fractional
option-pricing models, the option prices could be expressed in terms of rapidly convergent double
series (Aguilar et al. 2018). The main advantage of this approach is that the resulting prices can
be calculated without any advanced mathematical techniques or numerical methods, and with an
arbitrary degree of precision; the proof is based on Mellin transform and residue summation in
C2
,
and more details and applications can be found in Aguilar et al. (2018) and Aguilar and Korbel (2018).
Since, as already mentioned, the FMLS model is a special case of generic space-time fractional models,
this fruitful approach can be used for it as well to obtain efficient pricing tools. In this article, we will
complete these results by developing additional analytic tools for hedging, for computing various
market sensitivities and for P&L explanation, which constitute natural extensions to the double-series
pricing formula.
The paper is organized as follows: Section 2briefly summarizes the main aspects of the FMLS
option-pricing model. It also presents the double sum representation of the option price. Section 3
introduces explicit formulas for the corresponding risk sensitivities (i.e., the Greeks) Delta, Gamma,
and Theta. Section 4discusses expected profit and loss under several hedging strategies. The ultimate
section is devoted to conclusions.

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2. Lévy-Stable Option Pricing
2.1. Model Definition
Let
T>
0 and let the market spot price of some underlying financial asset be described by a
stochastic process
{S(t)}t≥0
on a filtered probability space
(Ω
,
F
,
P)
. Following Carr and Wu (2003),
we assume that there exists a risk-neutral measure
Q
under which the instantaneous variations of
S(t)
can be written in local form as:
dS(t)
S(t)=rdt+σdLα,−1(t)t∈[0, T],α∈(1, 2](1)
where
r∈R
is the (continuous) risk-free interest rate,
σ>
0 is the market volatility and
Lα,β(t)
is
a standardized Lévy-stable process (see Cont and Tankov (2004) and references therein). The fact
that the stochastic process is specified directly under the risk-neutral measure is clearly motivated by
the option-pricing purpose; it is also justified by former models defined under the physical measure
(see for instance McCulloch (1996) combining a Lévy-stable process under the physical measure, with
a utility maximization argument to achieve finite option prices).
The solution to the stochastic differential Equation (1) is the exponential Lévy process:
S(t) = S(0)e(r+µ)t+σLα,−1(t)(2)
where
µ
is the so-called “risk-neutral parameter”, which has its origin in the Esscher transform
(see details in Gerber and Shiu (1994); Kleinert and Korbel (2016)) and, for an exponential process
eX(t)with density g, is defined by
µ:=−log EPheXi=−log
+∞
Z
−∞
exg(x)dx. (3)
where
X=X(
1
)
. Note that
µ
is the negative cumulant-generating function
−log EPeλX
for
λ=
1.
Since all cumulants are finite, the cumulant-generating function is also finite. The fact that the spot
process admits the representation
(2)
shows that exponential Lévy models are a generalization of the
Black-Scholes model: when
α=
2 then for any
β
,
Lα,β(t)
degenerates into the usual Brownian motion
W(t)and (2) becomes a geometric Brownian motion
S(t) = S(0)e(r−σ2
2)t+σW(t), (4)
thus recovering the Black-Scholes framework. The risk-neutral parameter in this case is the well-known
−σ2
2term.
2.2. Stable Distributions
The probability distribution of the Lévy process
Lα,β(t)
is the
α
-stable distribution
Gα,β(x
,
t)
that
can be written under the form
Gα,β(x,t) = 1
t1
αgα,βx
t1
α
and is typically defined through the Fourier
transform as (Zolotarev 1986):
∞
Z
−∞
e−ikx gα,β(x)dx=e|k|α(1−iβsign(k)ω(k,α)), (5)

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where
ω(k
,
α) = tan πα
2
for
α6=
1 and
ω(k
, 1
) =
2
/πlog |k|
;
α
is called the stability parameter, and
β
the asymmetry. In general, the two-sided Laplace transform of Lα,βdoes not exist (which means that
its moments diverge), except for the case β= +1 (see e.g., Samorodnitsky and Taqqu 1994):
EP[e−λx] = e−λασα
cos πα
2(6)
From a symmetry argument, it follows from
(6)
and definition
(3)
that the risk-neutral parameter
µis finite when β=−1 and is equal to:
µ=σ
√2α
cos πα
2
(7)
which justifies the choice
β=−
1 in the model definition
(1)
. Please note that we have introduced the
√2-normalization so that we recover the Black-Scholes parameter µ(BS)=−σ2
2when α=2.
Under maximal negative asymmetry hypothesis
β=−
1, the asymptotic behavior of the
distribution is determined by the stability parameter:
−
If 1
<α<
2, then
gα,−1
decays exponentially on the positive real axis and has a heavy tail on the
negative real axis (that is, decays in |x|−α);
−
If
α=
2, then
ω(k
,
α) =
0 and in that case the transform
(5)
is independent of
β
and resumes to
e|k|2
, that is, the (re-scaled) Fourier transform of the heat kernel. Therefore,
Lα,β(t)
degenerates
into the usual Brownian motion W(t)and the process (2) is a geometric Brownian motion.
2.3. Mellin-Barnes Representation of the European Option
Let
τ=T−t
; the price of the European call option with strike
K
and maturity
T
is equal to the
discounted Q-expectation of the terminal payoff
Cα(S,K,r,µ,τ) = e−rτEQ[S(T)−K]+. (8)
The expectancy in
(8)
is equal to the convolution of all possible realizations for the payoff
[S(T)−K]+with the probability distribution (or Green function) of the process:
Cα(S,K,r,µ,τ) = e−rτ
(−µτ)1
α
+∞
Z
−∞hSe(r+µ)τ+y−Ki+gα,−1 y
(−µτ)1
α!dy. (9)
The ratio
1
α
is a temporal scaling exponent and allows to recover the Gaussian variance
σ√τ
when
α=
2 (see more details in Kleinert and Korbel (2016)). After some algebraic manipulations, it is
possible to re-write the Green function
(5)
as a Mellin-Barnes integral (see Flajolet et al. (1995) for a
precise introduction to the Mellin transform), that is, an integral over a vertical line in the complex
plane; precisely, it admits the representation:
gα,−1(X) = 1
α
c1+i∞
Z
c1−i∞
Γ(1−t1)
Γ(1−t1
α)Xt1−1dt1
2iπ0<c1<1 . (10)
The Green function
gα,−1
connects the FMLS model to fractional calculus, because it is the
fundamental solution to the space-fractional equation
∂g
∂τ +µDαg=0 (11)

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where
Dα
denotes the Riesz-Feller fractional derivative. When
α=
2, it degenerates into the usual
diffusion equation
∂g
∂τ −σ2
2
∂2g
∂x2=0 (12)
which drives the Black-Scholes model. More details can be found in Mainardi et al. (2001).
Let us introduce the log-forward moneyness
k:=log S
K+rτ(13)
so that we can re-write the payoff as
K[ek+µτ+y−
1
]+
. Plugging
(10)
into
(9)
, integrating by parts and
introducing the Mellin-Barnes representation for the exponential term (Bateman 1954)
ek+µτ+y=
c2+i∞
Z
c2−i∞
(−1)−t2Γ(t2) (k+µτ +y)−t2dt2
2iπc2>0 (14)
yields, after integration on the parameter y, the following representation for the call price:
Proposition 1.
Let
P
be the polyhedra
P:={(t1
,
t2)∈C2
,
Re(t2−t1)>
1 , 0
<Re(t2)<
1
}
; then, for
any vector c = (c1,c2)∈P,
Cα(S,K,r,µ,τ) =
Ke−rτ
αZ
c+iR2
(−1)−t2Γ(t2)Γ(1−t2)Γ(−1−t1+t2)
Γ(1−t1
α)(−k−µτ)1+t1−t2(−µτ)−t1
αdt1
2iπ∧dt2
2iπ(15)
2.4. Pricing Formulas
We now compute the double integral (15) by means of residue summation.
Theorem 1 (Pricing formula).The European call option price is equal to the double sum:
Cα(S,K,r,µ,τ) = Ke−rτ
α
∞
∑
n=0
m=1
1
n!Γ(1+m−n
α)(k+µτ)n(−µτ)m−n
α(16)
Proof.
Let
ω
denote the differential form under the integral sign in
(15)
; if we perform the change
of variables (u1:=−1−t1+t2
u2:=t2
(17)
then ωreads
ω= (−1)−u2Γ(u1)Γ(u2)Γ(1−u2)
Γ(1−−1−u1+u2
α)(−k−µτ)−u1(−µτ)−−1−u1+u2
αdu1
2iπ∧du2
2iπ. (18)
As the Gamma function is singular at every negative integer
−N
with residue
(−1)N
N!
(see Abramowitz and Stegun (1972) or any other monograph on special functions), it follows that in
the region
{Re(u1)<
0,
Re(u2)<
0
}
,
ω
has simple poles at every point
(u1
,
u2) = (−n
,
−m)
,
n
,
m∈N
with residue:
Res(−n,−m)ω=1
n!Γ(1+1+m−n
α)(k+µτ)n(−µτ)1+m−n
α. (19)

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As in this region, the integrand tends to in 0 at infinity (see Aguilar et al. (2017,2018) for technical
details), the integral
(15)
equals the sum of all residues
(19)
, which, after performing the change of
indexation m→m+1, is equal to the double sum (16) .
The pricing Formula
(16)
is a simple and efficient way of pricing European call options under
FMLS model; the convergence of partial sums is very fast and therefore only a few terms are needed to
obtain an excellent level of precision, as demonstrated in Table 1for a typical set of market parameters.
Table 1.
Numerical values for the
(n
,
m)
-term in the series (16) for the option price (
S=
3800,
K=
4000,
r=
1%,
σ=
20%,
τ=
1
Y
,
α=
1.7). The call price converges to a precision of 10
−3
after
summing only very few terms of the series.
n/m1 2 3 4 5 6 7
0 395.167 49.052 4.962 0.431 0.033 0.002 0.000
1−190.223 −32.268 −4.005 −0.405 −0.035 −0.003 −0.000
2 23.829 7.767 1.317 0.164 0.017 0.001 0.000
3 1.430 −0.649 −0.211 −0.036 −0.004 −0.000 −0.000
4−0.246 −0.029 0.013 0.001 0.000 0.000 0.000
5−0.046 0.004 0.000 −0.000 −0.000 −0.000 −0.000
6 0.001 0.000 −0.000 −0.000 0.000 0.000 0.000
7 0.001 −0.000 −0.000 0.000 0.000 −0.000 −0.000
8 0.000 −0.000 0.000 0.000 −0.000 −0.000 0.000
Call 229.914 253.790 255.866 256.024 256.035 256.035 256.035
The price of the put option is easily deduced from
(16)
and the call-put parity relation
C−P=
S−Ke−rτ; with our notations (13), we get:
Pα(S,K,r,µ,τ) = Cα(S,K,r,µ,τ)−S(1−e−k)(20)
Typical shape of call and put prices is depicted in Figure 1.
Particularly interesting situation occurs when the asset is at-the-money forward, that is, when
S=Ke−rτ
or equivalently with our notations
k=
0 in Equation
(16)
. In that case, it is immediate to
see that:
Corollary 1 (At-the-money price).When S =Ke−rτ, the European call option price is equal to:
CATM
α(S,µ,τ) = S
α
∞
∑
n=0
m=1
(−1)n
n!Γ(1+m−n
α)(−µτ)m+ (α−1)n
α(21)
=S
α"(−µτ)1
α
Γ(1+1
α)−(−µτ) + (−µτ)2
α
Γ(1+2
α)+O(−µτ)1+1
α#(22)
When α=2 (Black-Scholes model), then, by definition of µ, we are left with
CATM
B.S.(S,σ,τ) = S
2"1
Γ(3
2)
σ√τ
√2+O((σ√τ)3)#=1
√2πSσ√τ+O((σ√τ)3). (23)
As
1
√2π'
0.4 we have thus recovered the well-known Brenner-Subrahmanyam approximation
(that was first introduced in Brenner and Subrahmanyam (1994)):
CATM
B.S(S,σ,τ)'0.4Sσ√τ. (24)

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Figure 1.
Call (
left graph
) and put (
right graph
) option prices, as a function of
S
and for various
stability parameters
α
(parameters:
K=
4000,
r=
1% and
σ=
20%). In both the call and put cases, the
prices become higher as αdecreases.
3. Risk Sensitivities (Greeks)
The Greeks quantify the sensitivity of the option to market parameters such as asset (spot) price
or volatility, and are essential tools for portfolio management. In this section, we show that they admit
efficient representations, which can be easily obtained by differentiation of the pricing Formula (16).
3.1. Delta
From the definition of
k
, we have
∂k
∂S=1
S
and therefore, by differentiating
(16)
with respect to
S
and re-arranging the terms we obtain
∆Cα(S,K,r,µ,τ):=∂Cα
∂S=e−k
α
∞
∑
n=0
m=0
1
n!Γ(1+m−n
α)(k+µτ)n(−µτ)m−n
α(25)
This series is, again, very fast converging as demonstrated with typical values for the market
parameters in Table 2. The Delta of the put option is easily obtained by differentiation of the call-put
parity relation (20):
∆Pα(S,K,r,µ,τ) = ∆Cα(S,K,r,µ,τ)−1 (26)
When the asset is at-the-money forward (S=Ke−rτand therefore k=0), (25) reduces to:
∆ATM
Cα(S,µ,τ) = 1
α
∞
∑
n=0
m=0
(−1)n
n!Γ(1+m−n
α)(−µτ)m+ (α−1)n
α(27)
=1
α"1+(−µτ)1
α
Γ(1+1
α)−(−µτ)1−1
α
Γ(1−1
α)+O(−µτ)#(28)
When, moreover, α=2 (Black-Scholes model) then (27) becomes:
∆ATM
CBS (S,σ,τ) = 1
21−1
2√2πσ√τ+O(σ√τ)3(29)

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Table 2.
Numerical values for the
(n
,
m)
-term in the series (25) for the call option’s Delta (
S=
3800,
K=4000, r=1%, σ=20%, τ=1Y,α=1.7).
n/m0 1 2 3 4 5
0 0.613034 0.103991 0.0129082 0.001306 0.000113 8.8 ×10−6
1−0.153590 −0.050059 −0.008492 −0.001054 −0.000107 −9.3 ×10−6
2−0.013825 0.006271 0.002044 0.000347 0.000043 4.4 ×10−6
3 0.003174 0.000376 −0.000171 −0.000056 −9.4 ×10−6−1.2 ×10−6
4 0.000743 −0.000065 −7.6 ×10−63.5 ×10−61.1 ×10−61.9 ×10−7
5−0.000026 −0.000012 1.1 ×10−61.2 ×10−7−5.7 ×10−8−1.9 ×10−8
6−0.000023 3.5 ×10−71.7 ×10−7−1.4 ×10−8−1.7 ×10−97.7 ×10−10
7−1.3 ×10−62.7 ×10−7−4.1 ×10−9−1.9 ×10−91.7 ×10−10 2.0 ×10−11
Delta 0.449486 0.509990 0.516273 0.516819 0.516861 0.516864
In Figure 2, we make two different graphs to illustrate Formulas (25) and (27):
•
In left figure, we plot the value of
∆Cα
in function of the market price, for different cases of
α
; in
all cases, 0
<∆Cα<
1 for all
S
, and
∆Cα
admits an inflection in the “out-of-the-money” region
(
S<K
). However, we can observe that in this region,
∆Cα
grows faster when
α
decays, and the
inflection occurs for smaller market prices.
•
In the right figure, we choose 3 different values of
S
corresponding to the in, at or out-of-the-money
situation and we plot the evolution of
∆Cα
in function of
α
. We can observe that
∆Cα
is in all cases
a decreasing function of
α
(as could be expected from the overall
1
α
factor in
(25)
) meaning that
when
α
becomes smaller, then the options become more sensitive to variations of the underlying
price than in the Gaussian (
α=
2) case. This stronger sensitivity can be regarded as a conservative
feature of the FMLS model (similar features have also been observed in Robinson (2015)).
Figure 2.
(
Left graph
): Plot of the call’s Delta, in function of the market price
S
and for different
stability parameters
α
. (
Right graph
): Plot of the call’s Delta, in function of the stability parameter
α
and for different market configurations. In both cases, K=4000, r=1% and σ=20%.
3.2. Gamma
Differentiating (25) with respect to Sand re-arranging the terms, we obtain:
ΓCα(S,K,r,µ,τ):=∂2Cα
∂S2=e−k
αS
∞
∑
n=0
m=0 (−µτ)m−n−1
α
n!Γ(1+m−n−1
α)−(−µτ)m−n
α
n!Γ(1+m−n
α)!(k+µτ)n(30)
Differentiating the Call-Put relation for the Delta
(26)
with respect to
S
, it is immediate to see that
the Gamma is the same for the call and the put options:
ΓCα=ΓPα:=Γα(31)

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When the asset is at-the-money forward (S=Ke−rτand therefore k=0), (30) reduces to:
ΓATM
α(S,µ,τ) = 1
αS
∞
∑
n=0
m=0 (−µτ)m+ (α−1)n−1
α
n!Γ(1+m−n−1
α)−(−µτ)m+ (α−1)n
α
n!Γ(1+m−n
α)!(32)
=1
αS"1
Γ(1−1
α)
1
(−µτ)1
α−1
Γ(1+1
α)(−µτ)1
α−1
Γ(1−2
α)(−µτ)1−2
α+O(−µτ)#(33)
When, moreover, α=2 (Black-Scholes model) then (27) becomes:
ΓATM
BS (S,σ,τ) = 1
2S"r2
π
1
σ√τ−r2
πσ√τ+O(σ2τ)#(34)
3.3. Theta
By definition of
k
we have
∂k
∂τ =r
and therefore, by differentiating
(16)
with respect to
τ
and
re-arranging the terms:
ΘCα(S,K,r,µ,τ):=−∂Cα
∂τ =−Ke−rτ
α
∞
∑
n=0
m=1
Qn,m
n!Γ(1+m−n
α)(k+µτ)n−1(−µτ)m−n
α−1(35)
where
Qn,m=(−r(k+µτ) + n(r+µ))(−µτ)−µm−n
α(k+µτ)(36)
The Theta of the put option is easily obtained by differentiation of the call-put parity relation:
ΘPα(S,K,r,µ,τ) = ΘCα(S,K,r,µ,τ) + rKe−rτ(37)
When the asset is at-the-money forward (S=Ke−rτand therefore k=0), (35) reduces to:
ΘATM
Cα(S,r,µ,τ) = −Ke−rτ
α
∞
∑
n=0
m=1
(−1)n−1−rτ+m+(α−1)n
αµ+nr
n!Γ(1+m−n
α)(−µτ)m+ (α−1)n
α−1(38)
In Figure 3, we plot the evolution of
ΘCα(T−t)
. On the first graph, we fix
t=
1
Y
and we observe
that, as expected,
ΘCα(T−t)
is negative (as the value of the call can only decrease as time evolves),
and becomes even more negative as
α
decreases. Conversely, in graph 2 we show the time evolution
of a deeply in the moment put option; in this configuration, when
t→T
then the European call is
identically null, that is
ΘCα→
0 and therefore
ΘPα→rKe−rτ
and is positive. As illustrated by graph 2,
this situation is accentuated when αdecreases.
Figure 3.
(
Left graph
): Theta of an at-the-money call (
S=Ke−rτ)
. (
Right graph
): Theta of a deeply in
the money put (S=3500). In both cases K=4000, r=1%, σ=20% and T=2.

Risks 2019,7, 36 10 of 14
4. Expected P&L
Financial institutions are expected not only to compute the P&L of their trading desks, but also to
produce an explanation of this P&L, both daily. The explanation should include passage of time as
well as pure market effects such as price or volatility at first or second order (depending on the desired
precision). In this section, we show how the sensitivity Formulas
(25)
,
(30)
and
(35)
allow a fast and
efficient explanation of P&L, for various examples of portfolios.
In the following, we consider call and put options on the S&P 500 index with maturity
T=1Y=365d
and strike price
K=
3500. The calculation will be made for each of the 22 trading days
tj,j∈ {0, ..., 21}of January 2019. Corresponding (normalized) time to maturity τjis defined to be
τj:=T−tj
365 =1−tj
365 (39)
so that
τ0=
1,
τ1=359
365
etc. The underlying close prices
Sj
at every
tj
are easily obtained via historical
Bloomberg data for the ticker SPX Index.
4.1. Long-Call Position
The daily P&L of a portfolio constituted of a call option only is equal to:
P&L(j):=Cα(Sj,K,r,µ,τj)−Cα(Sj−1,K,r,µ,τj−1)(40)
for
j≥
1. If we assume that the market volatility remains constant, it follows from Taylor’s formula
that (40) can be approximated by:
P&L(j) = ΘCα(Sj−1,K,r,µ,τj−1)δtj+∆Cα(Sj−1,K,r,µ,τj−1)δSj+1
2Γ2
α(Sj−1,K,r,µ,τj−1)δS2
j(41)
where
δSj:=Sj−Sj−1δtj:=tj−tj−1(42)
for
j≥
1. We say that
(40)
is the real P&L, while
(41)
is the expected, or explained P&L. In Figure 4we
plot the evolution of the expected daily P&L for all the January trading period for different values of
α
.
The total P&L (real or expected) on the considered trading period is equal to:
P&L=
21
∑
j=1
P&L(j)(43)
and the total effects are obtained by summing the three components of the expected P&L (41):

























Time (Theta) effect =
21
∑
j=1
ΘCα(Sj−1,K,r,µ,τj−1)δtj
Spot (Delta) effect =
21
∑
j=1
∆Cα(Sj−1,K,r,µ,τj−1)δSj
Gamma effect =
21
∑
j=1
1
2Γ2
α(Sj−1,K,r,µ,τj−1)δS2
j
(44)

Risks 2019,7, 36 11 of 14
Figure 4.
Expected daily P&L of a long-call position, for various values of the stability parameter; one
observes that daily losses or gains are accentuated when αdeparts from 2.
In Table 3, we provide the monthly P&L explanation for the call option, as well as for the put;
unsurprisingly, it is largely driven by the spot price effect which, in both cases, grows when
α
decreases,
resulting in a bigger gain (resp. smaller loss) in the call (resp. put) option case.
Table 3.
Monthly P&L explain for various portfolios on the S&P 500 and different stability parameters.
Long-Call Position
Stability Total Expected P&L Time Effect Spot Price Effect Gamma Effect
α= 2 (Black-Scholes) 36.2548 −9.6994 45.9514 0.0027
α= 1.8 38.6730 −10.9094 49.5790 0.0034
α= 1.6 42.5902 −12.8088 55.3948 0.0042
α= 1.4 48.4861 −15.6892 64.1705 0.0049
Long Put Position
Stability Total Expected P&L Time Effect Spot Price Effect Gamma Effect
α= 2 (Black-Scholes) −127.1943 −6.8985 −120.2986 0.0027
α= 1.8 −124.7761 −8.1085 −116.6710 0.0034
α= 1.6 −120.8589 −10.0079 −110.8552 0.0042
α= 1.4 −114.9630 −12.8883 −102.0795 0.0049
Delta-Hedged Portfolio
Stability Total Expected P&L Time Effect Spot Price Effect Gamma Effect
α= 2 (Black-Scholes) −9.6966 −9.6994 - 0.0027
α= 1.8 −10.9060 −10.9094 - 0.0034
α= 1.6 −12.8045 −12.8088 - 0.0042
α= 1.4 −15.6843 −15.6892 - 0.0049
Gamma-Hedged Portfolio
Stability Total Expected P&L Time Effect Spot Price Effect Gamma Effect
All α163.449 −2.8009 166.25 -
4.2. Delta-Hedged Portfolio
If the portfolio is appropriately delta-hedged at every trading day
tj
, for instance by holding a
long-call position
C(Sj
,
K
,
r
,
µ
,
τj)
and a short position on the underlying
∆Cα(Sj
,
K
,
r
,
µτj)×Sj
then
the Delta-dependence is eliminated and the expected daily P&L resumes to:
P&L(j) = ΘCα(Sj−1,K,r,µ,τj−1)δtj+1
2Γ2
α(Sj−1,K,r,µ,τj−1)δS2
j. (45)

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In Figure 5we plot the expected daily P&L of the portfolio for various values of the stability.
As market volatility is assumed constant, the portfolio is essentially driven by the passage of time
with a typical daily P&L of
−
0.3, excepted when there are 3 days accrued (a situation occurring every
5 trading days and corresponding to weekends). Changes in
α
do not alter this feature but slightly
increase the daily loss of the portfolio. Total P&L and effects are presented in Table 3.
Figure 5.
Expected daily P&L of a delta-hedged portfolio; as before, the loss is accentuated when
departing from the Gaussian case.
4.3. Gamma-Hedged Portfolio (Synthetic Future)
When the portfolio is long of a call and short of a put of same strike and maturity, then the Gamma
and volatility risks are eliminated. The daily expected P&L is:
P&L(j) = (ΘCα(Sj−1,K,r,µ,τj−1)−ΘPα(Sj−1,K,r,µ,τj−1)) δtj
+ (∆Cα(Sj−1,K,r,µ,τj−1)−∆Pα(Sj−1,K,r,µ,τj−1)) δSj(46)
Please note that
(46)
is actually independent of
α
because of the call-put parity relations
(26)
and (37):
P&L(j) = −rKe−rτj−1δtj+δSj(47)
and therefore the P&L will not be affected by the change of model. This is no surprise because holding
a long-call and a short-put position is equivalent to holding a future contract, whose value is uniquely
determined by the discounted value of the strike price.
5. Conclusions
In this paper, we have provided efficient formulas for computing the prices and risk sensitivities
of European options under the FMLS model: they take the form of quickly converging double series,
whose terms are straightforward to calculate without the need for any numerical scheme. We have also
demonstrated that these formulas could be very conveniently used in the production and explanation
of P&L, and provided several examples of portfolio calculations: as expected, the FMLS model is more
“conservative” than the Black-Scholes model, in the sense that the options are more sensitive to stock
prices variations and to the passage of time, resulting in notable variations in P&L when the Lévy
parameter departs from 2.
We hope that the simple and efficient pricing and hedging tools provided in this paper will
help popularizing stable option pricing (and non-Gaussian fractional models in general) to a broader
community of capital market experts.
Author Contributions:
J.-P. A. established and tested the results, both J.-P. A. and J.K. contributed to writing and
revision of the manuscript.
Funding:
J.K. was supported by the Austrian Science Fund (FWF) under project I3073 and by the Czech Science
Foundation under grant 19-16066S.
Conflicts of Interest: The authors declare no conflict of interest.

Risks 2019,7, 36 13 of 14
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