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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

Risks 2019,7, 36; doi:10.3390/risks7020036 www.mdpi.com/journal/risks

Risks 2019,7, 36 2 of 14

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 EPeλX

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,βx

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

√2α

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)

Risks 2019,7, 36 6 of 14

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)

Risks 2019,7, 36 7 of 14

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

21−1

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

αµ+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)

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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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