The value of power-related options under spectrally negative Lévy processes
aa r X i v : . [ q -f i n . P R ] S e p The value of power-related options under spectrallynegative L´evy processes
Jean-Philippe Aguilar
This version: Nov. 19 th , 2019, revised Sept. 27 th , 2020. Abstract
We provide analytical tools for pricing power options with exoticfeatures (capped or log payoffs, gap options etc.) in the framework of expo-nential L´evy models driven by one-sided stable or tempered stable processes.Pricing formulas take the form of fast converging series of powers of the log-forward moneyness and of the time-to-maturity; these series are obtained viaa factorized integral representation in the Mellin space evaluated by means ofresidues in C or C . Comparisons with numerical methods and efficiency testsare also discussed. Keywords
L´evy Process · Stable Distribution · Tempered Stable Distribu-tion · Digital option · Power option · Gap option · Log option
Mathematics Subject Classification (2010) · Spectrally negative L´evy processes are L´evy processes (see the classical text-book [7] for a complete introduction to the theory of L´evy processes, and,among many other references, [23,43,15,45] for their applications in financialmodeling) whose L´evy measure is supported by the real negative axis, i.e., pro-cesses without positive jumps [31]; they include Brownian motion with drift,asymmetric α -stable [48,14] or asymmetric tempered-stable [40] processes andtheir particular cases, such as negative Gamma and Inverse Gamma processes.Such one-sided processes have been shown to be effective for modeling the priceof financial assets, because their heavy-tail induces a leptokurtosis in the dis-tribution of returns (whose empirical evidence is known since [19]), and theirskewed behavior introduces the asymmetry in the occurrence of upward and J. Ph. AguilarCov´ea Finance - Quantitative Research Team - 8 Rue Boissy d’Anglas - FR75008 ParisE-mail: jean-philippe.aguilar@covea-finance.fr Jean-Philippe Aguilar downward jumps (see [14,35,17] for more recent discussions and justifications).Moreover, in the context of exponential market models [43,15], they generatea wide range of dynamics for the log returns, from almost surely continuoustrajectories in the Brownian motion case [9], to highly discontinuous realiza-tions with a potentially infinite number of downward jumps on any given timeinterval.For the specific purpose of option pricing, spectrally negative L´evy pro-cesses have been introduced in [14] in the case of a totally skewed α -stabledynamics, the strong asymmetry of the model combined with the presence offat tails capturing volatility patterns for longer observable horizons more ac-curately than Gaussian models. They have subsequently been employed in thecalibration of index options on some major equity indices (it is shown in [17]that positive jumps are not needed for long term options on most index mar-kets); concerning path-dependent instruments, the impact of one-sided L´evydynamics on Asian and Barrier options has also been investigated [39,4]. Letus also mention that spectrally negative L´evy processes have been successfullyapplied in other areas of Quantitative Finance, notably in default modelingand credit exposure [35], as the default of a firm is often linked to brutal lossesin their assets’ value.When it comes to practical evaluation however, things are more compli-cated under L´evy dynamics than in the usual Black-Scholes framework; theliterature is dominated by numerical (finite difference) schemes for PartialIntegro-Differential Equations [16], by Monte Carlo simulations [40] or Fouriertransforms of option prices [11]. The latter approach is particularly popular,because in most exponential L´evy models, the characteristic function of theasset’s log price is available in a closed and relatively compact form; severalrefinements of the method have been introduced to accelerate the evaluation ofFourier integrals, notably by means of other integral transforms (among oth-ers, Fourier-cosine transform [20] or Hilbert transform [21]) or, more recently,by application of frame duality properties [28]. Let us also mention that, inthe case of European style options with non standard terminal payoffs, it isalso possible to use an integral representation decomposing the contract into asum of standard contracts (see [12]) and to evaluate numerically the involvedintegrals, at least for smooth (twice differentiable) payoffs. For discontinuouspayoffs, it is advantageous to refine this approach by assembling a series ofselected payoffs and by using the theory of frames, as developed in [29].In this paper, we would like to take profit of the properties of anotherFourier-related transform, namely the Mellin transform [22]. First, let us men-tion that the Mellin transform has been previously implemented in many areasof financial modeling, from providing representations for vanilla or basket op-tions in the Black-Scholes model [37], to quantifying the at-the-money impliedvolatility slope in various L´evy models [24]. In our approach, we will focus onexpressing Mellin integrals as a sum of residues in C or C , so as to obtainsimple series expansions for option prices. More precisely, we will show that,in the framework of exponential L´evy models driven by spectrally negativeprocesses, option prices have a factorized form in the Mellin space (in terms of he value of power-related options under spectrally negative L´evy processes 3 maturity and log-forward moneyness); inverting the transform, the prices canbe conveniently computed by a straightforward series of residues, allowing fora very simple and fast evaluation of the options.The Mellin residue technique has been used to derive fast convergent seriesfor European options prices and Greeks, in the Black-Scholes [2] and FiniteMoment Log Stable (FMLS) [3] models; in this article, we will show thatthe technique successfully applies to a more general range of exotic power-related options (Digital, Log, Gap, European with cap etc.). This family ofoptions offers a higher (and nonlinear) payoff than the vanilla options; it isused e.g. to increase the leverage ratio of strategies, or to lock the expositionto future volatility (see discussion in [46]). From a more theoretical point ofview, power payoffs have also been employed to determine the L´evy symbol ofthe underlying asset dynamics in [10]. In the Gaussian context, closed formulasfor pricing and hedging standard power options are known since [26], and havebeen recently generalized to include some barrier features [27]; studies havealso been made in the setup of local volatility models, or for more genericpolynomial options (decomposed a sum of power options) in [33]. The presentpaper will be devoted to establishing efficient pricing formulas in the context ofan asymmetric α -stable exponential L´evy model, and to show that it is possibleto extend them to the more generic class of tempered stable processes.The paper is organised as follows: in section 2 we start by recalling somebasic facts about option pricing in exponential L´evy models; then, in section 3,we establish a factorized form for option prices in the case of a spectrallynegative α -stable dynamics. This factorized form enables us to derive severalpricing formulas for power-related instruments in section 4, under the form offast convergent series of powers of the time-to-maturity and of the moneyness;in this section, we also test the results numerically, and provide efficiency tests.In section 5, we show that similar formulas can also be derived if the stabledistribution is tempered, and study the impact of the tempering parameter inthe case of a digital option. Finally, section 6 is devoted to concluding remarksand perspectives. Notations.
Given a filtered probability space ( Ω, F , {F t } t ≥ , P ), recall that aprocess { X t } t ≥ is a L´evy process [7,32] if there exists a triplet ( a, b, ν ) suchthat the characteristic exponent Ψ ( k ) := − log E P [ e ikX ] of X t admits therepresentation Ψ ( k ) = iak + 12 bk + Z R (1 − e ikx + ikx {| x | < } ) ν (d x ) , (1)where a, b ∈ R and ν is a measure concentrated on R \{ } satisfying Z R min(1 , x ) ν (d x ) < ∞ . (2) Jean-Philippe Aguilar
Equation (1) is known as the
L´evy-Khintchine formula; a is the drift , b is the Brownian (or diffusion ) coefficient and ν is the L´evy measure of the process.If ν ( R ) < ∞ , one speaks of a process with finite activity or intensity ; thiscorresponds to processes whose realizations have a finite number of jumps onevery finite interval, like in jump-diffusion models such as the Merton model[36] or the Kou model [30]. If ν ( R ) = ∞ , then one speaks of a process with infinite activity or intensity , and in this case an infinite number of jumpsoccur on every finite interval; this gives birth to a very rich dynamics and suchprocesses do not need a Brownian component to generate complex behaviors.When furthermore ν ( R + ) = 0 (resp. ν ( R − ) = 0), the process is said to be spectrally negative (resp. spectrally positive ).As a L´evy process has stationary independent increments, its characteristicfunction can be written down as F [ X t ]( k ) := E P [ e ikX t ] = e − tΨ ( k ) (3)and its moment generating function , whenever it converges, as: M [ X t ]( p ) := E P [ e pX t ] = e tφ ( p ) , φ ( p ) = − Ψ ( − ip ) . (4)The function φ ( p ) is the Laplace exponent or cumulant generating function of the process, and its existence depends on the asymptotic behavior of theL´evy measure; in particular, in the case of a spectrally negative process, theabsence of positive fat tail ensures that φ ( p ) exists in the whole complex half-plane { Re ( p ) > } . Exponential processes.
Let us now introduce the class of exponential L´evymodels, following the classical setup of [43,15] for instance. Let
T >
0, andlet S t denote the value of a financial asset at time t ∈ [0 , T ]; we assume thatit can be modeled as the realization of a stochastic process { S t } t ≥ on thecanonical space Ω = R + equipped with its natural filtration, and that, underthe risk-neutral measure Q , its instantaneous variations can be written downin local form as: d S t S t = ( r − q ) d t + d X t . (5)In the stochastic differential equation (5), r ∈ R is the risk-free interest rateand q ∈ R is the dividend yield, both assumed to be deterministic and con-tinuously compounded, and { X t } t ≥ is a L´evy process; for the simplicity ofnotations, we will assume that q = 0, but all the results of the paper remainvalid when replacing r by r − q .The solution to (5) is the exponential L´evy process defined by: S T = S t e ( r + µ ) τ + X τ (6)where τ := T − t is the horizon (or time-to-maturity), and µ is the martingale (or convexity ) adjustment computed in a way that the discounted stock priceis a Q -martingale, which reduces to the condition: E Q t (cid:2) e µτ + X τ (cid:3) = 1 , (7) he value of power-related options under spectrally negative L´evy processes 5 or, equivalently, in terms of the Laplace exponent: µ = − φ (1) . (8)2.2 Option pricingLet N ∈ N and P : R N + → R be a non time-dependent payoff functiondepending on the terminal price S T and on some positive parameters K n , n = 1 . . . N : P : ( S T , K , . . . , K N ) → P ( S T , K , . . . , K N ) := P ( S T , K ) . (9)The value at time t of an option with maturity T and payoff P ( S T , K ) is equalto the risk-neutral conditional expectation of the discounted payoff: C ( S t , K, r, µ, t, T ) = E Q t (cid:2) e − rτ P ( S T , K ) (cid:3) . (10)In the case where the L´evy process admits a density g ( x, t ), then, using (6),we can re-write (10) by integrating all possible realizations for the terminalpayoff over the probability density: C ( S t , K, r, µ, τ ) = e − rτ + ∞ Z −∞ P (cid:16) S t e ( r + µ ) τ + x , K (cid:17) g ( x, τ ) d x. (11)In all the following and to simplify the notations, we will forget the t depen-dence in the stock price S t . α -stable process (FMLS process) L´evy-stable process [42,48] is a L´evy process whose L´evy-Khintchine triplethas the form ( a, , ν stable ), with: ν stable ( x ) = γ − | x | α { x< } + γ + x α { x> } , (12)where α ∈ (0 ,
2) and γ ± ∈ R . It is known that, introducing γ and β defined by γ α := − ( γ + + γ − ) Γ ( − α ) cos πα β := γ + − γ − γ + + γ − , (13)then for α ∈ (0 , ∪ (1 ,
2] the characteristic exponent of the process admitsthe parametrization: Ψ stable ( k ) = γ α | k | α (cid:16) − iβ tan απ k (cid:17) + iηk (14) Jean-Philippe Aguilar for some constant η ∈ R (see, for instance, exercise 1.4 in the textbook [32]).A L´evy-stable process can therefore be represented as a 4-parameter process L ( η, γ α , β ): α controls the behavior of the tails and β ∈ [ − ,
1] their asymme-try , γ is a scale parameter, and η is a location parameter. In particular, when α ∈ (1 ,
2] then it follows from (14) that η equals the mean E Q [ X t ].It is interesting to note that when α = 2 and η = 0 then the characteris-tic function (14) degenerates into the characteristic function of the centerednormal distribution: L (0 , σ , β ) = N (0 , ( σ √ ) ∀ β ∈ [ − , , (15)and therefore the Black-Scholes model is a particular case of a L´evy-stablemodel for α = 2.3.2 Fully asymmetric processIt follows from the definition of the L´evy measure (12), that the momentgenerating function M [ X t ]( p ) of a L´evy-stable process exists if and only if γ + = 0, or equivalently β = − fully asymmetric process, and the condition β = − maximal negative asymmetry hypothesis . In this context, choosing η = 0 (process with zero mean), we have: − Ψ stable ( − ip ) = γ − Z −∞ ( e px −
1) d x | x | α = γ − Γ ( − α ) p α = − γ α cos πα p α (16)which is valid for p >
0. It follows from definition (8) that the martingaleadjustment reads: µ = γ α cos πα . (17)It is in [14] that an exponential L´evy model (5) for a process { X t } t ≥ beinga spectrally negative L´evy-stable process L (0 , σ α , −
1) was first introduced forthe purpose of option pricing. The authors gave it the name of
Finite MomentLog Stable (FMLS) process, in reference to the existence of the cumulant gen-erating function in this case. Note that the process has infinite activity, theintegral of the stable measure being divergent in 0.3.3 Self-similarity and option pricingWe now derive a Mellin-Barnes representation for the density of the FMLSprocess, and for the corresponding option price that we will denote by C α . he value of power-related options under spectrally negative L´evy processes 7 Lemma 1
Let σ > , α ∈ (1 , and X t ∼ L (0 , σ α , − . Then the density g α ( x, t ) of the process { X t } t ≥ admits the following Mellin-Barnes represen-tation: g α ( x, t ) = 1 αx c + i ∞ Z c − i ∞ Γ (1 − s ) Γ (1 − s α ) (cid:18) x ( − µt ) α (cid:19) s d s iπ (18) where c < and µ = σ α cos πα .Proof Using eqs. (16) and (17) and the Laplace inversion formula, we have: g α ( x, t ) = 12 iπ c p + i ∞ Z c p − i ∞ e − px e − µtp α d p (19)where c p >
0. Taking the Mellin transform and making the change of variables p α → p , we have: g ∗ α ( s , t ) := ∞ Z g ( x, t ) x s − d x = 1 α Γ ( s ) 12 iπ c p + i ∞ Z c p − i ∞ e − µtp p − sα − d p (20)for any s >
0. The remaining p -integral is equal to Γ (1+ s − α ) ( − µt ) − sα on thecondition that s > − α (see for instance [6] or any monograph on Laplacetransform); observe that, as α ∈ (1 , s reduce to s > s → − s . ⊓⊔ Equation (18) shows that the density is a function of the ratio x ( − µt ) α ,which is actually a consequence of the self-similarity property [18] of stableprocesses (a scaling of time is equivalent to an appropriate scaling of space).This property allows for a nice factorization of the option price in the Mellinspace; indeed, let us denote G ∗ α ( s ) := 1 α Γ (1 − s ) Γ (1 − s α ) (21)and K ∗ ( s ) := + ∞ Z −∞ P (cid:16) Se ( r + µ ) τ + x , K (cid:17) x s − d x, (22)and let us assume that the integral (22) converges for Re ( s ) ∈ ( c − , c + ) forsome real numbers c − < c + . Then, as a direct consequence of the pricingformula (11) and of lemma 1, we have: Jean-Philippe Aguilar
Proposition 1 (Factorization in the Mellin space)
Let c ∈ (˜ c − , ˜ c + ) where (˜ c − , ˜ c + ) := ( c − , c + ) ∩ ( −∞ , − is assumed to be nonempty. Then, underthe hypothesis of lemma 1, the value at time t of an option with maturity T and payoff P ( S T , K ) is equal to: C α ( S, K, r, µ, τ ) = e − rτ c + i ∞ Z c − i ∞ K ∗ ( s ) G ∗ α ( s ) ( − µτ ) − s α d s iπ . (23)The factorized form (23) turns out to be a very practical tool for optionpricing. Indeed, as an integral along a vertical line in the complex plane, itcan be conveniently expressed as a sum of residues associated to the singu-larities of the integrand. As Gamma functions are involved, we can controlthe behavior of the integrand when the contour goes to infinity by using theStirling asymptotic formula for the Gamma function [1]: if a k , b k , c j , d j arereal numbers, if δ := P k a k − P j c j and if δ < (cid:12)(cid:12)(cid:12)(cid:12) Π k Γ ( a k s + b k ) Π j Γ ( c j s + d j ) (cid:12)(cid:12)(cid:12)(cid:12) | s |→∞ −→ s ∈ ( − π , π ), and the same holds for arg s ∈ ( π , π ) if δ > e − rτ × X h residues of K ∗ ( s ) G ∗ α ( s ) × powers of ( − µτ ) α i . (25)The only technical difficulty will in fact lie in the evaluation of K ∗ ( s ): de-pending on the payoff’s complexity, it can be either computed directly, or viathe introduction of a second Mellin complex variable s . α -stable environment In all this section, α ∈ (1 , σ > X t ∼ L (0 , σ α , −
1) and u >
0; the log-forward moneyness is defined to be: k u := log SK u + rτ (26)and we will use the standard notation X + := X { X> } .4.1 One complex variable payoffs Digital power options (cash-or-nothing).
The call’s payoff is: P ( C/N ) ( S, K ) := { S u − K> } . (27) he value of power-related options under spectrally negative L´evy processes 9 Proposition 2
The value at time t of a digital power cash-or-nothing calloption is: C ( C/N ) α ( S, K, r, µ, τ ) = e − rτ α ∞ X n =0 n ! Γ (cid:0) − nα (cid:1) ( k u + µτ ) n ( − µτ ) − nα . (28) Proof
As we can write: P ( C/N ) ( Se ( r + µ ) τ + x , K ) = { e u ( ku + µτ + x ) > } = { x> − k u − µτ } , (29)then, with the notation (22), the K ∗ ( s ) function reads: K ∗ ( s ) = − ( − k u − µτ ) s s (30)for s < −
1. Using proposition 1 and the functional relation Γ ( − s ) = − s Γ (1 − s ), the option price is: C ( C/N ) α ( S, K, r, µ, τ ) = e − rτ α c + i ∞ Z c − i ∞ Γ ( − s ) Γ (1 − s α ) ( − k u − µ τ ) s ( − µτ ) − s α d s iπ (31)which converges for s <
0. We can note that: δ = 1 α − α >
1, thus, it follows from the Stirling formula (24) thatthe analytic continuation of the integrand vanishes at infinity in the righthalf plane. Therefore, the integral (31) equals the sum of residues at the poleslocated in this half plane; these poles are induced by the Γ ( − s ) term at everypositive integer n , and the associated residues are:( − n n ! 1 Γ (1 − nα ) ( − k u − µτ ) n ( − µτ ) − nα . (33)Simplifying and summing all residues yields (28). ⊓⊔ Log power options.
These options were introduced in [47] in the case u = 1,and are basically options on the rate of return of the underlying asset. Thecall’s payoff is: P ( Log ) ( S, K ) := (cid:20) log (cid:18) S u K (cid:19)(cid:21) + (34) Proposition 3
The value at time t of a Log power call option is: C ( Log ) α ( S, K, r, µ, τ ) = ue − rτ α ∞ X n =0 n ! Γ (cid:0) − nα (cid:1) ( k u + µτ ) n ( − µτ ) − nα . (35) Proof
As we can write: P ( Log ) ( Se ( r + µ ) τ + x , K ) = u [ k u + µτ + x ] + , (36)then the K ∗ ( s ) function reads: K ∗ ( s ) = u ( − k u − µτ ) s s (1 + s ) (37)for s < −
1. Using proposition 1 and the functional relation Γ (1 − s ) = − s Γ ( − s ), the option price is: C ( Log ) α ( S, K, r, µ, τ ) = ue − rτ α c + i ∞ Z c − i ∞ − Γ ( − s )(1 + s ) Γ (1 − s α ) ( − k u − µ τ ) s ( − µτ ) − s α d s iπ (38)which converges for s < −
1. Again, δ <
0, and the analytic continuation ofthe integrand in the right half-plane has: – a simple pole in s = − − µτ ) α Γ (1 + α ) ; (39) – a series of poles at every positive integer s = n with residues: − ( − n ( n + 1)! 1 Γ (1 − nα ) ( − k u − µτ ) n ( − µτ ) − nα . (40)Summing the residues (39) and (40) for all n and re-ordering yields (35). ⊓⊔ Capped power options (cash-or-nothing).
For K − < K + , the call’s payoff is: P ( C/N,cap ) ( S, K + , K − ) := { K −
The value at time t of a capped cash-or-nothing call option is: C ( C/N,cap ) α ( S, K + , K − , r, µ, τ ) = e − rτ α ∞ X n =0 n ! Γ (cid:0) − nα (cid:1) (cid:0) ( k − u + µτ ) n − ( k + u + µτ ) n (cid:1) ( − µτ ) − nα . (42) he value of power-related options under spectrally negative L´evy processes 11 Proof
We can write: P ( C/N,cap ) ( Se ( r + µ ) τ + x , K + , K − ) = {− k − u − µτ The call’s payoff is: P ( A/N ) ( S, K ) := S u { S u − K> } (46) Proposition 5 The value at time t of a digital power asset-or-nothing calloption is: C ( A/N ) α ( S, K, r, µ, τ ) = Ke − rτ α ∞ X n =0 m =0 n ! Γ (cid:0) m − nα (cid:1) u m ( k u + µτ ) n ( − µτ ) m − nα . (47) Proof We can write: P ( A/N ) ( Se ( r + µ ) τ + x , K ) = Ke u ( k u + µτ + x ) { x> − k u − µτ } . (48)Introducing a Mellin-Barnes representation for the exponential term: e u ( k u + µτ + x ) = c + i ∞ Z c − i ∞ ( − − s u − s Γ ( s )( k u + µτ + x ) − s d s iπ (49)for c > x variable, the K ∗ ( s ) function reads: K ∗ ( s ) = K c + i ∞ Z c − i ∞ ( − − s u − s Γ ( s ) Γ (1 − s ) Γ ( − s + s ) Γ (1 − s ) ( − k u − µτ ) s − s d s iπ (50) and converges for ( s , s ) in the triangle { Re ( s ) ∈ (0 , , Re ( s ) < Re ( s ) } .From proposition 1, the option price is: C ( A/N ) α ( S, K, r, µ, τ ) = Ke − rτ α c + i ∞ Z c − i ∞ c + i ∞ Z c − i ∞ ( − − s Γ ( s ) Γ (1 − s ) Γ ( − s + s ) Γ (1 − s α ) u − s ( − k u − µτ ) s − s ( − µτ ) − s α d s d s (2 iπ ) . (51)Poles of the integrand occur when Γ ( s ) and Γ ( − s + s ) are singular; per-forming the change of variables − s + s → U , s → V allows to compute theassociated residues, which read:( − m ( − n n ! ( − m m ! Γ (1 + m ) Γ (1 + m − nα ) u m ( − k u − µτ ) n ( − µτ ) m − nα (52)Simplifying and summing the residues yields the series (47). The fact thatone can close the C contour in (51) is a consequence of the multidimensionalgeneralization of the Stirling estimate (24) (see [38] or the appendix of [2] fordetails). ⊓⊔ Gap power options. A gap option [44], also called gap risk swap , offers anonzero payoff on the condition that a trigger price is attained at t = T .More precisely, the call’s payoff is: P ( Gap ) ( S, K , K ) := ( S u − K ) { S u − K > } (53)where K is the strike price and K the trigger price; if the trigger is lowerthan the strike then a negative payoff is possible (which would not be the casewith a classical knock-in barrier). From the definition of the payoff (53), thevalue of the gap call option is equal to: C ( Gap ) α ( S, K , K , r, µ, τ ) = C ( A/N ) α ( S, K , r, µ, τ ) − K C ( C/N ) α ( S, K , r, µ, τ ) . (54) European power options. The classical European power option is a gap poweroption with equal strike and trigger prices ( K = K = K ); the payoff thereforereads P ( E ) ( S, K ) := [ S u − K ] + . (55)Observing that (28) is actually a particular case of (47) for m = 0, it followsimmediately from (54) that the value of the European power call is: C ( E ) α ( S, K, r, µ, τ ) = Ke − rτ α ∞ X n =0 m =1 n ! Γ (cid:0) m − nα (cid:1) u m ( k u + µτ ) n ( − µτ ) m − nα . (56) he value of power-related options under spectrally negative L´evy processes 13 When the asset is at-the-money (ATM) forward, that is when S = K u e − rt ,or, equivalently, k u = 0, then (56) becomes: C ( E,AT M ) α ( S, K, r, µ, τ ) = Ke − rτ α " u ( − µτ ) α Γ (1 + α ) − u ( − µτ ) + u ( − µτ ) α Γ (1 + α ) + O (cid:16) u ( − µτ ) α (cid:17) . (57)In particular, if we choose α = 2 and the normalization γ = σ √ in the definitionof the martingale adjustment (17), then (57) reads: C ( E,AT M )2 ( S, K, r, σ, τ ) = Ke − rτ (cid:20) u σ √ τ √ π − u (1 − u ) σ τ + O (cid:0) u ( σ √ τ ) (cid:1)(cid:21) = u → √ π Sσ √ τ + O (cid:0) ( σ √ τ ) (cid:1) (58)which is the well-known approximation for the ATM Black-Scholes call. Capped power options (asset-or-nothing, European). For K − < K + , the payoffof a capped power asset-or-nothing call is: P ( A/N,cap ) ( S, K + , K − ) := S u { K − The value at time t of a capped power asset-or-nothing calloption is: C ( A/N,cap ) α ( S, K + , K − , r, µ, τ ) = e − rτ α S u e u ( r + µ ) τ × ∞ X n =0 m =0 ( − u ) m (cid:0) ( k − u + µτ ) n + m − ( k + u + µτ ) n + m (cid:1) (1 + n + m ) n ! m ! Γ (cid:0) − nα (cid:1) ( − µτ ) − nα . (60)The value of the capped European power option is easily deduced from thevalues of the capped cash-or-nothing (42) and asset-or-nothing (60) options: C ( E/N,cap ) α ( S, K + , K − , r, µ, τ ) = C ( A/N,cap ) α ( S, K + , K − , r, µ, τ ) − K − C ( C/N,cap ) α ( S, K + , K − , r, µ, τ ) . (61)When K + → ∞ , the value of the capped option (61) coincides with the classi-cal uncapped option (56); this situation is displayed in figure 1. We can observethat the convergence to the uncapped price is quicker when α decreases, whichis no surprise given the overall α factor. Fig. 1 Convergence of capped European call to the uncapped price when the cap K + goesto infinity, for different tail index parameters α ; when α decreases, the European (uncapped)price grows higher, given the presence of a left fat tail as soon as α < 2. Parameters: strike K − = 4000 and horizon τ = 2 Y ; market parameters are set to S = 4200, r = 1% and σ = 1%. α = 2, u = 1 (i.e., in the Black-Scholes setup); we also provide comparisonswith numerical evaluation of Fourier integrals when α = 2. Except otherwisestated, we choose r = 1%, σ = 20%, K = 4000, τ = 2 years and we make thenormalization γ = σ √ in the martingale adjustment (17), so as to recover theBlack-Scholes adjustment − σ when α = 2. Log options When α = 2 and u = 1, a closed pricing formula exists for theLog option [25]: C ( Log )2 ( S, K, r, σ, τ ) = e − rτ σ √ τ [ n ( d ) + d N ( d )] , d k − σ τσ √ τ , (62)where k := k , n ( x ) = √ π e − x is the Gaussian density and N ( x ) is theNormal cumulative distribution function. In table 1, we compare this formulato various truncations of the series (35) for α = 2 and u = 1, in several marketsituations (out-of-the money, at-the-money and in-the-money). Power options ( α = 2 ) For u > 0, recall the formula by Heynen and Kat [26]for European power options in the Black-Scholes setup: C ( E )2 ( S, K, r, σ, τ ) = S u e ( u − r + u σ ) τ N ( d ) − Ke − rτ N ( d ) , (63) he value of power-related options under spectrally negative L´evy processes 15 Table 1 Log call ( P ( Log ) ( S, K ) = [log S u − log K ] + ): comparisons between the series (35)truncated at n = n max and the closed formula (62) in the case α = 2, u = 1. We observethat very few terms are needed to obtain an excellent degree of precision, even in deeply outor in the money situations. n max = 3 n max = 5 n max = 10 Formula (62) S = 5000 0.238691 0.237465 0.237525 0.237525 S = 4200 0.125287 0.125286 0.125286 0.125286ATM 0.092106 0.092104 0.092104 0.092104 S = 3800 0.079177 0.079158 0.079158 0.079158 S = 3000 0.025250 0.018797 0.019488 0.019487 where d := k u + ( u − ) σ τσ √ τ , d := d − uσ √ τ . (64)In table 2, values obtained with formula (63) are compared to various trunca-tions of the series (56), for various powers u > k u grows when u = 1(for instance, if S = 4500, k = 0 . 14 but k . = 2 . 90 and k = 5 . k µ + τ ) n in the numerator are less quickly neutralizedby the factorial/Gamma terms of the denominator. Table 2 European power call ( P ( E ) ( S, K ) = [ S u − K ] + ): comparisons between the series(56) for the truncated at n max = m max := max , and the values obtained via the formula(63), for α = 2 and various positive powers u . max = 3 max = 5 max = 10 Heynen & Kat (62) u = 1 439.65 440.93 440.94 440.94 u = 1 . u = 2 1057.71 1080.49 1081.64 1081.64 u = 3 1908.17 2034.41 2049.37 2049.39 European options ( α = 2 ) As a consequence of the Gil-Pelaez inversion for-mula for the characteristic functions, the price of an European call can bedecomposed into a sum of Arrow-Debreu securities of the form (see detailse.g. in [5]): C ( E ) α ( S, K, r, µ, τ ) = SΠ − Ke − rτ Π . (65)The price of each security can be expressed in terms of the stable characteristicfunction and of the log-forward moneyness: Π = 12 + 1 π ∞ Z Re (cid:20) e iuk Φ ( u − i, τ ) iu (cid:21) d u (66) and Π = 12 + 1 π ∞ Z Re (cid:20) e iuk Φ ( u, τ ) iu (cid:21) d u, (67)where Φ ( u, t ) is the risk-neutral characteristic function Φ ( u, t ) := e iµut e − tΨ stable ( u ) (68)satisfying the martingale condition Φ ( − i, t ) = 1. Given the simple form ofthe stable characteristic exponent (14), the integrals in (66) and (67) can becarried out very easily via a classical recursive algorithm on the truncatedintegration region (typically, u ∈ [0 , − ). In table 3, we compare the values obtained with this method withseveral truncations of the series (56), for a tail-index α = 1 . Table 3 European call ( P ( E ) ( S, K ) = [ S u − K ] + ): comparisons between the series (56)truncated at n max = m max := max , and the values obtained by the Gil-Pelaez method(65), in the case α = 1 . u = 1. The convergence is very fast, in particular for ITM longterm options. max = 3 max = 10 max = 20 max = 30 Gil-Pelaez (65) Long term options ( τ = 2) S = 5000 1302.92 1309.86 1309.86 1309.86 1309.86 S = 4200 679.32 681.56 681.56 681.56 681.56ATM 496.87 498.07 498.07 498.07 498.07 S = 3800 425.76 426.44 426.44 426.44 426.44 S = 3000 128.50 92.46 96.50 96.50 96.50 Short term options ( τ = 0 . S = 5000 1089.70 1075.64 1075.63 1075.63 1075.63 S = 4200 383.17 383.30 383.30 383.30 383.30ATM 230.47 203.49 203.49 203.49 203.49 S = 3800 143.53 143.09 143.09 143.09 143.09 S = 3000 211.44 -27.24 1.04 1.39 1.39 Like before, the convergence is very fast, and goes even faster in the ITMregion; this is because the log-forward moneyness (26) is positive in this zone,and therefore, as µ < 0, ( k µ + µτ ) is closer to 0 than in the OTM zone, whichaccelerates the convergence of the series (56). This situation is displayed infigure 2. Tempered stable L´evy processes, which are known in Physics as truncated L´evyflights , combine α -stable and Gaussian trends, and are an alternative solution he value of power-related options under spectrally negative L´evy processes 17 Fig. 2 Convergence of partial sums of the series (56) to the option price ( α = 1 . S ∈ (3000 , n max = m max = 5 to obtain an excellent level of precision. to achieve finite moments (see details and further references in [41]). TheirL´evy-Khintchine triplet has the form ( a, , ν T S ) where ν T S ( x ) = γ − e − λ − | x | | x | α { x< } + γ + e − λ + x x α { x> } (69)for γ ± , λ ± ≥ < α ± < 2. When γ − = γ + and α − = α + , we recover the CGMY process [13] (sometimes named classical tempered stable process ) andwhen furthermore α − = α + = 0, the Variance Gamma process [34]. In the casewhere λ ± = 0, there is no more tempering and the process is simply a L´evy-stable process like in section 3. When α ± ∈ (0 , ∪ (1 , p ∈ ( − λ − , λ + ) onehas φ ( p ) = ηp + γ − Γ ( − α − )( − λ α − − +( λ − + p ) α − ) + γ + Γ ( − α + )( − λ α + + +( λ + − p ) α + )(70)where η is a constant depending on the drift a and the choice of truncationfunction for the characteristic function of the process; without loss of generalitywe choose it to be equal to 0.5.1 Tempered stable densitiesLet us denote by ν − T S ( x ) (resp. ν + T S ( x )) the negative (resp. positive) part ofthe L´evy measure (69), and by T S ± ( γ ± , λ ± , α ± ) the associated one-sided tem-pered stable processes. Lemma 2 Let α ± ∈ (1 , and µ ± := − γ ± Γ ( − α ± ) . (i) If X t ∼ T S − ( γ − , λ − , α − ) , then its density g − ( x, t ) admits the Mellin-Barnes representation: g − ( x, t ) = e λ α −− µ − t + λ − x α − x c + i ∞ Z c − i ∞ Γ (1 − s ) Γ (1 − s α − ) x ( − µ − t ) α − ! s d s iπ (71) for any c < ;(ii) If X t ∼ T S + ( γ + , λ + , α +) , then its density g + ( x, t ) admits the Mellin-Barnes representation: g + ( x, t ) = − e λ + α + µ + t − λ + x α + x c + i ∞ Z c − i ∞ Γ (1 − s ) Γ (1 − s α + ) − x ( − µ + t ) α + ! s d s iπ (72) for any c < .Proof It follows from (70) and from the Laplace inversion formula that: g − ( x, t ) = e λ α −− µ − t c p + i ∞ Z c p − i ∞ e − px e − µ − t ( λ − + p ) α − d p iπ (73)for c p > 0. From the frequency shifting property of the Laplace transform, wecan write: g − ( x, t ) = e λ α −− µ − t e λ − x g α − ( x, t ) (74)where g α − ( x, t ) is the stable density (18), and (i) is proved. A similar approachcan be used to prove (ii). ⊓⊔ α − ∈ (1 , 2) and X t ∼ T S − ( γ − , λ − , α − ); from definition (8) and theLaplace exponent (70), the martingale adjustment reads: µ = (cid:0) ( λ − + 1) α − − λ α − − (cid:1) µ − (75)where µ − = − γ − Γ ( − α − ) corresponds to the FMLS martingale adjustment(16), and as expected µ → µ − when λ − → 0. From the pricing formula (11)and using the notation (21) and (22), the value at time t of an option withmaturity T and payoff P ( S T , K ) is equal to: C α − ,λ − ( S, K, r, µ − , τ ) = e − ( r − λ α −− µ − ) τ c + i ∞ Z c − i ∞ K ∗ λ − ( s ) G ∗ α − ( s ) ( − µ − τ ) − s α − d s iπ (76) he value of power-related options under spectrally negative L´evy processes 19 where we have defined K ∗ λ − ( s ) := + ∞ Z −∞ e λ − x P (cid:16) Se ( r + µ ) τ + x , K (cid:17) x s − d x, (77)and where µ is given by (75). The K ∗ λ − ( s ) function can be expressed in termsof the K ∗ ( s ) function (22) by introducing a Mellin-Barnes representation forthe exponential term: K ∗ λ − ( s ) = c + i ∞ Z c − i ∞ ( − − s λ − s − Γ ( s ) K ∗ ( s − s ) d s iπ (78)for c > 0, and therefore, replacing in (76), we obtain: Proposition 7 (Factorization) If X t ∼ T S − ( γ − , λ − , α − ) and if α − ∈ (1 , ,then the value at time t of an option with maturity T and payoff P ( S T , K ) isequal to C α − ,λ − ( S, K, r, µ − , τ ) = e − ( r − λ α −− µ − ) τ × c + i ∞ Z c − i ∞ c + i ∞ Z c − i ∞ ( − − s λ − s − Γ ( s ) K ∗ ( s − s ) G ∗ α − ( s ) ( − µ − τ ) − s α − d s d s (2 iπ ) . (79) Example: digital power option (cash-or-nothing) In that case, we know from(30) that: K ∗ ( s − s ) = − ( − k u − ρ − µ − τ ) s − s s − s (80)where ρ − := (cid:0) ( λ − + 1) α − − λ α − − (cid:1) , and therefore it follows from (79) that thedigital cash-or-nothing call reads: C ( C/N ) α − ,λ − ( S, K, r, µ − , τ ) = 1 α − e − ( r − λ α −− µ − ) τ c + i ∞ Z c − i ∞ c + i ∞ Z c − i ∞ ( − − s × Γ (1 − s ) Γ ( s )( s − s ) Γ (1 − s α − ) λ − s ( − k u − ρ − µ − τ ) s − s ( − µ − τ ) − s α − d s d s (2 iπ ) . (81)The double integral (81) has a simple pole in (0 , 0) with residue 1, and a seriesof simple poles in (1 + n, m ), n, m ∈ N induced by the singularities of the Γ (1 − s ) and Γ ( s ) functions. Summing all these residues yields: C ( C/N ) α − ,λ − ( S, K, r, µ − , τ ) = e − ( r − λ α −− µ − ) τ α − [1+ ∞ X n =0 m =0 ( − λ − ) m (1 + n + m ) n ! m ! Γ (1 − nα − ) ( k u + ρ − µ − τ ) n + m ( − µ − τ ) − nα − . (82) Fig. 3 At the money stable (28) and tempered stable (82) prices, and linear approximation(84); the tempered stable price intercepts the stable price when λ − = 0. Note that when λ − = 0, only the terms for m = 0 survive and (82) degeneratesinto the α -stable price (28), as expected. In the ATM forward case ( k u =0),the first few terms of the series (82) read: e − ( r − λ α −− µ − ) τ α − " − ρ − Γ (1 − α − ) ( − µ − τ ) − α − + O (cid:16) ( − µ − τ ) − α − (cid:17) (83)and can be Taylor-expanded for small λ − : e − rτ α − " − ( − µ − τ ) − α − Γ (1 − α − ) − α − ( − µ − τ ) − α − Γ (1 − α − ) λ − + O (cid:0) λ α − − (cid:1) . (84)In the linear approximation (84), the intercept is the stable price, while theslope is governed by the negative left tail parameter − α − ; the tempered stableprice is therefore lower than the stable price (which is due to the temperingof the heavy tail), and the difference increases when α grows. This situationis displayed on fig. 3 for α = 1 . K = 4000, r = 1%, σ = 20%, τ = 2 Y , theseries (28) and (82) being truncated to n max = m max = 10. In this article, we have derived generic representations in the Mellin space forpath-independent options with arbitrary payoff, in the setup of exponentialL´evy models driven by spectrally negative stable or tempered stable processes.These representations have allowed us to obtain simple series expansions forthe price of options with an exotic power-related payoff (Power Digital, Logor Gap Power options, Capped Power European options), by means of residuesummation in C or C . These series contain only simple terms and convergevery fast, in particular when calls are in-the-money and for longer maturities; he value of power-related options under spectrally negative L´evy processes 21 they can be very easily used for practical evaluation without requiring anyhelp from numerical schemes.Future work will include the investigation of path-dependent options, likeBarrier or Lookback options; spectrally negative α -stable processes are par-ticularly interesting in this context, because the law of the supremum on aperiod of time is known to be [8]: P " sup t ∈ [0 ,T ] X t ≥ x = α P [ X T ≥ x ] (85)which generalizes the reflection principle for the Wiener process ( α = 2).Regarding path-independent instruments, the extension of the Mellin residuetechnique to two-sided L´evy processes is currently in progress, with a partic-ular focus on the Variance Gamma and the Normal Inverse Gamma (NIG)processes; the technique is very well adapted to these models too, because,like in the spectrally negative case, their density functions can be expressedunder the form of Mellin integrals. 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