A series representation for the Black-Scholes formula
PPaper • November 2, 2017
A series representation for theBlack-Scholes formula
Jean-Philippe Aguilar
BRED Banque Populaire, Modeling Department, 18 quai de la Râpée, Paris - 75012 [email protected] ontents
I Introduction 1II Pricing formula 2
I The Green function approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3II The call price as a complex integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3III Residue summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5IV Brenner-Subrahmanyam and refinement . . . . . . . . . . . . . . . . . . . . . . . . . 7
III Numerical tests 8
I Quickness of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8II Comparisons between the series and the BS formula . . . . . . . . . . . . . . . . . . 9
IV Concluding remarks 9A APPENDIX: Mellin transforms and residues 10
I One-dimensional Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10II Multidimensional Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Abstract
We prove and test an efficient series representation for the European Black-Scholes call (formula (38) ),which generalizes and refines previously known approximations, and works in every market configuration.
Key words—
Option pricing, Black-Scholes approximation, Mellin transform, Multidimensionalcomplex analysis
I. I ntroduction
The celebrated Black-Scholes formula [5] V ( S , K , r , σ , τ ) = SN ( d + ) − Ke − r τ N ( d − ) d ± = σ √ τ (cid:18) log SK + r τ (cid:19) ± σ √ τ (1)where N ( . ) is the normal distribution function, is a popular tool for the pricing of Europeanoptions of strike K and time-to-maturity τ = T − t , where the market conditions are described byan underlying (spot) asset price S , volatility σ and risk-free interest rate r .It is an interesting question to know whether the Black-Scholes formula can be approximated by a r X i v : . [ q -f i n . P R ] O c t I PRICING FORMULA S (cid:39) Ke − r τ (2)then there exists an approximation (see the paper by Brenner and Subrahmanyam [4], cited by[14]): V ( S , K , r , σ , τ ) (cid:39) S σ √ τ (3)This approximation can be useful in certain situations but has some major drawbacks:- It is not very precise: for instance when K = r = σ = τ = Y then theapproximation (3) yields V ( × e − × = ) (cid:39) × × √ = V ( ) = SK . But, as alreadynoticed by Estrella in [7], the Black-Scholes formula mixes two components of strongly differentnatures: the logarithmic function, possessing an expansion converging very fast but only fora small range of arguments, and the normal distribution, whose expansion (in odd powers ofthe arguments) converges on the whole real axis but with fewer level of accuracy. As shown byEstrella, this leads to situations where, for a plausible range of parameter values, the Taylor seriesfor (1) diverges.The main motivation of this note is therefore to derive a simple, efficient and compact approxi-mation formula for the Black-Scholes formula (1), that works in every market situation and canbe made as more precise as one wishes; such a formula is obtained under the form of a fastconverging series (38). This series, whose terms are very simple and straightforward to compute,converges as quickly as σ √ τ is small, and, in any case, is absolutely convergent.The paper is organized as follows: first we derive the series formula for the pricing of the Europeancall. The starting point is the Green representation for the solution of the Black-Scholes PDE; theproof uses tools from complex analysis in C . We discuss our formula in the ATM-forward case,and show that is also constitutes a refinement of the Brenner-Subrahmanyam approximation inthis case. Then we demonstrate the quickness of convergence of the series and show that theresults are extremely close to the Black-Scholes formula after only a few iterations, in any marketsituation (out of, at or in the money). At the end of the paper, we make a short review of complexanalysis in one and several dimensions, so that the reader who would be unfamiliar with theseconcepts can find them conveniently presented, and be provided with some classic references inthe literature. II. P ricing formula
The Black-Scholes model is a Gaussian model into which the underlying asset price S is assumedto be described by a geometric Brownian motion of drift r and volatility σ . From standard The Green function approach V ( S , K , r , σ , t ) of anEuropean call option of strike K and maturity T satisfies the Black-Scholes equation, which is apartial differential equation (PDE) with terminal condition [5]: ∂ V ∂ t + σ S ∂ V ∂ S + rS ∂ V ∂ S − rV = t ∈ [ T ] V ( S , K , r , σ , t = T ) = [ S − K ] + (6) I. The Green function approach
It is known (see the classic textbook by Wilmott [14] for instance) that, with the change of variables x : = log S + ( r − σ ) ττ : = T − tV ( S , K , r , σ , t ) : = e − r τ W ( x , K , r , σ , τ ) (7)then the Black-Scholes PDE (6) resumes to the diffusion (or heat) equation ∂ W ∂τ − σ ∂ W ∂ x = g ( x , K , r , σ , τ ) : = σ √ πτ e − x σ τ (9)In this new set of variables the terminal condition becomes an initial condition: W ( x , K , r , σ , τ = ) = [ e x − K ] + (10)and therefore, by the method of Green functions, we know that we can express W as W ( x , K , r , σ , τ ) = + ∞ (cid:90) − ∞ [ e x + y − K ] + g ( y , K , r , σ , τ ) d y (11)Turning back to the initial variables (we keep the notation for time to maturity τ ): V ( S , K , r , σ , τ ) = e − r τ + ∞ (cid:90) − ∞ [ Se ( r − σ ) τ + y − K ] + σ √ πτ e − y σ τ d y (12) II. The call price as a complex integral
Let us introduce the notations z : = σ √ τ , [ log ] : = log SK + r τ (13)Then we can write: [ Se ( r − σ ) τ + y − K ] + = K [ e [ log ] − z + y − ] + (14) I The call price as a complex integral V ( S , K , r , σ , τ ) = Ke − r τ √ π ∞ (cid:90) z − [ log ] ( e [ log ] − z + y − ) z e − y z d y (15)Let us introduce a Mellin-Barnes representation for the heat kernel-term in (15) (see (50) inAppendix, and [8, 6] or any monograph on integral transforms):1 z e − y z = z c + i ∞ (cid:90) c − i ∞ Γ ( t ) (cid:18) y z (cid:19) − t d t i π ( c > ) (16)We thus have: V ( S , K , r , σ , τ ) = Ke − r τ √ π c + i ∞ (cid:90) c − i ∞ t Γ ( t ) ∞ (cid:90) z − [ log ] ( e [ log ] − z + y − ) y − t d y z t − d t i π (17)Integrating by parts in the y -integral yields: V ( S , K , r , σ , τ ) = Ke − r τ √ π c + i ∞ (cid:90) c − i ∞ t Γ ( t ) t − ∞ (cid:90) z − [ log ] e [ log ] − z + y y − t d y z t − d t i π (18)Let us introduce another Mellin-Barnes representation for the remaining exponential term (again,see (50) in Appendix): e [ log ] − y + y = c + i ∞ (cid:90) c − i ∞ ( − ) − t Γ ( t ) (cid:18) [ log ] − z + y (cid:19) − t d t i π ( c > ) (19)Therefore the call price is: V ( S , K , r , σ , τ ) = Ke − r τ √ π × c + i ∞ (cid:90) c − i ∞ c + i ∞ (cid:90) c − i ∞ ( − ) − t t t − Γ ( t ) Γ ( t ) ∞ (cid:90) z − [ log ] y − t (cid:18) [ log ] − z + y (cid:19) − t d y z t − d t i π ∧ d t i π (20)The y -integral is a particular case of Bêta-integral [1] and equals: ∞ (cid:90) z − [ log ] y − t (cid:18) [ log ] − z + y (cid:19) − t d y = (cid:18) z − [ log ] (cid:19) − t − t Γ ( − t ) Γ ( − + t + t ) Γ ( t − ) (21)and converges on the conditions Re ( t ) < Re ( t + t ) >
2; plugging into (20), using theGamma function functional relation ( t − ) Γ ( t − ) = Γ ( t ) and the Legendre duplicationformula for the Gamma function [1]: Γ ( t ) Γ ( t ) = − t √ π Γ ( t + ) (22) II Residue summation V ( S , K , r , σ , τ ) = Ke − r τ × c + i ∞ (cid:90) c − i ∞ c + i ∞ (cid:90) c − i ∞ ( − ) − t − t Γ ( t ) Γ ( − t ) Γ ( − + t + t ) Γ ( t + ) (cid:18) z − [ log ] (cid:19) − t − t z t − d t i π ∧ d t i π (23) III. Residue summation
Let us introduce the notations t : = (cid:20) t t (cid:21) c : = (cid:20) c c (cid:21) d t : = d t ∧ d t (24)and the complex differential 2-form ω = ( − ) − t − t Γ ( t ) Γ ( − t ) Γ ( − + t + t ) Γ ( t + ) (cid:18) z − [ log ] (cid:19) − t − t z t − d t ( i π ) (25)then we can write the call price under the form: V ( S , K , r , σ , τ ) = Ke − r τ (cid:90) c + i R ω (26)where c is located in the polyhedra of convergence of the double Mellin-Barne integral (23) P = { t ∈ C , Re ( t + t ) > Re ( t ) > Re ( t ) < } (27)This complex integral can be performed by means of summation of C -residues, by virtue of amultidimensional analogue to the residue theorem valid for this specific class of integrals (see(66) and references in Appendix). This configuration is shown on fig. 1. Indeed, the characteristicquantity (see definition (58) in appendix) associated to the differential form (25) is: ∆ = (cid:20) − − + (cid:21) = (cid:20) (cid:21) (28)and the admissible half-plane is Π ∆ : = (cid:110) t ∈ C , Re ( ∆ . t ) < ∆ . c (cid:111) (29)in the sense of the euclidean scalar product. This half-plane is therefore located under the line t = − t + c + c (30)In this half-plane, the cone Π as shown of fig. 1 and defined by Π : = (cid:110) t ∈ C , Re ( t ) ≤ Re ( t + t ) ≤ (cid:111) (31)contains and is compatible with the two family of divisors D = (cid:110) t ∈ C , − + t + t = − n , n ∈ N (cid:111) D = (cid:110) t ∈ C , t = − n , n ∈ N (cid:111) (32) II Residue summation D (oblique lines) are induced by the Γ ( − + t + t ) term, and D (horizontal lines) by the Γ ( t ) term. The intersection set D ∩ D (the dots), located in thecompatible green cone Π , gives birth to residues whose sum in the whole cone equals the integral(26).induced by Γ ( − + t + t ) and Γ ( t ) respectively. To compute the residues associated to everyelement of the singular set D ∩ D , we change the variables: (cid:40) u : = − + t + t u : = t −→ t = ( + u − u ) t = u dt ∧ dt = du ∧ du (33)so that in this new configuration ω reads ω = ( − ) − u u − u − Γ ( u ) Γ ( − u ) Γ ( u ) Γ ( + u − u + ) (cid:18) z − [ log ] (cid:19) − u z u − u + d u i π ∧ d u i π (34)With this new variables, the divisors D and D are induced by the Γ ( u ) and Γ ( u ) functionsin ( u , u ) = ( − n , − m ) , n , m ∈ N . From the singular behavior of the Gamma function around asingularity (48), we can write ω ∼ ( u , u ) → ( − n , − m ) ( − ) − u ( − ) n + m n ! m ! 2 u − u − Γ ( − u ) Γ ( + u − u + ) (cid:18) z − [ log ] (cid:19) − u z u − u + d u i π ( u + n ) ∧ d u i π ( u + m ) (35)and therefore the residues are, by the Cauchy formula:Res ( u = − n , u = − m ) = ( − ) n n − m − − n ! Γ ( + m − n + ) (cid:18) z − [ log ] (cid:19) n z m − n + (36) V Brenner-Subrahmanyam and refinement V ( S , K , r , σ , τ ) = Ke − r τ ∞ ∑ n , m = ( − ) n n − m − − n ! Γ ( + m − n + ) (cid:18) z − [ log ] (cid:19) n z m − n + (37)We can further simplify by changing the index m → m + Z : = z √ = σ √ τ √ , andwe finally obtain:Let [ log ] : = log SK + r τ and Z : = σ √ τ √ , then the Black-Scholes price of the European call is givenby the absolutely convergent double series: V ( S , K , r , σ , τ ) = Ke − r τ ∞ ∑ n = m = ( − ) n n ! Γ ( + m − n ) (cid:16) Z − [ log ] (cid:17) n Z m − n (38) IV. Brenner-Subrahmanyam and refinement
By defintion of [ log ] , the ATM forward configuration implies: S = Ke − r τ = ⇒ [ log ] = V ( S , K , r , σ , τ ) = S ∞ ∑ n = m = ( − ) n n ! Γ ( + m − n ) Z n + m (40)This series is now a series of positive powers of Z . It starts for n = m = V ( S , K , r , σ , τ ) = S Γ ( ) Z + O ( Z ) (41)Recalling that Z = σ √ τ √ and that Γ ( ) = √ π [1], we get V ( S , K , r , σ , τ ) = S √ π σ √ τ + O ( σ τ ) (42)As: 1 √ π (cid:39) (cid:39) V ( S , K , r , σ , τ ) = S σ √ τ + O ( σ τ ) (44)Note that the series (40) is indeed a refinement of the Brenner-Subrahmanyam approximation. Inthis precise market situation, the call price can therefore be expressed as a power series of σ √ τ ,which is not the case in the general case as shown in (38), where negative powers of σ √ τ andpowers of [ log ] arise. II NUMERICAL TESTS III. N umerical tests
I. Quickness of convergence
It is interesting to note that in the series (38), only the very first few terms contribute and alreadygive an excellent approximation to the option price. See for instance Tab. 1, where the set ofparameters used is S = K = r = σ = τ =
1. The convergence is a little bit lessfast when τ grows, but remains very efficient, see for instance Table 2. In Fig. 2 we compute thepartial terms of the series (38) as a series of m , that is, the sum of vertical columns in tables 1 and 2,for a time to maturity τ being 1 or 5 years. We observe that, in both cases, the convergence is fast. τ =1Y Table 1: Table containing the numerical values for the ( n , m ) -term in the series (38) for the optionprice ( S = K = r = σ = τ = Y ). The call price converges to a precision of10 − after summing only very few terms of the series. τ =5Y Table 2: Same set of parameters as in Table 1, but for a longer maturity τ = 5 years. The convergenceis slightly slower but remains very efficient.Figure 2: Convergence of partial terms of the series (38) to the call price, for short and long timesto maturity (1Y and 5Y). I Comparisons between the series and the BS formula II. Comparisons between the series and the BS formula
In the following array, we compare the results obtained via an application of the Black-Scholesformula and via the expansion (38) truncated at n = m =
20. We show that for every time tomaturity and every market situation (in, at or out of the money), this truncation is sufficientto agree with the Black-Scholes formula at a level of precision of 10 − . Other parameters are K = r = σ = Out of the money (S=3800) τ = Y τ = Y τ = Y τ = Y τ = Y Black-Scholes Formula 235.5135954 376.3907685 488.2564760 584.4538077 670.3385381Series (38), 20 (n,m)-iterations 235.5135954 376.3907685 488.2564760 584.4538077 670.3385381
At the money (S=400) τ = Y τ = Y τ = Y τ = Y τ = Y Black-Scholes Formula 337.3327476 486.1060719 603.0304375 703.1314722 792.2680293Series (38), 20 (n,m)-iterations 337.3327476 486.1060719 603.0304375 703.1314722 792.2680293
In the money (S=4200) τ = Y τ = Y τ = Y τ = Y τ = Y Black-Scholes Formula 458.7930654 609.5901660 728.9304673 831.3409473 922.6298574Series (38), 20 (n,m)-iterations 458.7930654 609.5901660 728.9304673 731.3409473 922.6298574
IV. C oncluding remarks
To the author’s knowledge, formula (38) is new. It can be efficiently used in practice as analternative of the Black-Scholes formula, for instance in an excel sheet or a Mathematica file.More interesting, the technique used for obtaining this result applies to a wider class of problems.Indeed, as soon as one is able to write the option price under the form of an integral over a Greenvariable y : V = (cid:90) Payo f f × Green f unction d y (45)and that one can write a suitable Mellin-Barnes representation for the Green function, then thetechnology applies. This is notably the case for the class of Lévy processes (of which Black-Scholesis a special case), where the Green functions are all known under the form of Mellin-Barnesrepresentations (see for instance [11], [10] and references therein). This work is currently inprogress [3] . APPENDIX: MELLIN TRANSFORMS AND RESIDUES A. APPENDIX: M ellin transforms and residues
We briefly present here some of the concepts used in the paper. The theory of the one-dimensionalMellin transform is explained in full detail in [8]. An introduction to multidimensional complexanalysis can be found in the classic textbook [9], and applications to the specific case of Mellin-Barnes integrals is developped in [12, 13, 2].
I. One-dimensional Mellin transforms
1. The Mellin transform of a locally continuous function f defined on R + is the function f ∗ defined by f ∗ ( s ) : = ∞ (cid:90) f ( x ) x s − d x (46)The region of convergence { α < Re ( s ) < β } into which the integral (46) converges is often calledthe fundamental strip of the transform, and sometimes denoted < α , β > .2. The Mellin transform of the exponential function is, by definition, the Euler Gamma function: Γ ( s ) = ∞ (cid:90) e − x x s − d x (47)with strip of convergence { Re ( s ) > } . Outside of this strip, it can be analytically continued,expect at every negative s = − n integer where it admits the singular behavior Γ ( s ) ∼ s →− n ( − ) n n ! 1 s + n n ∈ N (48)3. The inversion of the Mellin transform is performed via an integral along any vertical line in thestrip of convergence: f ( x ) = c + i ∞ (cid:90) c − i ∞ f ∗ ( s ) x − s d s i π c ∈ ( α , β ) (49)and notably for the exponential function one gets the so-called Cahen-Mellin integral : e − x = c + i ∞ (cid:90) c − i ∞ Γ ( s ) x − s d s i π c > f ∗ ( s ) is a ratio of products of Gamma functions of linear arguments: f ∗ ( s ) = Γ ( a s + b ) . . . Γ ( a n s + b n ) Γ ( c s + d ) . . . Γ ( c m s + d m ) (51)then one speaks of a Mellin-Barnes integral , whose characteristic quantity is defined to be ∆ = n ∑ k = a k − m ∑ j = c j (52) I Multidimensional Mellin transforms ∆ governs the behavior of f ∗ ( s ) when | s | → ∞ and thus the possibility of computing (49) bysumming the residues of the analytic continuation of f ∗ ( s ) right or left of the convergence strip: ∆ < f ( x ) = − ∑ Re ( s N ) > β Res S N f ∗ ( s ) x − s ∆ > f ( x ) = ∑ Re ( s N ) < α Res S N f ∗ ( s ) x − s (53)For instance, in the case of the Cahen-Mellin integral one has ∆ = e − x = ∑ Re ( s n ) < Res s n Γ ( s ) x − s = ∞ ∑ n = ( − ) n n ! x n (54)as expected from the usual Taylor series of the exponential function. II. Multidimensional Mellin transforms
1. Let the a k , c j , be vectors in C ,and the b k , d j be complex numbers. Let t : = (cid:20) t t (cid:21) and c : = (cid:20) c c (cid:21) in C and "." represent the euclidean scalar product. We speak of a Mellin-Barnes integral in C when one deals with an integral of the type (cid:90) c + i R ω (55)where ω is a complex differential 2-form who reads ω = Γ ( a . t + b ) . . . Γ ( a n . t n + b n ) Γ ( c . t + d ) . . . Γ ( c m . t m + b m ) x − t y − t d t i π ∧ d t i π x , y ∈ R (56)The singular sets induced by the singularities of the Gamma functions D k : = { t ∈ C , a k . t k + b k = − n k , n k ∈ N } k = n (57)are called the divisors of ω . The characteristic vector of ω is defined to be ∆ = n ∑ k = a k − m ∑ j = c j (58)and the admissible half-plane : Π ∆ : = { t ∈ C , ∆ . t < ∆ . c } (59)2. Let the ρ k in R , the h k : C → C be linear aplications and Π k be a subset of C of the type Π k : = { t ∈ C , Re ( h k ( t k )) < ρ k } (60)A cone in C is a cartesian product Π = Π × Π (61)where Π and Π are of the type (60). Its faces ϕ k are ϕ k : = ∂ Π k k =
1, 2 (62)
EFERENCES distinguished boundary , or vertex is ∂ Π : = ϕ ∩ ϕ (63)3. Let 1 < n < n . We group the divisors D = ∪ nk = D k of the complex differential form ω intotwo sub-families D : = ∪ n k = D k D : = ∪ nk = n + D k D = D ∪ D (64)We say that a cone Π ⊂ C is compatible with the divisors family D if:- Its distinguished boundary is c ;- Every divisor D and D intersect at most one of his faces: D k ∩ ϕ k = ∅ k =
1, 2 (65)4. Residue theorem for multidimensional Mellin-Barnes integral [12, 13]: If ∆ (cid:54) = Π ⊂ Π ∆ is a compatible cone located into the admissible half-plane, then (cid:90) c + i R ω = ∑ t ∈ Π ∩ ( D ∩ D ) Res t ω (66)and the series converges absolutely. The residues are to be understood as the "natural" generaliza-tion of the Cauchy residue, that is:Res (cid:34) f ( t , t ) d t i π t n ∧ d t i π t n (cid:35) = ( n − ) ! ( n − ) ! ∂ n + n − ∂ t n − ∂ t n − f ( t , t ) | t = t = (67) R eferences [1] Abramowitz, M. and Stegun, I., Handbook of mathematical functions, 1972, Dover Publica-tions[2] Aguilar, J.-Ph., Représentation de Mellin-Barnes et anomalie magnétique du muon, 2008, PhDThesis CPT-CNRS Marseille[3] Aguilar, J.-Ph., Coste, C., and Korbel, J., Non-Gaussian Analytic option Pricing: a closedformula for the Lévy-stable model, arXiv:1609.00987[4] Brenner, M. and Subrahmanyam, M.G., A simple approach to option valuation and hedgingin the Black-Scholes model, 1994, Financial Analysts Journal, [5] Black, F. and Scholes, M., The pricing of options and corporate liabilites, 1973, Journal ofPolitical Economy, , 637[6] Erdélyi, O. F., Magnus, W., Tricomi, G., Table of integral transforms, 1954, McGraw & Hill[7] Estrella, A., Taylor, Black and Scholes: series approximations and risk management pitfalls,1995, Federal Reserve Bank of New-York, Research paper EFERENCES , 3–58[9] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, 1978, Wiley & sons[10] Kleinert, H., Korbel, J., Option pricing beyond Black-Scholes based on double fractionaldiffusion, Physica A, 449, (2016) pp. 200 – 214[11] Mainardi, F, Pagnini, G. and Saxena, R., Fox H-functions in fractional diffusions, Journal ofcomputational and applied mathematics, 2005, , 321-331[12] Passare, M., Tsikh, A., Zhdanov, O., A multidimensional Jordan residue lemma with anapplication to Mellin-Barnes integrals, 1994, Aspects of Mathematics,
E26 , 233-241[13] Passare, M., Tsikh, A., Zhdanov, O., Multiple Mellin-Barnes integrals as periods of Calabi-Yaumanifolds, 1997, Theor. Math. Phys.,109