Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions
IInternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company
ANALYTICAL PATH-INTEGRAL PRICING OFMOVING-BARRIER OPTIONS UNDER NON-GAUSSIANDISTRIBUTIONS
ANDRÉ CATALÃO
Instituto de Física Teórica, Universidade Estadual Paulista - UNESPR. Dr. Bento Teobaldo Ferraz, 271São Paulo, SP, 01140-070, [email protected]
ROGERIO ROSENFELD
Instituto de Física Teórica, Universidade Estadual Paulista - UNESPSouth American Institute for Fundamental ResearchR. Dr. Bento Teobaldo Ferraz, 271São Paulo, SP, 01140-070, [email protected]
Received (Day Month Year)Revised (Day Month Year)In this work we present an analytical model, based on the path-integral formalism of Sta-tistical Mechanics, for pricing options using first-passage time problems involving bothfixed and deterministically moving absorbing barriers under possible non-gaussian distri-butions of the underlying object. We adapt to our problem a model originally proposedto describe the formation of galaxies in the universe of De Simone et al. (2011), whichuses cumulant expansions in terms of the Gaussian distribution, and we generalize it totake into acount drift and cumulants of orders higher than three. From the probabilitydensity function, we obtain an analytical pricing model, not only for vanilla options (thusremoving the need of volatility smile inherent to the Black & Scholes (1973) model), butalso for fixed or deterministically moving barrier options. Market prices of vanilla op-tions are used to calibrate the model, and barrier option pricing arising from the modelis compared to the price resulted from the relative entropy model.
Keywords : non-gaussian distribution; stochastic processes; first-passage time; mov-ing barrier, Black and Scholes model; cumulant expansion; path integral; Breeden-Litzenberger theorem; relative entropy. 1 a r X i v : . [ q -f i n . M F ] A p r nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld
1. Introduction
In Stochastic Processes, the first passage time τ f , defined as the time a systemtakes to cross a barrier for the first time - usually associated to survival analysis -appears in several branches of science: from Biology, in cell transport phenomena,to Economics, in credit default events; Sociology, in group decisions; Physics, inStatistical Mechanics, Optics, Solid State; Chemistry, in reactions and corrosion;and Cosmology. In this latter case, to form a galaxy, the concentration of massneeds to reach a critical value, which can be seen as a barrier, not necessarily fixed,and possibly subject to a stochastic process.The problem of finding the probability distribution of first passage time was firststudied by Schrödinger (1915) in the context of a Brownian motion in a physicalmedium, and it was shown to be given by the Inverse Normal distribution. InStatistics, the distribution was first obtained by Wald (1947) in likelihood-ratiotests. In stochastic calculus, the problem can be studied in terms of the transitionprobability distribution between states emerging from boundary conditions imposedto the Fokker-Planck equation (Gardiner (2004), Risken (1989)), from which thecumulative probability distribution that the first passage time occurs after a giveninstant T , that is, P ( τ f > T ) , is derived.The study of first passage time depends on the distribution of the underlyingprocess that is assumed. In the case of galaxy formation, Maggiore & Riotto (2010a)and Maggiore & Riotto (2010b) discusses the treatment of Gaussian distributions,while Maggiore & Riotto (2010c) and De Simone et al. (2011) treat non-Gaussiandiffusion, the latter including the case of moving barriers. Usually, the non-Gaussianapproach is developed in terms of expansions based on a benchmark distribution,which is commonly taken as the Gaussian one.In Finance, the first passage time problem may arise in derivative contractsthat establish deactivation or activation conditions, upon the passage of a timedependent variable P t through a barrier B : τ f = min { t | P t > B } . (1.1)The most frequently used distribution in Finance is the lognormal distributionfor prices, in the context of the Black & Scholes (1973) model hypothesis. A singlebarrier knock-up-and-out european call option (KUO european call) is then acontract that enables its holder to buy a certain asset S T , the underlying asset,at maturity date T , paying the contract strike price K , as long as the underlyingasset does not cross a contractual barrier level B . Mathematically, the payoff ofsuch contract is Call
KUO ( T ) = 1 S t
2. Cumulant expansion and the path integral formalism
In this section, we present the cumulant expansion, connecting it to the pathintegral formalism. Let ω = ω ( t ) = ω t be the stochastic variable whose distributionwe wish to model. In our case, ω = σ ln S t S , where σ is the volatility parameter, S and and S t are the underlying values at t = 0 and t , respectively. A pathbegins at t = 0 , with ω = 0 , and evolves until the final instant of time t n , where ω n = ω ( t n ) = ω ( T ) . We assume the time discretization ∆ t = (cid:15) , with t k = k(cid:15) . Aprice path is a collection { ω , ..., ω n } , such that ω ( t k ) = ω k . If there is no absorbingbarrier, ω t ∈ ( −∞ , ∞ ) . The probability density in the space of trajectories can bedescribed by the expected value of a product of Dirac delta functions: W n = W ( ω , ω , ..., ω n ; t n ) ≡ (cid:104) δ ( ω ( t ) − ω ) ...δ ( ω ( t n ) − ω n ) (cid:105) , (2.1)which follows from (cid:104) δ ( x − ¯ x ) δ ( x − ¯ x ) ...δ ( x n − ¯ x n ) (cid:105) = ˆ ∞−∞ ... ˆ ∞−∞ p ( x , x , ..., x n ) δ ( x − ¯ x ) δ ( x − ¯ x ) ...δ ( x n − ¯ x n ) dx dx ...dx n = p (¯ x , ¯ x , ..., ¯ x n ) , (2.2)which is the probability density. a In terms of W , the probability that the variableassumes the value ω n at instant t n , from ω , at t = 0 , in trajectories that neverexceed ω c , is given by: Π (cid:15) ( ω , ω n ; t n ) = ˆ ω c −∞ dω ... ˆ ω c −∞ dω n − W ( ω , ω , ..., ω n ; t n ) . (2.3)And the probability that the path remains in the region ω < ω c , for all instantslower than t n , is: Π ( ω ; t n ) = ˆ ω c −∞ dω n Π (cid:15) ( ω , ω n ; t n ) . (2.4) a See Risken (1989), section 2.4. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company
Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions This equation represents the sum over all possible paths, thus representing thepath integral that computes the probability function. We will express it in termsof the cumulants of the distribution. The characteristic function is the Fouriertransform of the distribution b : C n ( u , ..., u n ) = (cid:10) e iu ω + ··· + iu n ω n (cid:11) = ˆ . . . ˆ e iu ω + ··· + iu n ω n W ( ω , ω , ..., ω n ; t n ) dω ...dω n . (2.5)The joint moment function is defined by M m ,...,m n = (cid:104) ω m . . . ω m n n (cid:105) = (cid:18) ∂∂ ( iu ) (cid:19) m . . . (cid:18) ∂∂ ( iu n ) (cid:19) m n C n ( u , ..., u n ) | u = ... = u n =0 . (2.6)The joint moments are the coefficients of the Taylor expansion of thecharacteristic function: C n ( u , ..., u n ) = (cid:88) m ,...,m n M m ,...,m n ( iu ) m m ! · · · ( iu n ) m n m n ! . (2.7)The joint cumulants κ m ,...,m n of a distribution are related to the characteristicfunction by C n ( u , ..., u n ) = exp (cid:32) ∞ (cid:88) m ,...,m n κ m ,...,m n ( iu ) m m ! · · · ( iu n ) m n m n ! (cid:33) (2.8) κ m ,...,m n = (cid:18) ∂∂ ( iu ) (cid:19) m . . . (cid:18) ∂∂ ( iu n ) (cid:19) m n ln [ C n ( u , ..., u n ) | u = ... = u n =0 (cid:3) . (2.9) Π (cid:15) ( ω , ω n ; t n ) can be expressed in terms of the cumulants. To see this we usethe following representation of the Dirac delta function δ ( ω ) = ˆ ∞−∞ du π · e − iuω . (2.10) b See Risken (1989), section 2.3. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld
Substituting in (2.1), W ( ω , ω , ..., ω n ; t n ) = (cid:42) ˆ ∞−∞ du π · · · du n π · e − i n (cid:80) j =1 u j ( ω ( t j ) − ω j ) (cid:43) = ˆ ∞−∞ du π · · · du n π · e i n (cid:80) j =1 u j ω j (cid:42) e − i n (cid:80) j =1 u j ω ( t j ) (cid:43) . (2.11)We define the integration measure ˆ ∞−∞ Du ≡ ˆ ∞−∞ du π · · · du n π . (2.12)Therefore, using the definition (2.5) in (2.11) we can write W ( ω , ω , ..., ω n ; t n ) = ˆ ∞−∞ Du ·· exp i n (cid:88) j =1 u j ω j + ∞ (cid:88) m ,...,m n κ m ,...,m n ( − iu ) m m ! · · · ( − iu n ) m n m n ! . (2.13)This is the expansion of the probability of a given path in terms of the jointcumulants.Keeping only terms with m i is equal to or , that is, m = 0 , m =0 , ...m n = 0 , (we will justify this shortly) results in: W ( ω , ω , ..., ω n ; t n ) = ˆ ∞−∞ Du · exp i n (cid:88) j =1 u j ω j − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ... . (2.14)In this notation, κ ≡ κ m =1 ,m =0 ...,m n =0 , κ ≡ κ m =0 ,m =1 ,m =0 ,...,m n =0 , κ ≡ κ m =1 ,m =1 ,m =0 ,...,m n =0 , κ ≡ κ m =1 ,m =0 ,m =1 ,m =0 ,...,m n =0 , and so on. (2.14)will be the version of equation (2.13) that we will use.Had we considered other values of m , m , etc , different from zero and one, wewould have included generalized moments, beyond the usual covariance betweentwo variables. For instance, the covariance between the fourth power of a variable i nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions and the cube of another variable j , in the case of m i = 2 and m j = 3 , etc, and theycontribute at higher orders. In our case, where we seek to calibrate market data,it will be enough to consider just the usual moments (variance, kurtosis, etc) and,thus, we will not consider covariances and its generalizations in the combinations ofthe several orders of the variables. In this notation, for example, in κ ij , when i = j ,we have the cumulant linked to the variance; in κ ijk , when i = j = k , the cumulantrelated to asymmetry, etc.Using this expansion in (2.3) one obtains Π (cid:15) ( ω , ω n ; t n ) = ˆ ω c −∞ dω ... ˆ ω c −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) j =1 u j ω j − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ... . (2.15)This equation is the path integral representation of the probability distributionin termos of the cumulants.
3. Gaussian Fluctuations
In this Section we show that our formalism reproduces the well-known formulaefor the price of barrier options for Gaussian fluctuations as a sanity check. In thecase of Gaussian fluctuations, the cumulants are zero, except those satisfying m + m + ... + m r ≤ c : (cid:42) e i n (cid:80) j =1 ( − u j ) ω ( t j ) (cid:43) = exp − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij . (3.1)In this case equations (2.14) and (2.15) assume the form (we put a superscritp“g" to indicate Gaussian) W g ( ω , ω , ..., ω n ; t n ) = ˆ ∞−∞ Du · exp i n (cid:88) j =1 u j ω j − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij ; (3.2) c Risken (1989), section 2.3.3. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Π g(cid:15) ( ω , ω n ; t n ) = ˆ ω c −∞ dω ... ˆ ω c −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij . (3.3)Besides, κ ij = σ ij , where σ ij is the covariance between i and j .Consider the case of Markovian processes, where just the previous state of thevariable influences the present state: Π ( ω ( t n ) ≤ ω n | ω ( t n − ) , ..., ω ( t )) = Π ( ω ( t n ) ≤ ω n | ω ( t n − )) . (3.4)We will denote Π gm(cid:15) and W gm the Gaussian probability and probability densityunder the Markov hypothesis, that is, when the particle executes a MarkovianGaussian Brownian motion. In a stationary stochastic process, the moments areconstant along time, and their values only depend on the least instante betweenperiods. If the variable ω is standard Gaussian, as in a Wiener process, σ ij = < ω i ω j > = (cid:15)min ( i, j ) ≡ (cid:15)A ij . (3.5)The probability density (3.2) becomes: W gm ( ω , ω , ..., ω n ; t n ) = ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − (cid:15) n (cid:88) i,j =1 u i u j A ij = ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ( ω i − κ i ) − (cid:15) n (cid:88) i,j =1 u i u j A ij . (3.6)To illustrate, consider one variable ω i = ω . Then, W g ( ω ) = 12 π ˆ ∞−∞ du · e iu ( ω − κ ) − u κ = 1 √ π · κ e − ( ω − κ κ , (3.7)where κ = < ω > = (cid:15) .nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions In the case of n Gaussian Markovian variables, with κ = (cid:15) and κ = (cid:15)α , where α is the drift : W gm ( ω , ω , ..., ω n ; t n ) = 1(2 π(cid:15) ) n/ e − n − (cid:80) i =0 ( ωi +1 − ωi + α(cid:15) ) (cid:15) . (3.8)Therefore we can write W gm ( ω , ω , ..., ω n ; t n ) = Ψ (cid:15) ( ω n − ω n − ) W gm ( ω , ω , ..., ω n − ; t n − ) (3.9) Ψ (cid:15) (∆ ω ) ≡ √ π(cid:15) e − (∆ ω + α(cid:15) )22 (cid:15) (3.10) ∆ ω = ω n − ω n − . (3.11)Thus, Π gm(cid:15) ( ω , ω n ; t n ) = ˆ ω c −∞ dω n − Ψ (cid:15) ( ω n − ω n − ) Π gm(cid:15) ( ω , ω n − ; t n − ) . (3.12)In the presence of a fixed barrier, the probability density in the case of and upabsorbing barrier B , to be used in the call KUO pricing, under the Black-Scholesassumptions, is (Shreve (2004)): Π gm(cid:15) → ( ω , ω n ; t n ) = 1 √ πt n e α ( ω n − ω ) − α t n (cid:20) e − ( ωn − ω tn − e − (2 ωc − ωn − ω tn (cid:21) . (3.13) ω n = ω ( t n ) = 1 σ ln S t S ; ω c = b = 1 σ ln BS . (3.14)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld
4. Analytical expansion for non-Gaussian distributions withmoving barrier in the path-integral formalism
In this Section, the non-Gaussian distribution with absorbing moving barrieris obtained from the path integral formulation. As in the work of De Simone etal. (2011), we present two alternative approaches in the expansion: (i) first, thehypothesis of Sheth & Tormen (2002), which states that instants t i < t n areinsignificant compared to t n in derivatives higher than the first order and (ii) second,barrier moves slowly. The latter we call “adiabatic barriers”.The accomplishment of this task involves expanding the non-Gaussiandistribution with moving barrier, Π (cid:15) → ( ω , ω n ; t n ) , in an expression of the form: Π (cid:15) → ( ω , ω n ; t n ) = Π mb(cid:15) → ( ω , ω n ; t n ) + derivatives of Π mb(cid:15) → ( ω , ω n ; t n ) , (4.1)where, in each approach (Sheth-Tormen and adiabatic barriers), Π mb(cid:15) → ( ω , ω n ; t n ) assumes different formats, both involving the Gaussian distribution with fixedbarrier (3.13), plus terms regarding moving barriers. The Sheth-Tormen approach
Consider the expansion in cumulants (2.15), in the case of a barrier that movesaccording to a deterministic rule B ( t i ) , i = 1 , ..., n − : Π (cid:15) → ( ω , ω n ; t n ) = ˆ B ( t ) −∞ dω ... ˆ B ( t n − ) −∞ dω n − W ( ω , ω , ..., ω n ; t n ) , (4.2)with W ( ω , ω , ..., ω n ; t n ) given by (2.14)Next, we assume that the barrier does not change significantly and expand in aTaylor series around B ( t n ) ≡ B n . Therefore, B ( t i ) = B ( t n )+ ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p (4.3) B ( p ) n = d p B ( t n ) dt pn . (4.4)Redefining the variables ω i , i = 1 , ..., n − : (cid:36) i ≡ ω i − ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions ∴ (cid:36) i = ω i − ( B ( t i ) − B ( t n )) d(cid:36) i = dω i . (4.5)Thus, Π (cid:15) → ( ω = 0 , (cid:36) n ; t n ) = ˆ B n −∞ d(cid:36) ... ˆ B n −∞ d(cid:36) n − ˆ ∞−∞ Du · e Z (4.6) Z = i n (cid:88) i =1 u i (cid:36) i + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ... (4.7)Since (cid:36) i is a dummy variable, we will use the notation ω i again. We work withthe expansion until the 5th order, generalizing it later. Z = i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ( − i ) n (cid:88) i,j,k,l,m =1 u i u j u k u l u m κ ijklm + ... + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p . (4.8)The first line of this equation is the Gaussian term. Applying the Taylorexpansion to the exponential term of the non-Gaussian part (2nd and 3rd linesof (4.8)), one can write: Π (cid:15) → ( ω = 0 , ω n ; t n ) = ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p + ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ( − i ) n (cid:88) i,j,k,l,m =1 u i u j u k u l u m κ ijklm + .... . (4.9)The summation term involving the barrier can also be expanded in Taylor series.We also consider up to second order: exp (cid:32) i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p (cid:33) (cid:39) i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p − n − (cid:88) i,j =1 u i u j ∞ (cid:88) p,q =1 B ( p ) n B ( q ) n p ! q ! ( t i − t n ) p ( t j − t n ) q + ... (4.10)(4.9) can be rewritten as: Π (cid:15) → ( ω = 0 , ω n ; t n ) = ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij · (cid:32) i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p (cid:33) + ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij · − n − (cid:88) i,j =1 u i u j ∞ (cid:88) p,q =1 B ( p ) n B ( q ) n p ! q ! ( t i − t n ) p ( t j − t n ) q + ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ( − i ) n (cid:88) i,j,k,l,m =1 u i u j u k u l u m κ ijklm + .... . (4.11)Using Π gm(cid:15) → ( ω , ω n ; t n ) as given by (B.22), we can decompose Π (cid:15) → ( ω = 0 , ω n ; t n ) as Π (cid:15) → ( ω = 0 , ω n ; t n ) = Π gm(cid:15) → ( ω , ω n ; t n ) + Π (1) (cid:15) → ( ω n , t n ) + Π (2) (cid:15) → ( ω n , t n )+ ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij + i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · ( − i ) n (cid:88) i,j,k =1 u i u j u k κ ijk + ( − i ) n (cid:88) i,j,k,l =1 u i u j u k u l κ ijkl + ( − i ) n (cid:88) i,j,k,l,m =1 u i u j u k u l u m κ ijklm + .... , (4.12)where the gaussian markovian piece Π gm(cid:15) → ( ω , ω n ; t n ) was already given in Eq.(3.13).The remainder of this Section is devoted to the computation of the differentterms in Equation (4.12).Consider (3.2): W gm ( ω , ω , ..., ω n ; t n ) = ˆ ∞−∞ Du · exp i n (cid:88) j =1 u j ω j − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij ≡ ˆ ∞−∞ Du · exp ( Z gm ) (4.13)with Z gm = i n (cid:88) j =1 u j ω j − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij . (4.14)Defining ∂/∂ω i = ∂ i , we note that iu i e iu i ω i = ∂ i e iu i ω i . (4.15)Then, the second term of (4.10), in the first term of (4.9) can be rewritten as Π (1) (cid:15) → ( ω n , t n ) = ˆ B n −∞ dω ... ˆ B n −∞ dω n − nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions (cid:34) ˆ ∞−∞ Du · (cid:32) i n − (cid:88) i =1 u i ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij = ˆ B n −∞ dω ... ˆ B n −∞ dω n − (cid:34) n − (cid:88) i =1 ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · ∂ i W gm ( ω , ω , ..., ω n ; t n ) (cid:35) . (4.16)The third term of (4.10) also in the first term of (4.9), observing the rule (4.15), Π (2) (cid:15) → ( ω n , t n ) = ˆ B n −∞ dω ... ˆ B n −∞ dω n − ˆ ∞−∞ Du · − n − (cid:88) i,j =1 u i u j ∞ (cid:88) p,q =1 B ( p ) n B ( q ) n p ! q ! ( t i − t n ) p ( t j − t n ) q · exp i n (cid:88) i =1 u i ω i − i n (cid:88) i =1 u i κ i − n (cid:88) i,j =1 u i u j κ ij = ˆ B n −∞ dω ... ˆ B n −∞ dω n − · n − (cid:88) i,j =1 ∞ (cid:88) p,q =1 B ( p ) n B ( q ) n p ! q ! ( t i − t n ) p ( t j − t n ) q · ∂ j ∂ i W gm ( ω , ω , ..., ω n ; t n )) . (4.17)To evaluate (4.16), we use (A.5) and the transformation (B.30): Π (1) (cid:15) → ( ω n , t n ) = 1 (cid:15) ˆ t n dt i (cid:34) ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · (Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i )) (cid:35) . (4.18)Now we use (B.41) and (B.42), with ω = 0 :nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Π (1) (cid:15) → ( ω n , t n ) = 1 (cid:15) ˆ t n dt i (cid:34) ∞ (cid:88) p =1 B ( p ) n p ! ( t i − t n ) p · (cid:32) √ (cid:15) √ π e α ( ω n − B n ) B n − ω t / i e − [ Bn − ω − αti ] ti (cid:33) · (cid:32) √ (cid:15) √ π e α ( ω n − B n ) B n − ω n ( t n − t i ) / e − [ ( Bn − ωn ) − α ( tn − ti )] ( tn − ti ) (cid:33)(cid:35) = ( B n − ω n ) ( B n − ω ) π e α ( ω n − B n ) ∞ (cid:88) p =1 ( − p B ( p ) n p ! ·· ˆ t n dt i ( t n − t i ) p − t / i e − [ ( Bn − ω − αti ] ti e − [ ( Bn − ωn ) − α ( tn − ti )] ( tn − ti ) (4.19)To solve Π (1) (cid:15) → ( ω n , t n ) and Π (2) (cid:15) → ( ω n , t n ) , we adopt at this point anapproximation due to Sheth & Tormen (2002), abbreviated by “ST”, which impliesthat t n (cid:29) t i in higher than first order derivatives in (4.3): ( t n − t i ) p − (cid:39) ( t n ) p − (4.20)to obtain Π (1) (cid:15) → ( ω n , t n ) = 2 ( B n − ω n ) √ πt / n e − tn [ αt n − (2 B n − ω − ω n )] e α ( ω n − B n ) ∞ (cid:88) p =1 ( − p B ( p ) n p ! . (4.21)We develop now (4.17), Π (2) (cid:15) → ( ω n , t n ) , also using (ST), (4.20). To do so, we willuse (A.14), with (A.16): n − (cid:88) i,j =1 ∂ i ∂ j = 2 (cid:88) i Slowly moving barrier In this section, the hypothesis is that the barrier moves slowly with t . For futurereferences we may call it abiabatic barrier . Under this assumption, we may discusswhich terms in (4.3) are relevant. We will work with up to second order terms in thebarrier time derivatives: ∂ B n /∂t n e ( ∂B n /∂t n ) . In a certain way, this hypothesisand the ST one work in the same direction: in (4.3), the (ST) acts in the terms ( t i − t n ) p , while the slow varying barrier is related to the derivatives.Under these conditions, (4.16) breakes into two terms, denoted by the followingexpressions: Π ( a ) (cid:15) → ( ω n , t n ) = n − (cid:88) i =1 B (cid:48) n ( t i − t n ) ˆ B n −∞ dω ... ˆ B n −∞ dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) (4.43) Π ( b ) (cid:15) → ( ω n , t n ) = 12 n − (cid:88) i =1 B (cid:48)(cid:48) n ( t i − t n ) ˆ B n −∞ dω ... ˆ B n −∞ dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) . (4.44)Equation (4.17) enables us to write: Π ( c ) (cid:15) → ( ω n , t n ) = 12 n − (cid:88) i =1 (cid:16) B (cid:48) n (cid:17) ( t i − t n ) ( t j − t n ) · ˆ B n −∞ dω ... ˆ B n −∞ dω n − ∂ i ∂ j W gm ( ω , ω , ..., ω n ; t n ) , (4.45)where B (cid:48) n = ∂B ( t n ) /∂t n and B (cid:48)(cid:48) n = ∂ B ( t n ) /∂t n .Starting by Π ( a ) (cid:15) → ( ω n , t n ) , from (4.19), with p = 1 , Π ( a ) (cid:15) → ( ω n , t n ) == − ( B n − ω n ) ( B n − ω ) π e α ( ω n − B n ) · dB n dt n · ˆ t n dt i ( t n − t i ) − t / i e − [ ( Bn − ω − αti ] ti e − [ ( Bn − ωn ) − α ( tn − ti )] ( tn − ti ) . (4.46)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions ∴ Π ( a ) (cid:15) → ( ω n , t n ) = − (cid:114) π dB n dt n ( B n − ω n ) t / n e α ( ω n − B n ) e − [(2 Bn − ω − ωn ) − αtn ]22 tn . (4.47)We now value Π ( b ) (cid:15) → ( ω n , t n ) . In (4.19), with p = 2 , Π ( b ) (cid:15) → ( ω n , t n ) == ( B n − ω n ) ( B n − ω ) π e α ( ω n − B n ) · d B n dt n · ˆ t n dt i ( t n − t i ) t / i e − [ ( Bn − ω − αti ] ti e − [ ( Bn − ωn ) − α ( tn − ti )] ( tn − ti ) . (4.48) ∴ Π ( b ) (cid:15) → ( ω n , t n ) =12 π d B n dt n ( B n − ω n ) e α ( ω n − B n ) e α (2 B n − ω − ω n ) − α tn · (cid:20) √ πt / n e − (2 Bn − ω − ωn )22 tn − π ( B n − ω ) Erf c (cid:18) B n − ω − ω n √ t n (cid:19)(cid:21) . (4.49)Finally, now computing Π ( c ) (cid:15) → ( ω n , t n ) , in (4.45), which refers to Π (2) (cid:15) → ( ω n , t n ) ,from (4.24), with p = q = 1 : Π ( c ) (cid:15) → ( ω n , t n ) = ( B n − ω ) ( B n − ω n ) π √ π (cid:18) dB n dt n (cid:19) e α ( ω n − B n ) e α ( B n − ω ) e − α tn · ˆ t n dt i ( t n − t i ) e − ( Bn − ω ti t / i ˆ t n t i dt j e − ( Bn − ωn )22 ( tn − tj ) ( t j − t i ) / ( t n − t j ) / . (4.50) ∴ Π ( c ) (cid:15) → ( ω n , t n ) = − (cid:114) π ( B n − ω n ) t / n (cid:18) dB n dt n (cid:19) e α ( ω n − B n ) e − [(2 Bn − ω − ωn ) − αtn ]22 tn . (4.51)Therefore, in the hypothesis of this section, the equivalent to (4.27) isnternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Π mb(cid:15) → ( ω n , t n ) = ˆ B n −∞ dω ... ˆ B n −∞ dω n − W mb ( ω , ω , ..., ω n ; t n ) = Π gm(cid:15) → ( ω n ; t n ) +Π ( a ) (cid:15) → ( ω n , t n ) + Π ( b ) (cid:15) → ( ω n , t n ) + Π ( c ) (cid:15) → ( ω n , t n ) , (4.52)with Π gm(cid:15) → ( ω n ; t n ) , Π ( a ) (cid:15) → ( ω n , t n ) , Π ( b ) (cid:15) → ( ω n , t n ) and Π ( b ) (cid:15) → ( ω n , t n ) given by (B.22),(4.47), (4.49) and (4.51), respectively, taking ω = 0 . (4.42) is still valid, with thisspecification for Π mb(cid:15) → ( ω n , t n ) .Therefore, in the presence of fixed or moving barriers, we have expressed adistribution in terms of a cumulant expansion based on the Gaussian distributionwith fixed barrier. Non-Gaussian distribution with constant barrier The constant barrier B n = b = σ ln BS , in a non-Gaussian distribution impliestime derivatives of the barrier equal to zero, in both (ST) and adiabatic barrierhypothesis: lim dBndtn → lim d Bndt n → Π mb(cid:15) → ( ω n , t n ) = Π gm(cid:15) → ( ω n ; t n ) , (4.53)where Π gm(cid:15) → ( ω n ; t n ) is given by (B.22). This distribution is used in (4.42) to get Π (cid:15) → ( ω = 0 , ω n ; t n ) related to this case. Non-Gaussian corrections emerge fromthe derivatives related to ω n and B n . Linearly moving barrier The application to the moving barrier case depends on derivatives, which appearin Π mb(cid:15) → ( ω n , t n ) components, with respect to the value of the barrier at the finalinstant t n . We will consider the case of a barrier tha evolves linearly, to exemplifythe use of the notation: B i = B ( t i ) = B + ξt i . (4.54)We first find the value at t n : B n = B + ξt n (4.55)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions and write the derivatives. In our case, only the first order one takes non-null value: B ( p ) n = d p B n d ( t n ) p = ξ p = 10 p > . (4.56) Non-Gaussian distribution in the absence of barriers The absence of barriers is a particular case of the probability densities of sections(4.1) or (4.2). Specifically, add an extra limit in (4.53), setting the barrier at infinity: lim B n →∞ lim dBndtn → lim d Bndt n → Π mb(cid:15) → ( ω n , t n ) = Π gm(cid:15) → ( ω n , t n ) . (4.57) Π gm(cid:15) → ( ω n , t n ) = 1 √ πt n e αω n − α t n e − ω n tn . (4.58)Second, in (4.42), since the derivative related to the barrier (cid:0) ∂ i /∂B in (cid:1) arelimit operations, they can be applied after those of (4.57). At the infinity, thedistribution converges to the Gaussian density, without barrier, according to (4.57).As a consequence, the derivative operator ∂ i /∂B in nullifies the term. In the case offirst order derivative, lim B n →∞ lim dBndtn → lim d Bndt n → Π mb(cid:15) → ( ω n , t n ) = lim B n →∞ lim dBndtn → lim d Bndt n → lim ∆ B n →∞ ∆Π mb(cid:15) → ( ω n , t n )∆ B n = lim ∆ B n →∞ ∆ lim B n →∞ lim dBndtn → lim d Bndt n → Π mb(cid:15) → ( ω n , t n ) ∆ B n = lim ∆ B n →∞ ∆Π gm(cid:15) → ( ω n , t n )∆ B n = 0; (4.59)and so on, for higher order derivatives in the barrier. Nevertheless, in (4.42) remainthe derivative terms ∂ i /∂ω in , which assure the survival of non-Gaussian terms ofthe expansion.In a nutshell, the non-barrier case of non-Gaussian distribution is givenby substituting Π mb(cid:15) → ( ω n , t n ) → Π gm(cid:15) → ( ω n , t n ) in (4.42), and excluding barrierderivatives of the distribution. We rewrite (4.42) in this case:nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Π inf(cid:15) → ( ω = 0 , ω n ; t n ) = Π gm(cid:15) → ( ω n , t n ) − κ ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) + 14! κ ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) − κ ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) + (cid:34) · (cid:18) κ (cid:19) + 16! κ (cid:35) ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) − (cid:20)(cid:18) κ (cid:19) (cid:18) κ (cid:19) + 17! κ (cid:21) ∂ ∂ω n Π gm(cid:15) → ( ω n , t n )+ (cid:34) · (cid:18) κ (cid:19) − (cid:18) κ (cid:19) (cid:18) κ (cid:19) + 18! κ (cid:35) ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) − (cid:20)(cid:18) κ (cid:19) (cid:18) κ (cid:19) + (cid:18) κ (cid:19) (cid:18) κ (cid:19) + 19! κ (cid:21) ∂ ∂ω n Π gm(cid:15) → ( ω n , t n ) + .... (4.60)where Π gm(cid:15) → ( ω , ω n ; t n ) = 1 √ πt n e α ( ω − ω ) − α t n e − ( ω − ω tn . (4.61)From now on, when we refer to the vanilla case under non-Gaussian distribution,we will deal with (4.60) and (4.61), the infinite barrier case. Martingale condition for drift Under the risk-neutral measure Q , the martingale condition establishes thenon-arbitrage drift condition e − r ( t n − t n − ) E Q (cid:2) S n |F t n − (cid:3) = S n − . (4.62)In terms of the distribution (4.60), S = e − rt n ˆ ∞−∞ S e σω n Π inf(cid:15) → ( ω n , t n ) dω n , ∴ e − rt n ˆ ∞−∞ e σω n Π inf(cid:15) → ( ω n , t n ) dω n . (4.63)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions When we keep up to the 15th order in derivatives in (4.60), we get the equation(C.1) of Appendix C. Analytical option pricing To price call vanilla european options, under the probability density Π inf(cid:15) → ( ω n , t n ) , P ( S , K, t n , α, κ , κ , ... ) = ˆ ∞−∞ max [( S e σω n − K ) , 0] Π inf(cid:15) → ( ω n , t n ) dω n , (4.64)Barrier options are priced with Π (cid:15) → ( ω n , t n ) in (4.42), specifying Π mb(cid:15) → ( ω n , t n ) for fixed (4.53) or moving barriers, either in (ST), or in adiabatic barrierapproximations. The price of a knock-up-and-out call is given by: P = ˆ bk ( S e σω n − K ) Π (cid:15) → ( ω n , t n ) dω n . (4.65)and that of a knock-up-and-out put by d : P = ˆ k −∞ ( K − S e σω n ) Π (cid:15) → ( ω n , t n ) dω n . (4.66) 5. Calibration In order to price barrier options, the parameters σ and κ i are calibrated witheuropean vanilla call options, for each maturity, meaning a piecewise constant set.In our example, the daily data consist of foreign exchange (FX) european calloptions of Brazilian Real against US dollar (BRL/USD), ranging from 05/2009 to05/2014. Each day includes implied volatility rates corresponding to five standarddelta values { , , , , } , for twenty-four months { , , , ..., } .Therefore, a volatility σ ijk is indexed by the date t i , delta ∆ j and maturity T k : σ ijk = σ ( t i , ∆ j , T k ) ; i = 1 , ..., ; j = 1 , ..., ; k = 1 , ..., . Deltas are convertedto strikes in the usual manner. d The valuation of these integrals result in closed-form expressions. However, if a high number ofcumulants is included in the specification, numerical valuation may become less costly. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Concerning the model specification, the number of parameters in the case ofvanilla options depends on the ability to fit the smile delta range. On the left sideof figure 1, the maximum order was in κ , while, on the right side of the figure, themaximum was κ . However, higher order corrections gained with higher derivativeorders in the expansion are done at the expense of adding more parameters tothe model. Such improvements are necessary near the barrier region in pricingbarrier options. Further, just including higher derivative orders is not enoughif combinations of parameters are not considered (we took up to second ordercombinations, κ i κ j , with i + j = n , n = order of derivative correction in theterm). For instance, in the upper plot in figure 2, we used 18th order in derivativesfor different values of the barrier, but didn’t include the second order combinations.When we included the second order combinations of parameters, the 15th order wasenough to improve precision in the barrier region, as shown in the lower plot. Figure 1. Smile calibration for vanillas. The first graph uses up to the 7th cumulant, while thesecond uses up to the 15th one. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Thus, in order to price knock-up-and-out calls, according to the integration limitsin the pricing equation (4.65), one should guarantee a good distribution fitting inthe interval [ k, b ]] . Therefore, although price calibration is possible, because of thelarge amount of terms in the pricing formula, we calibrated the parameters of themodel density (4.60) by fitting the probability density retrieved from the Breeden& Litzenberger (1978) theorem, taking into account the smile, as in Shimko (1993): Π inf(cid:15) → ( ω n = k, t n ≡ T ) = e rT Kσ d CdK . (5.1)The total derivative d C/dK was computed numerically and analitically, usingthe parameters of cubic spline interpolation, both producing the same results. Anexample of calibration is given in figure 3. Figure 3. Plot representing (i) the non-Gaussian density in the path-integral (“non-Gaussian (PI)”);(ii) the distribution according to the Breeden & Litzenberger (1978) theorem, where derivativesare computed with cubic spline (red) and numerically (green); and (iii) Gaussian density using theat-the-money volatility. Data for 1-month maturity of the 85th sample point. The fitting of model prices P model to 300 vanilla call option market prices P market nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions can be summarized by fitting the linear regression ( e is the residual): P Model = a p · P market + b p + e, (5.2)where we hope to get a p = 1 and b p = 0 . The result is in figure 4, which displays R ∼ , and a p = 1 , b p = − × − . Figure 4. Price regression: dependent variable is the model price and the independent one is themarket prices of vanilla call options. 6. Barrier option pricing To analyze in a standardized way the effect of an absorbing up barrier on thecall option price, according to its proximity to the initial underlying price or to thestrike, we set fixed barrier levels defined by a group of multiplicative factors Θ = { . 1; 1 . 2; 1 . 3; 1 . } , to be applied to data base strikes, which cover the deltas from 10nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld to 90 in each maturity. In order to assure that the price does not start deactivatedby the barrier, when multiplication results in a barrier below initial underlyingfuture price, we change the rule, setting the fixed barrier to a multiplicative factorregarding the future price, given maturity. Thus, to each strike i and maturity j ,the barrier B k is defined, resulting in price P ijk = P ijk ( B k , K i , T j ) ; B k = (cid:26) Θ k · K i , Θ k · K i > F j Θ k · F j , Θ k · K i ≤ F j (6.1)where k = 1 , ..., and F j is the future value related to the maturity T j .Usually, data providers of barrier option prices rely on market-to-model values.Therefore, we have chosen to compare ours results to the relative entropy modelof Avellaneda et al. (2001). When the absorbing up barrier is near the strike ofthe call or the inital underlying value, that is, when Θ k = 1 . , prices are closerto zero. In this situation, in longer maturities, we have found greater divergencecomparing the path-integral approach, the lognormal model (Black-Scholes, withstrike-related volatility) and the relative entropy model, as in figure 5. In figure6, in the 18-month case, we notice that, although the average underlying priceis higher in the path-integral model, the Black-Scholes model presents a greaterdispersion, meaning that the barrier region has greater chance of being reached,thus displaying lower prices in the case of barrier close to initial underlying value.In shorter maturities, such as 1-month, the underlying process has less time toreach, the closer the barrier; consequently, prices are closer in the model comparison.Therefore, difference between path-integral and entropy models is expected, sincethe first approach includes the density behaviour in the vicinity of the barrier.nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Figure 5. 18-month model comparison: path integral, relative entropy and Black-Scholes. nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions In the same way as vanilla pricing, we summarize pricing differences betweenthe path integral and the relative entropy models by fitting a linear regression: P P ath integral = a p · P Entropy + b p + e, (6.2)In figure 8, we see that the higher divergence in lower barrier levels at longermaturities corresponds to the dispersion in the region of low prices in the graph(target < . and output < . ).nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Figure 8. Linear regression between the relative entropy model (independent variable) and thepath integral model (dependent variable) of barrier option pricing. 7. Conclusion In this article we presented a non-gaussian probability distribution model, basedon cumulant expansion in the the well-known path integral formalism of StatisticalMechanics, including an absorbing, deterministically moving, barrier. The idea relieson the work of De Simone et al. (2011) for galaxy formation, in Cosmology, whichwe extend to include drift and more cumulants than the original authors do. Inapplying the model to option pricing in Finance, we find the condition for a risk-neutral drift, and present an analytical method to price deterministically movingabsorbing barrier options. The development encompasses analisys of the behaviourof the distribution in the vicinity of the barrier. Usually, general distribution modelsused in such products’ pricing demand numerical and simulation methods; the workof Kunitomo & Ikeda (1992) being a case of closed-form solution, but under thelognormal distribution hypothesis of Black & Scholes (1973).In the case of constant barrier, we obtain an analytic non-gaussian pricing modelto price standard knock-up-and-out (KUO) barrier call options. And, in the limit ofnternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions infinite barrier, it becomes a non-gaussian probability distribution model to pricevanilla options. Since the model parameters, volatility and cumulants, belong toboth barrier and vanilla versions of the model, we calibrate them with vanilla optiondata and then price constant barrier KUO calls. Given that barrier option pricingdata contributors often provide market-to-model values, we compare the constantbarrier option prices of the path integral model with the ones obtained from therelative entropy model. We adapted the approach of Avellaneda et al. (2001) toanalytical constant barrier option pricing.The results demonstrate that our model reproduces those obtained by theEntropy model. The KUO barrier call option pricing presents larger differenceswhen, in long maturity contracts, the barrier is set close the initial underlyingprice, and the delta is near 90% (low strikes). However, in such situation, theunderlying process has enough time to reach the barrier, deactivating the contractand, therefore, prices are expected to be rather small in such combination, evenmore when there is the additional condition that the call strike is high. So, theselarger differences between models refer to small prices; and it does not happen,for instance, when the barrier, in a short maturity contract, is close to the initialunderlying price, since there is not enough time to reach the barrier. In addition, wenotice that such discrepancies between the relative entropy and the path integralmodel we presented also happened in the region of higher deltas, where the relativeentropy model did not fit properly to vanillas and, as we have emphasized, it isimportant to have a good calibration in the delta region associated to the barrierposition.Another point is that barrier option pricing requires the choice of a largernumber of cumulants than vanilla option pricing do, because polynomial fine-tunecorrections are important near the barrier.Finally, as a future development, the model might include double barriercontracts, stochastic barriers, and might also be extended to other classes ofproducts, such as interest rates.nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld Appendix A. Useful relations involving the path integral Let F ( t i ) be a function. Consider objects such as n − (cid:88) i =1 F ( t i ) ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) , (A.1)where F is a generic function and ∂ i ≡ ∂/∂ω i . Because W gm is Gaussian, it islimited and tends to zero at −∞ . Integrating, ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) = ˆ ω c −∞ dω ...d ˆ ω i ...dω n − W gm ( ω , ω , ..., ω i = ω c , ..., ω n − , ω n ; t n ) , (A.2)in which ˆ ω i denotes a variable that is no longer in the integral, because it wasintegrated. The density W gm , by (3.9), satisfies W gm ( ω , ω , ..., ω i = ω c , ..., ω n − , ω n ; t n ) = W gm ( ω , ω , ..., ω i − , ω i = ω c ; t i ) W gm ( ω i = ω c , ω i +1 , ..., ω n − , ω n ; t n − t i ) . (A.3)In (A.2), ˆ ω c −∞ dω ...dω i − ˆ ω c −∞ dω i +1 d ˆ ω i ...dω n − W gm ( ω , ω , ..., ω i − , ω i = ω c ; t i ) · W gm ( ω i = ω c , ω i +1 , ..., ω n − , ω n ; t n − t i ) = Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i ) . (A.4)In (A.1), n − (cid:88) i =1 F ( t i ) ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n )= n − (cid:88) i =1 F ( t i ) Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i ) . (A.5)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions Another relationship, analogous to (A.2), is ˆ ω c −∞ dω ...dω n − ∂ ij W gm ( ω , ω , ..., ω n ; t n )= ˆ ω c −∞ dω ...dω i − ˆ ω c −∞ dω i +1 d ˆ ω i ...d ˆ ω j ...dω n − · W gm ( ω , ω , ..., ω i − , ω i = ω c , ..., ω j − , ω j = ω c , ..., ω n − , ω n ; t n ) . (A.6)Because W gm ( ω , ω , ..., ω i − , ω i = ω c , ..., ω j − , ω j = ω c , ..., ω n − , ω n ; t ,n )= W gm ( ω , ω , ..., ω i − , ω i = ω c ; t i ) · W gm ( ω c , ω i +1 , ..., ω j − , ω j = ω c ; t j − t i ) · W gm ( ω c , ω j +1 , ..., ω n − , ω n ; t n − t j ) , (A.7)we have: ˆ ω c −∞ dω ...dω n − ∂ ij W gm ( ω , ω , ..., ω n ; t n )= ˆ ω c −∞ dω ...dω i − ˆ ω c −∞ dω i +1 d ˆ ω i ...d ˆ ω j ...dω n − W gm ( ω , ω , ..., ω i − , ω i = ω c ; t i ) · W gm ( ω c , ω i +1 , ..., ω i − , ω i = ω c ; t j − t i ) · W gm ( ω c , ω j +1 , ..., ω n − , ω n ; t n − t j )= Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω c ; t j − t i ) Π gm(cid:15) ( ω c , ω n ; t n − t j ) . (A.8)Besides, as in (A.2), we calculate the following integral: ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld = ˆ ω c −∞ dω ...d ˆ ω i ...dω n − ∂ i W gm ( ω , ω , ..., ω i = ω c , ..., ω n − , ω n ; t n )= ∂ i ˆ ω c −∞ dω ...d ˆ ω i ...dω n − W gm ( ω , ω , ..., ω i = ω c , ..., ω n − , ω n ; t n )= ∂ i [Π gm(cid:15) ( ω , ω i ; t i ) Π gm(cid:15) ( ω i , ω n ; t n − t i )] ω i = ω c . (A.9)We also want to analyze the derivatives with respect to the barrier ω c . Considerthe derivative ∂ Π gm(cid:15) ∂ω c ( ω = 0 , ω n ; t n ) : ∂ Π gm(cid:15) ∂ω c ( ω = 0 , ω n ; t n ) = ∂∂ω c (cid:18) ˆ ω c −∞ dω ... ˆ ω c −∞ dω n − W gm ( ω , ω , ..., ω n ; t n ) (cid:19) , (A.10)where we used the definition (2.3). The limits of the integral depend on the variablewith respect to which we derive, ω c . By Leibniz rule: ddx ˆ b ( x ) a ( x ) dtf ( x, t ) = f ( b, t ) db ( x ) dx − f ( a, t ) da ( x ) dx + ˆ b ( x ) a ( x ) dt ddx f ( x, t ) , (A.11) ∂ Π gm(cid:15) ∂ω c ( ω = 0 , ω n ; t n ) = Π gm(cid:15) ( ω n = ω c , t n ) ∂ω c ∂ω c + n − (cid:88) i =1 ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n ) . (A.12)where Π gm(cid:15) ( ω n = ω c , t n ) is the Gaussian density with barrier, (3.13), which becomeszero at the barrier. The first term on the RHS of (A.12) is zero, therefore. Then, ∂ Π gm(cid:15) ∂ω c ( ω = 0 , ω n ; t n ) = n − (cid:88) i =1 ˆ ω c −∞ dω ...dω n − ∂ i W gm ( ω , ω , ..., ω n ; t n )= n − (cid:88) i =1 ˆ ω c −∞ dω ...d ˆ ω i ...dω n − W gm ( ω , ω , ..., ω i = ω c , ..., ω n − , ω n ; t n )= n − (cid:88) i =1 Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i ) , (A.13)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions where we have used (A.2), (A.4).The second derivative with respect to the barrier, ∂ Π gm(cid:15) ∂ω c ( ω = 0 , ω n ; t n ) , canalso be computed. Initially, we note that n − (cid:88) i,j =1 ∂ i ∂ j = 2 (cid:88) i In this appendix we analyze, in the presence of barrier ω c , the behaviourof the Gaussian density of probability, given by (3.3), when the variable ω n approximates ω c . The first discussed situation is when the variable starts at ω ,reaching the barrier ω c , in t n , Π gm(cid:15) → ( ω , ω n = ω c ; t n ) . The second situation is whenthe variables starts in the barrier vicinity, in t , and arrives at ω n (cid:54) = ω c in t n , Π gm(cid:15) → ( ω = ω c , ω n ; t n ) . The third situation corresponds to the case in which thevariable starts in the barrier vicinity and stays near it, Π gm(cid:15) → ( ω = ω c , ω n = ω c ; t n ) .To do so, the behaviour of the Gaussian distribution near the barrier, as thevariable approaches it, is obtained by the Taylor expansion in the element ∆ ω = ω n − ω n − . We will see that there is a regime change in the relation of theprobability with time discretization (cid:15) as the variable tends to the barrier: ω n → ω c .To arrive at the expressions Π gm(cid:15) → ( ω , ω n = ω c ; t n ) , Π gm(cid:15) → ( ω = ω c , ω n ; t n ) and Π gm(cid:15) → ( ω = ω c , ω n = ω c ; t n ) , we expand the Gaussian density in powers of √ (cid:15) . B.1. Behaviour of the Gaussian distribution near the barrier We start by getting some relations from Π gm(cid:15) ( ω , ω n ; t n ) (3.12). From (3.11), ω n − = ω n − ∆ ω . In (3.12), changing the variable to ∆ ω , d (∆ ω ) = − dω n − , for agiven ω n , fixed. Thus, the limits of integration are ∆ ω | = ω n − ω n − | ω C = ω n − ω c and ∆ ω | = ω n − ω n − | −∞ = ∞ ; inverting them, due to the negative sign comingfrom d (∆ ω ) , we have: Π gm(cid:15) ( ω , ω n ; t n = t n − + (cid:15) ) = − ˆ ω n − ω c ∞ d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ; t n − )= ˆ ∞ ω n − ω c d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ; t n − ) nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions = ˆ ∞ ω n − ω c d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ∆ ω ; t n − ) . (B.1)Notice that lim (cid:15) → Ψ (cid:15) (∆ ω ) = δ (∆ ω ) . (B.2)If ω n − ω c < (that is, if the variable crosses the barrier after t n ), it includesthe support of the Dirac δ function. If ω n − ω c > , the integral is zero because itis outside the support. And, if ω n = ω c , with the condition of the continuum limit (cid:15) → , it goes to zero because of the initial condition. Therefore, Π gm(cid:15) → ( ω , ω n ; t n ) = 0 , if ω n ≥ ω c . (B.3)In the case ω n < ω c , we analyze (B.1), starting by expanding its LHS, in termsof t n − : Π gm(cid:15) ( ω , ω n ; t n − + (cid:15) ) = Π gm(cid:15) ( ω , ω n ; t n − )+ (cid:15) ∂ Π gm(cid:15) ( ω , ω n ; t n − ) ∂t n − + (cid:15) ∂ Π gm(cid:15) ( ω , ω n ; t n − ) ∂t n − + ... (B.4)Expanding in Taylor series the RHS (B.1), in terms of ∆ ω , in the region of ∆ ω = 0 , that is, when ω n → ω c : ˆ ∞ ω n − ω c d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ∆ ω ; t n − ) == ∞ (cid:88) i =0 ( − i i ! ∂ i Π gm(cid:15) ( ω , ω n ; t n − ) ∂ω in ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) (B.5)where, when expanding in ∆ ω , which involves ∂ Π gm(cid:15) ( ω ,ω n − ∆ ω ; t n − ) ∂ ∆ ω , we applied thechain rule (also considering the expansion in the region ∆ ω = 0 ): ∂∂ ∆ ω = ∂∂ ( ω n − ∆ ω ) ∂ ( ω n − ∆ ω ) ∂ ∆ ω = ∂∂ ( ω n − ∆ ω ) · ( − (cid:12)(cid:12)(cid:12)(cid:12) ∆ ω =0 = ∂∂ω n · ( − . (B.6)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld For second order, ∂ ∂ ∆ ω = ∂∂ ∆ ω (cid:18) ∂∂ ∆ ω (cid:19) = ∂∂ ( ω n − ∆ ω ) · (cid:18) ∂∂ ( ω n − ∆ ω ) · ( − (cid:19) ∂ ( ω n − ∆ ω ) ∂ ∆ ω = ∂ ∂ ( ω n − ∆ ω ) · ( − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ω =0 = ∂ ∂ω n · ( − . (B.7)Then, for i − th order, ∂ i ∂ ∆ ω i = ∂ i ∂ω in · ( − i . (B.8)Back to (B.5), we see that, changing to the variable y = ∆ ω √ (cid:15) , ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) = (2 (cid:15) ) i/ √ π ˆ ∞− ( ωc − ωn ) √ (cid:15) dy · y i e − y . (B.9)Consider the cases i = 0 and i = 1 , with ω n → ω c : ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) (cid:12)(cid:12)(cid:12)(cid:12) i =0 ,ω n → ω c = ⇒ ˆ ∞ d (∆ ω ) Ψ (cid:15) (∆ ω ) = 12 (B.10) ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) (cid:12)(cid:12)(cid:12)(cid:12) i =1 ,ω n → ω c = ⇒ ˆ ∞ d (∆ ω ) (∆ ω ) Ψ (cid:15) (∆ ω )= (2 (cid:15) ) / √ π ˆ ∞ dy · y · e − y = (cid:16) (cid:15) π (cid:17) / (B.11)where we integrated by parts. Back to (B.5), taking, as mentioned, the terms i = 0 and i = 1 , ˆ ∞ d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ∆ ω ; t n − )= 12 Π gm(cid:15) ( ω , ω c ; t n − ) − (cid:16) (cid:15) π (cid:17) / ∂∂ω n Π gm(cid:15) ( ω , ω n ; t n − ) (cid:12)(cid:12)(cid:12)(cid:12) ω n = ω c + ... (B.12)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions Therefore, when ω n − ω c < , the behaviour when ω n → ω c , with (cid:15) → , dependson √ (cid:15) . Consider the case ω n − ω c < , but when we are not in the situation ω n → ω c .In this case, in the continuum (cid:15) → , the inferior limit − ( ω c − ω n ) √ (cid:15) of the integral (B.9)becomes −∞ . Then, the new integral (B.9), with new limits, is ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) = (2 (cid:15) ) i/ √ π ˆ ∞− ( ωc − ωn ) √ (cid:15) dy · y i e − y → (2 (cid:15) ) i/ √ π ˆ ∞−∞ dy · y i e − y . (B.13)Because ˆ ∞−∞ dy · y n e − y ∼ = 1 + ( − n √ π n/ ( n − , (B.14)then ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) → (cid:15) i/ ( n − i even i odd . (B.15)In this case, the expansion B.5, with the analogous terms to (B.10) and (B.11),respectively, for i = 0 and i = 2 are: ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) (cid:12)(cid:12)(cid:12)(cid:12) i =0 ,ω n <ω c = ⇒ ˆ ∞−∞ d (∆ ω ) Ψ (cid:15) (∆ ω ) = 1 (B.16) ˆ ∞ ω n − ω c d (∆ ω ) (∆ ω ) i Ψ (cid:15) (∆ ω ) (cid:12)(cid:12)(cid:12)(cid:12) i =2 ,ω n <ω c = (cid:15). (B.17)Then, in the case ω n < ω c , (cid:15) → , but ω n not near ω c , ˆ ∞−∞ d (∆ ω ) Ψ (cid:15) (∆ ω ) Π gm(cid:15) ( ω , ω n − ∆ ω ; t n − )= Π gm(cid:15) ( ω , ω c ; t n − ) + (cid:15) ∂ ∂ω n Π gm(cid:15) ( ω , ω n ; t n − ) (cid:12)(cid:12)(cid:12)(cid:12) ω n = ω c + ... (B.18)Thus, there is a regime change with respect to (cid:15) in the passage from ω n < ω c to the case ω n < ω c , with the extra condition ω n → ω c : in the first case, the nextnternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld leading term of the expansion (B.18) behaves as (cid:15) , while in the second, the nextleading term dominating the expansion (B.12) follows √ (cid:15) . Such transition is ruledby the inferior limit ( ω c − ω n ) √ (cid:15) in integral (B.9). To tackle this, we define η = ( ω c − ω n ) √ (cid:15) . (B.19)We write Π gm(cid:15) in the form Π gm(cid:15) ( ω , ω n ; t n ) = C (cid:15) ( ω , ω n ; t n ) v ( η ) . (B.20)Here, C is the smooth part of the function, while v ( η ) is responsible for theregime transition of the function Π gm(cid:15) . We must impose lim η →∞ v ( η ) = 1 (B.21)so that C is the solution of Π gm(cid:15) when ω c − ω n is finite and positive. Consider theGaussian probability density, which, in the continuum limit ( (cid:15) → , in the presenceof a barrier, is the solution (3.13): Π gm(cid:15) → ( ω , ω n ; t n ) = 1 √ πt n e α ( ω n − ω ) − α t n (cid:20) e − ( ωn − ω tn − e − (2 ωc − ωn − ω tn (cid:21) . (B.22)Isolating ω n in (B.19), Π gm(cid:15) → ( ω , ω n ; t n ) = 1 √ πt n e α ( ω c − η √ (cid:15) − ω ) − α t n (cid:20) e − ( ωc − η √ (cid:15) − ω ) tn − e − ( ωc + η √ (cid:15) − ω ) tn (cid:21) . (B.23)Expanding in powers of √ (cid:15) , which means that η → ∞ and, consequently, (B.21): exp (cid:18) − t n (cid:16) ω c ∓ η √ (cid:15) − ω (cid:17) (cid:19) = e − tn ( ω c − ω ) (cid:34) ± η √ t n ω c − ω ) √ (cid:15) + . . . (cid:35) (B.24)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions exp (cid:16) α (cid:16) ω c − η √ (cid:15) − ω (cid:17)(cid:17) = e α ( ω c − ω ) (cid:104) − η √ α √ (cid:15) + . . . (cid:105) (B.25) ∴ C (cid:15) ( ω = 0 , ω n , T ) = 1 √ πt n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) (cid:104) − η √ α √ (cid:15) + . . . (cid:105) ·· (cid:34) η √ t n ( ω c − ω ) √ (cid:15) − η √ t n ( ω c − ω ) √ (cid:15) + . . . (cid:35) = 1 √ πt n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) (cid:104) − η √ α √ (cid:15) + . . . (cid:105) (cid:34) η √ t n ( ω c − ω ) √ (cid:15) + . . . (cid:35) = √ (cid:15) η √ π ( ω c − ω ) t / n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) + O ( (cid:15) ) . (B.26)In (B.20), when ω n → ω c , Π gm(cid:15) ( ω = 0 , ω n ; t n = T ) = √ (cid:15)γ ( ω c − ω ) t / n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) + O ( (cid:15) ) (B.27) γ = 2 √ π lim η → ηv ( η ) . (B.28)In this equation, the limit is taken as η → , because it corresponds to ω n → ω c in definition (B.19). Next, we will show that γ = 1 √ π e α ( ω n − ω c ) . (B.29)Now consider the derivative with respect to the barrier, given by (A.13). In thelimit (cid:15) → , n − (cid:88) i =1 → (cid:15) ˆ t n dt i . (B.30)In (A.13),nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld ∂ Π gm(cid:15) → ∂ω c ( ω = 0 , ω n ; t n ) = n − (cid:88) i =1 Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i )= ˆ t n dt i lim (cid:15) → (cid:15) Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i ) . (B.31)The LHS of (B.31) is computed by (B.22). In the case ω (cid:54) = 0 , ∂ Π gm(cid:15) → ∂ω c ( ω , ω n ; t n ) = 2 √ πt n e α ( ω n − ω ) − α t n (cid:20) (2 ω c − ω n − ω ) t n e − (2 ωc − ωn − ω tn (cid:21) = (cid:18) π (cid:19) / (2 ω c − ω n − ω ) t / n e α ( ω n − ω ) − α t n e − (2 ωc − ωn − ω tn = (cid:18) π (cid:19) / (2 ω c − ω n − ω ) t / n e α ( ω n − ω c ) e − (2 ωc − ωn − ω − αtn )22 tn . (B.32)The RHS (B.31) is given by (B.27). We notice that Π gm(cid:15) → ( ω c , ω n ; t n ) = Π gm(cid:15) → ( ω n , ω c ; t n ) . (B.33)This can be seen by (2.3) and (3.8): Π gm(cid:15) → ( ω c , ω n ; t n ) ⇒ W gm ( ω c , ω , ..., ω n ; t n ) = 1(2 π(cid:15) ) n/ e − ( ω − ωc )22 (cid:15) − n − (cid:80) i =1 (cid:20) ( ωi +1 − ωi ) (cid:15) (cid:21) − ( ωn − ωn − ) (cid:15) (B.34) Π gm(cid:15) → ( ω n , ω c ; t n ) ⇒ W gm ( ω n , ω , ..., ω c ; t n ) = 1(2 π(cid:15) ) n/ e − ( ω − ωn )22 (cid:15) − n − (cid:80) i =1 (cid:20) ( ωi +1 − ωi ) (cid:15) (cid:21) − ( ωc − ωn − ) (cid:15) . (B.35)This happens because ω (= ω c ) and ω n are not integration variables in (2.3),that is, ω c to be with ω n − or ω doen not matter, because they are integrated innternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions the same integration limits. Therefore, (B.33) is valid. Equation (B.27), with ω (cid:54) = 0 is written as: Π gm(cid:15) → ( ω , ω c ; t n ) = √ (cid:15)γ ω c − ω t / n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) . (B.36)Then, Π gm(cid:15) → ( ω c , ω n ; t n ) = Π gm(cid:15) → ( ω n , ω c ; t n )= √ (cid:15)γ ω c − ω n t / n e − α t n e α ( ω c − ω n ) e − tn ( ω c − ω n ) . (B.37)Returningto (B.31), in Π gm(cid:15) ( ω , ω c ; t i ) we use (B.27), and, in Π gm(cid:15) ( ω c , ω n ; t n − t i ) , (B.37).We also use ˆ c dx x / ( c − x ) / e − a x − b c − x ) = √ π a + bab c / e − ( a + b )22 c , (B.38)so, ˆ t n dt i lim (cid:15) → (cid:15) Π gm(cid:15) ( ω , ω c ; t i ) Π gm(cid:15) ( ω c , ω n ; t n − t i )= ˆ t n dt i lim (cid:15) → (cid:15) (cid:32) √ (cid:15)γ ω c − ω t / i e − α t i e α ( ω c − ω ) e − ti ( ω c − ω ) (cid:33) ·· (cid:32) √ (cid:15)γ ω c − ω n ( t n − t i ) / e − α ( t n − t i ) e α ( ω c − ω n ) e − ( tn − ti ) ( ω c − ω n ) (cid:33) = ˆ t n dt i γ ( ω c − ω ) ( ω c − ω n )( t n − t i ) / t / i e α ( ω c − ω ) e α ( ω c − ω n ) e − ti ( ω c − ω ) e − ( tn − ti ) ( ω c − ω n ) e − α t n = √ π (2 ω c − ω n − ω ) t / n γ e − [(2 ωc − ωn − ω − αtn ]22 tn . (B.39)Equating with (B.32),nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld γ = 1 √ π e α ( ω n − ω c ) . (B.40)Back to (B.36), with ω (cid:54) = 0 , Π gm(cid:15) ( ω , ω c , t n = T ) = √ (cid:15) √ π e α ( ω n − ω c ) ω c − ω t / n e − α t n e α ( ω c − ω ) e − tn ( ω c − ω ) = √ (cid:15) √ π e α ( ω n − ω ) ω c − ω t / n e − α t n e − tn ( ω c − ω ) . ∴ Π gm(cid:15) ( ω , ω c , t n = T ) = √ (cid:15) √ π e α ( ω n − ω c ) ω c − ω t / n e − [( ωc − ω − αtn ]22 tn . (B.41)In the case of ω n < ω c , we use (B.33) and (B.41) to analyze the situation wherethe process starts near the barrier and finishes at ω n : Π gm(cid:15) ( ω c , ω n , t n = T ) = √ (cid:15) √ π e α ( ω n − ω c ) ω c − ω n t / n e − [( ωc − ωn ) − αtn ]22 tn ; ω n < ω c . (B.42)In the case of Π gm(cid:15) ( ω c , ω c , t n = T ) , we must have translation invariance, and Π gm(cid:15) ( ω , ω c , t n = T ) can only depend on ω and ω c with ω c − ω . The expansion(B.27), already developed in the case of (B.36), with ω n → ω c (or η → ), tends tozero, as in (B.41). The next term of the expansion (B.26) must be proportional to (cid:15)/t / n and to the drift term: Π gm(cid:15) ( ω c , ω c , t n = T ) = c (cid:15)t / n e − α tn . (B.43)In Maggiore & Riotto (2010a), the identification of the constant is done by(3.12), and it is sufficient to study the case of n = 2 variables. Π gm(cid:15) ( ω , ω , t ) = ˆ ω c −∞ dω π(cid:15) e − (cid:15) [ ( ω − ω − (cid:15)α ) +( ω − ω − (cid:15)α ) ] , (B.44)We can solve this integral in Mathematica, taking into account that in two steps t n = 2 (cid:15) : Π gm(cid:15) ( ω , ω ; t ) = 12 π(cid:15) e − (2 (cid:15)α + ω − ω (cid:15) ) √ π √ (cid:15) (cid:18) Erf (cid:20) ω − ω √ (cid:15) (cid:21)(cid:19) . (B.45)nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions e − (2 (cid:15)α + ω − ω (cid:15) ) = e − (cid:15) α (cid:15) ) − (cid:15)α ( ω − ω ) (cid:15) − ( ω − ω (cid:15) ) We make ω − ω ∼ , because we are in the case ω → ω c and ω n → ω c : e − ( ω − ω (cid:15) ) ∼ (cid:20) ω − ω )2 (2 (cid:15) ) e − ( ω − ω (cid:15) ) (cid:21) ω − ω =0 ( ω − ω )+ 12 (cid:34) 22 (2 (cid:15) ) e − ( ω − ω (cid:15) ) − (cid:18) ω − ω )2 (2 (cid:15) ) (cid:19) e − ( ω − ω (cid:15) ) (cid:35) ω − ω =0 ( ω − ω ) = 1 + 12 t n ( ω − ω ) → Keeping the term in (cid:15) and, given that Erf ( ω − ω √ (cid:15) ∼ 0) = 0 , Π gm(cid:15) ( ω = ω c , ω = ω c ; t ) = 1 √ π √ (cid:15) . (cid:15) (cid:15) e − tnα = 1 √ π (cid:15)t / n e − α tn . (B.46)Hence, c = 1 √ π (B.47) Π gm(cid:15) ( ω c , ω c , t n = T ) = 1 √ π (cid:15)t / n e − α tn . (B.48)In a nutshell, we analyzed the behaviour near the barrier: Π gm(cid:15) → ( ω , ω c ; t n = T ) , Π gm(cid:15) ( ω c , ω n , t n = T ) and Π gm(cid:15) ( ω c , ω c , t n = T ) , described by (B.41), (B.42) and(B.48), respectively.nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld B.2. Analysis of divergent and finite terms in n − (cid:80) i,j =1 ∂ i ∂ j In (4.17) there is the term n − (cid:80) i,j =1 ∂ i ∂ j , which was used in (A.14) in the followingway: n − (cid:88) i,j =1 ∂ i ∂ j = 2 (cid:88) i Drift of the non-gaussian distribution: the specificcase of κ In the case of the density of probability (4.60) be expanded up to the th orderin derivatives, we evaluate the integral (4.63). α = 1 t n σ (cid:26)(cid:20) ( r − r f ) − t n σ (cid:21) + { , , , , · [1 , , , , σ (cid:2) , , , κ + 1 , , , · (cid:0) κ + κ (cid:1) σ +259 , , · (35 κ κ + κ ) σ +32 , , · (cid:0) κ + 56 κ κ + κ (cid:1) σ + 3 , , · (126 κ κ + 84 κ κ + κ ) σ +360 , · (cid:0) κ + 6 · (cid:0) κ + 35 κ κ + 20 κ κ (cid:1)(cid:1) σ +32 . · ( κ + 33 · (14 κ κ + 10 κ κ + 5 κ κ )) σ nternational Journal of Theoretical and Applied Finance© World Scientific Publishing Company A. Catalão & R. Rosenfeld +2 , · (cid:0) κ + 11 · (cid:0) κ + 72 κ κ + 45 κ κ + 20 κ κ (cid:1)(cid:1) σ +210 · ( κ + 143 · (2 κ κ + 12 κ κ + 9 κ κ + 5 κ κ )) σ +15 · (cid:0) κ + 13 · (cid:0) κ κ + 11 · (cid:0) κ κ + 12 κ + 21 κ κ + 14 κ κ (cid:1)(cid:1)(cid:1) σ + ( κ + 13 · (35 κ κ + 105 κ κ + 231 κ κ + 495 κ κ + 385 κ κ )) σ + 10 , , , · σ · (5 κ + κ σ )]] − (cid:111)(cid:111) . (C.1) References M. Avellaneda, R. Buff, C. Friedman, N. Grandechamp, L. Kruk & J. Newman (2001)Weighted Monte Carlo: a new technique for calibrating asset-pricing models, International Journal of Theoretical and Applied Finance (01), 91-119.R. Balieiro & R. Rosenfeld (2004) Testing Option Pricing with Edgworth Expansion, Physica A , 484-490.F. Black & M. Scholes (1973) The pricing of options and corporate liabilities, Journal ofPolitical Economy (3), 637–654.D. Breeden & R. Litzenberger (1978) Prices of state-contingent claims implicit in optionprices, Journal of Business (4), 621-651.A. Catalão & R. Rosenfeld (2010). 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