PRICING DOUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS

Transcription

1 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS CÉLINE LABART AN JÉRÔME LELONG Abstract In this work, we study a double barrier version of the standard Parisian options We give closed formulae for the Laplace transforms of their prices with respect to the maturity time We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion Key words: double barrier, Parisian option, Laplace transform, numerical inversion, Brownian excursions, Euler summation Introduction The pricing and hedging of vanilla options is now part of the common knowledge and the general interest has moved on to more complex products So, practitioners need to be able to price these new products Among them, there are the so-called path-dependent options The ones we study in this paper are called double barrier Parisian options They are a version with two barriers of the standard Parisian options introduced by Marc Chesney, Monique Jeanblanc and Marc Yor in 997 (see Chesney et al(997chesney, Jeanblanc-Picqué, and Yor Before introducing double barrier Parisian options, we first recall the definition of Parisian options Parisian options can be seen as barrier options where the condition involves the time spent in a row above or below a certain level, and not only an exiting time ouble barrier Parisian options are options where the conditions imposed on the asset involve the time spent out of the range defined by the two barriers The valuation of single barrier Parisian options can be done by using several different methods: Monte Carlo simulations, lattices, Laplace transforms or partial differential equations As for standard barrier options, using simulations leads to a biased problem, due to the choice of the discretisation time step in the Monte Carlo algorithm The problem of improving the performance of Monte Carlo methods in exotic pricing has drawn much attention and has particularly been developed by Andersen and Brotherton-Ratcliffe(996 Concerning lattices, we refer the reader to the work Avellaneda and Wu(999 The idea of using Laplace transforms to price single barrier Parisian options is owed to Chesney et al(997chesney, Jeanblanc-Picqué, and Yor The Formulae of the Laplace transforms of all the different Parisian option prices can be found in Labart and Lelong(5 Schröder(3 and Hartley( have also studied these options using Laplace transforms An approach based on partial differential equations has been developed by Haber et al(999haber, Schonbucher, and PWilmott and Wilmott(998 ate: March, We wish to thank B Lapeyre for thoroughly reading the paper We also would like to address our deepest gratitude to A Alfonsi for the numerous comments he made on a previous version of this work and the fruitful discussions we had about complex analysis

2 C LABART AN J LELONG ouble Parisian options have already been priced by Baldi et al(baldi, Caramellino, and Iovino using Monte Carlo simulations corrected by the means of sharp large deviation estimates In this paper, we compute the prices of double barrier Parisian options by using Laplace transforms First, we give a detailed computation of the Laplace transforms of the prices with respect to the maturity time Then, we establish a formula for the inverse of the Laplace transforms using contour integrals Since it cannot be computed exactly, we give an upper bound of the error between the approximated price and the exact one We improve the approximation by using the Euler summation to get a fast and accurate numerical inversion The paper is organised as follows In section, we introduce the general framework and give precise definitions of double barrier Parisian option prices In section 3, we establish a Call Put parity relationship, which enables us to deduce the price of put options from the prices of call options In section 4, we carry out the computation of the Laplace transforms of double barrier Parisian option prices In section 5, we give a formula for the inversion of the Laplace transforms and state some results concerning the accuracy of the method The technique we use to prove these results is based on the regularity of option price (see Appendix A In section 6, we draw some graphs and compare the Laplace transform technique with the corrected Monte Carlo method of Baldi et al(baldi, Caramellino, and Iovino For the comparison, we have used the implementation of the algorithm of Baldi et al(baldi, Caramellino, and Iovino available in PREMIA efinitions Some notations Let S = {S t, t } denote the price of an underlying asset We assume that under the risk neutral measure Q, the dynamics of S is given by ds t = S t ((r δdt σdw t, S = x where W = {W t, t } is aqbrownian motion, x >, the volatility σ is a positive constant, r denotes the interest rate The parameter δ is the dividend rate if the underlying is a stock or the foreign interest rate in case of a currency We assume that both r and δ are constant It follows that S t = x e (rδσ /tσw t We introduce ( m = σ ( r δ σ Under Q, the dynamics of the asset is given by S t = x e σ(mtwt From now on, we consider that every option has a finite maturity time T Relying on Girsanov s Theorem (see Revuz and Yor(999, we can introduce a new probability P defined by dp dq FT = e mz T m T which makes Z = {Z t = W t mt, t T } a P-Brownian motion Thus, S rewrites S t = x e σzt under P Without any further indications, all the processes and expectations are considered under P PREMIA is a pricing software developed the MathFi team of INRIA Rocquencourt, see

3 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 3 ouble barrier Parisian option There are two different ways of measuring the time spent above or below a barrier Either, one only counts the time spent in a row and resets the counting each time the stock price crosses the barrier(s we call it the continuous manner or one adds the time spent in the relevant excursions without resuming the counting from each time the stock price crosses the barrier(s we call it the cumulative manner In practice, these two ways of counting time raise different questions about the paths of Brownian motion In this work, we only focus on continuous style options Knock Out A knock out double barrier Parisian call (respectively put is lost if S makes an excursion outside the range (L, L older than before T otherwise it pays at maturity time T (S T K (respectively (K S T where K is the strike We introduce b and b the barriers corresponding to L and L for the Brownian motion Z b = ( σ log L, b = ( x σ log L x For some level b, let us introduce the following notations g b t = g b t(z = sup {u t Z u = b}, T b = T b (Z = inf {t > (t gb t {Z t<b} > }, T b = T b (Z = inf {t > (t gb t {Zt>b} > } 9 5 b 3 b T b T b Figure Brownian paths Hence, the price of a knock out double barrier Parisian call (POC is given by m ( ( POC(x, T ; K, L, L ; r, δ = e rt E e mz T (S T K {T >T } {T >T } b b The two indicators can be rewritten {T b >T } {T b >T } = {T b <T } {T b <T } {T b <T } {T b <T }

4 4 C LABART AN J LELONG Since the rv T b and T b have a density wrt the Lebesgue measure (see Appendix B, one can use either strict or non-strict inequalities in the previous formula ealing with inequalities of the type {T ± b <T } is much simpler than {T ± >T }since we can b condition wrt F T ± and use the Strong Markov property Consequently, Equation ( can b be split into four terms using the prices of single barrier Parisian options To describe single barrier Parisian options, we use the following notations : POC means Parisian own and Out Call, whereas PUIP stands for Parisian Up and In Put and so on BSC simply denotes the price of a standard call option (3 where POC(x, T ; K, L, L ; r, δ =BSC(x, T ; K; r, δ PIC(x, T ; K, L ; r, δ (4 A = E m ( PUIC(x, T ; K, L ; r, δ e rt A, e mz T (S T K {T <T } {T <T }, b b For any function f of the maturity T, we introduce the star notation f(t = e (r m T f(t The computation of POC will be done using numerical inversion of its Laplace transform with respect to T Explicit formulae for the Laplace transforms of the first three terms in (3 BSC, PIC, PUIC can be found in Labart and Lelong(5 and are recalled in Appendix for the sake of clearness We only need to compute  = E e mzu (S u K {T <u} {T <u} e λu du A detailed computation can be found in b b Section 4 Knock In A knock in double barrier Parisian call (respectively put pays at maturity time T (S T K (respectively (K S T if S makes an excursion outside the range (L, L longer than before T and is lost otherwise The price of such an option (PIC is given by m ( PIC(x, T ; K, L, L ; r, δ = e rt E e mz T (S T K ( {T b <T } {T b <T } {T b <T } {T b <T } It is quite obvious that PIC can be expressed in terms of single barrier Parisian option prices (5 PIC(x, T ; K, L, L ; r, δ = PIC(x, T ; K, L ; r, δ where A is defined by (4 m ( PUIC(x, T ; K, L ; r, δ e rt A, 3 A Call Put parity relationship As for single barrier Parisian options, a parity relationship between calls and puts holds The basic idea of the relationship is that the processes Z and Z have the same law

5 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 5 Therefore, introducing the new Brownian motion Z = Z enables to rewrite the price of double barrier Parisian puts ( (3 POP(x, T ; K, L, L,, r, δ = Kxe r m T ( Let us introduce E e (mσ Z T ( x eσ Z T K {T b >T } {T b >T } m = (m σ, δ = r, r = δ, b = b, b = b One can easily check that m = σ (r δ σ and that r m = r m Moreover, by noticing that the barrier L corresponding to b = b is, it becomes clear that the L expectation on the right hand side of (3 can be interpreted as ( xk POC x, T ; K,,,, δ, r L L The same kind of relation holds for knock in options PIP(x, T ; K, L, L,, r, δ = xk PIC ( x, T ; K,,,, δ, r L L 4 Computation of Laplace transforms The computation of POC will be done using numerical inversion of its Laplace transform with respect to maturity time As explained above, the computation of the Laplace transform of POC boils down to the one of A First, we split A into two terms depending on the relative position of Tb and T b A =E {T <T } E b {T T <T } emz T (xe σz T K F T b b b E {T <T } E b {T T <T } emz T (xe σz T K = F T A A b b b The computation of  being quite similar to the one of Â, we only focus on  See Appendix C for a computation of  The computation of  is quite lengthy, so we split it into two separate steps First, we give a global formula for  (see Theorem 4 Then, we carry out a detailed computation of the different terms appearing in the expression of  in the case K L The reader is referred to Appendix C for the other cases Computations are quite long but not difficult, that s why we omit further details 4 Global formula for  Before giving a global formula for Â, we state a theorem, which ensues a corollary giving the global formula for  The rest of the paragraph is devoted to the proof of the theorem Theorem 4 In the case L x L (ie b b, we have A = k e my (xe σy Kh(T, ydy,

6 6 C LABART AN J LELONG where k = σ ln( K x The function h(t, y is characterised by its Laplace transform e (b b λ ĥ(λ, y = λψ ( λ ψ( λ xe x λ xb (4 y dx, where (4 ψ(z = xe x zx dx = z πe z N (z Corollary 4 In the case L x L (ie b b, we have (43 where H(z = Â = e (b b λ λψ ( λ ψ( λ xe x λ xz dx k e my (xe σy KH(y b dy, Proof of Theorem 4 In the first part of the proof, we show that A can be written as k e my (xe σy Kh(T, ydy for a certain function h Then, we compute the Laplace transform of h wrt t to get (4 Step : Computation of A We can write (44 A = E {T <T } E em(zt Z T Z T σ(z T Z T Z T b {T T b b <T } (xe b b K F b b T b = E {T b <T } A Let us introduce a new Brownian motion B = {B t = Z tt Z b T, t } independent of b F T thanks to the Strong Markov property On the set {T b b < T }, the indicator {T T <T } b b can be rewritten using B (45 A = E {T b Z T (B T T } e m(b T T Z T σ(b T T Z T b b (xe b b K F b T b b = E {T b z (B T τ}em(b T τ z (xe σ(b T τ z K ( = E Z T, Tb b z=z T b,τ=t b, Once again, conditioning wrt F T and introducing a new Brownian motion B = { B t = b z B tt B b z T, t } yields b z E(z, τ = E {T b z (B T τ}e e m( B T τt yz (xe σ( B T τt yz K, y=b T,t=T b b z z = E {T b z (B T τ}p T τt (f x (B b z T z, b z,

7 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 7 where f x (z = e mz (xe σz K, and P t (f x (z = πt f x (ue (uz t du As recalled by Chesney et al(997chesney, Jeanblanc-Picqué, and Yor, the random variables B T b z, we have T b z are independent Let ν (du denote the law of B T b z where h b (t, y = where h(t, y = of Z T b E(z, τ = = A = E = E {T b z (B T τ}p T τt (f x (u z b z f x (yh b z(t τ, y zdy, e (yu ν (du, (tt E b {T b t} π(t T b ν (du By using (44 and (45, we get {T b <T } f x (yh(t, ydy, f x (yh b Z T b (T T b, y Z T b dy, and E {T <T } h b z(t T b b, y z ν (dz and ν (dz denotes the density Step : Laplace transform of h wrt t Before computing ĥ(λ, y, we give a more explicit formula for the function h Using the law of B T (see Chesney et al(997chesney, Jeanblanc-Picqué, and Yo b z we have h b z(t, y = du u (b z e (u(b z γ(t, u y, b z where γ(t, x = E formula of ν (dz yields x {T b z (B t} e (tt b z (B π(tt b z (B b Using the expression of h and the explicit h(t, y = dz du b z u (b z e (b z e (u(b (46 z γ (t, u (y z, b z where γ (t, x = E {T t} γ(t T b b, x In view of (46, computing ĥ(λ, y boils down more or less to computing γ (λ, u (y z By doing some changes of variables, we get γ (λ, x = E e λt b e λv γ(v, xdv = E e λt λt b E e b z (B x λu e u e du πu

8 8 C LABART AN J LELONG x λu e u One can easily prove that e du = e λ x Furthermore, the values of πu λ E e λt b ande e λt b z (B are explicitly known (see Appendix Chesney et al(997chesney, Jeanblanc-P for a proof Then, γ (λ, x = e(b x b z λψ (, and Equation (4 follows λ λ 4 Computation of  For the sake of clearness, in the following we write θ = λ In this part, we state and prove a theorem giving the value of  in the case K L We refer the reader to Appendix C for the case K > L Theorem 4 In the case K L, we have the following result  = Ke(b b θ θψ (θ ψ (θ e (mθk m θ m θ σ e(b b θ ψ (θ ψ(θ e mb Kψ(m m θ L ψ((m σ (m σ θ Proof We want to compute Â, so we need to evaluate (47 I = k e my (xe σy KH(y b dy Standard computations lead to the following formula for the function H (see Corollary 4 for the definition of H e θx ψ(θ if x, H(x = e θx ψ(θ θ { πe λ N (θ x e θx N (θ x e θx} otherwise Using this result, we can compute I b (48 I = e my (xe σy KH(y b dy e my (xe σy KH(y b dy = I I k b Computation of I In view of the definition of H, this case is the simpler one Easy computations give I = ψ(θ { } e θb e (mθb L m σ θ K Ke (mθk m θ Computation of I The second integral in (48 can be split into three terms I = ψ(θ b I = θ πe λ e my (xe σy Ke θ(b y dy, b I 3 = θ πe λ b m θ m σ θ e my (xe σy Ke θ(b y N (θ b y dy, e my (xe σy Ke θ(yb N (θ b y dy

9 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 9 For I, we simply get I = ψ(θ e mb K mθ L mσθ I and I 3 are computed in the following way: we change variables (we introduce v = θ b y (for the valuation of I and we use the following equality a N (vebv dv = b (N (aeab e b N (a b, for a, b R, b We get I = θ πe mb e λ N (θ K m θ L m σ θ θ πe mb K m θ N (m e m L m σ θ N ((m σ e (mσ, I 3 = θ πe mb e λ N (θ K m θ L m σ θ θ πe mb K m θ N (m e m L m σ θ N ((m σ e (mσ Summing I, I and I 3 and using the definition of ψ (see (4 yield I = e mb ψ(θ K m θ L θe mb Kψ(m m σ θ m θ L ψ((m σ (m σ θ We sum I and I to get I =K ψ(θ e (mθk e θb m θ m σ θ θe mb Kψ(m m θ L ψ((m σ (m σ θ From the definitions of  and I (see (43, (47, we complete the proof of Theorem 4 5 The inversion of Laplace transforms This section is devoted to the numerical inversion of the Laplace transforms computed previously We recall that the Laplace transforms are computed with respect to the maturity time We explain how to recover a function from its Laplace transform using a contour integral The real problem is how to numerically evaluate this complex integral This is done in two separate steps involving two different errors First, as explained in Section 5 we replace the integral by a series The first step creates a discretisation error, which is handled by Proposition 5 Secondly, one has to compute a non-finite series This can be achieved by simply truncating the series but it leads to a tremendously slow convergence Here, we prefer to use the Euler acceleration as presented in Section 5 Proposition 5 states an upper-bound for the error due to the accelerated computation of the non finite series Theorem 5 gives a bound for the global error 5 The Fourier series representation Thanks to Widder(94, Theorem 9, we know how to recover a function from its Laplace transform Theorem 5 Let f be a continuous function defined on R and α a positive number If the function f(t e αt is integrable, then given the Laplace transform ˆf, f can be recovered from

10 C LABART AN J LELONG the contour integral (5 f(t = πi αi αi e st ˆf(sds, t > The variable α has to be chosen greater than the abscissa of convergence of ˆf The abscissa of convergence of the Laplace transforms of the double barrier Parisian option prices computed previously is smaller than (m σ / Hence, α must be chosen strictly greater than (m σ / For any real valued function satisfying the hypotheses of Theorem 5, we introduce a trapezoidal discretisation of Equation (5 (5 f π/t (t = eαt t f(α eαt ( ( ( k Re f α i kπ t t k= Proposition 5 If f is a continuous bounded function satisfying f(t = for t <, we have e αt (53 e π/t (t = f(t f π/t (t f e αt To prove Proposition 5,we need the following result adapted from Abate et al(999abate, Choudhury, and Theorem 5 Lemma 5 For any continuous and bounded function f such that f(t = for t <, we have (54 e π/t (t = f π/t (t f(t = f (t( k e kαt k = k Proof of Proposition 5 By performing a change of variables s = α iu in the integral in (5, we can easily obtain an integral of a real variable f(t = eαt f(α iu(cos(ut i sin(utdu π Moreover, since f is a real valued function, the imaginary part of the integral vanishes f(t = eαt ( ( Re f(α iu cos(ut Im f(α iu sin(utdu π We notice that ( ( ( ( Im f(α iu = Im f(α iu, Re f(α iu = Re f(α iu So, (55 f(t = eαt ( ( Re f(α iu cos(ut Im f(α iu sin(utdu π Using a trapezoidal integral with a step h = π t leads to Equation (5 Remembering that f(t = for t <, we can easily deduce from Lemma 5 that e π/t (t = f (t( k e kαt k=

11 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS Taking the upper bound of f yields (53 Remark 5 For the upper bound in Proposition 5 to be smaller than 8 f, one has to choose αt = 84 In fact, this bound holds for any choice of the discretisation step h satisfying h < π/t Simply truncating the summation in the definition of f π/t to compute the trapezoidal integral is far too rough to provide a fast and accurate numerical inversion One way to improve the convergence of the series is to use the Euler summation 5 The Euler summation To improve the convergence of a series S, we use the Euler summation technique as described by Abate et al(999abate, Choudhury, and Whitt, which consists in computing the binomial average of q terms from the p-th term of the series S The binomial average obviously converges to S as p goes to infinity The following proposition describes the convergence rate of the binomial average to the infinite series f π/t (t when p goes to Proposition 5 Let f be a function of class C q4 such that there exists ǫ > st k q 4, f (k (s = O(e (αǫs We define s p (t as the approximation of f π/t (t when truncating the non-finite series in (5 to p terms (56 s p (t = eαt t f(α eαt t p ( k Re ( ( f α i πk t k= and E(q, p, t = q k= Ck q q s pk (t Then, f π/t (t E(q, p, t teαt f ( ( αf( (p! q! π q (p q! O p q3 when p goes to infinity Using Propositions 5 and 5, we get the following result concerning the global error on the numerical computation of the price of a double barrier Parisian call option Theorem 5 Let f be the price of a double barrier Parisian call option Using the notations of Proposition 5, we have e αt (57 f(t E(q, p, t S e αt eαt t f ( ( αf( (p! q! π q O (p q! p q3 where α is defined in Theorem 5 Proof of Theorem 5 f being the price of a double barrier Parisian call option, we know that f is bounded by S Moreover, f is continuous (actually of class C, see Appendix A Hence, Proposition 5 yields the first term on the right-hand side of (57 Relying on Proposition A, we know that f is of class C and f (k (t = O(e (mσ t, k Since f(t = e (rm /t f(t, it is quite obvious that f is also of class C and f (k (t = O (e ((mσ /(rm /t, k Since α > (mσ, we can apply Proposition 5 to get the result Proof of Proposition 5 We compute the difference between two successive terms,

12 C LABART AN J LELONG where E(q, p, t E(q, p, t = eαt q t (58 a p = q Cq k (pk a pk, k= ( p e αs cos t πs f(sds Let g(s = e αs f(s Since g (k ( = for k q 3 and g (q4 is integrable, we can perform (q 3 integrations by parts in (58 to obtain a Taylor expansion when p goes to infinity (59 a p = c p c ( 4 p 4 c q p (q3/ O p q4 with c = 4t (f (αf( π We can rewrite (59 a p = c p(p c 3 p(p (p c ( q p(p (p q O p q4 Some elementary computations show that for j q Cq k ( pk (p k(p k (p k j = p! (q j! (p j!(p q j! k= Computing q k= Ck q ( pk a pk leads to Moreover, E(q, p, t E(q, p, t = ( p e αt ( p! (q! c q t (p q! O p! (q! (pq! is decreasing wrt p, so e αt p! (q! E(q,, t E(q, p, t c q t (p q! O p q4 ( p q3 Remark 5 Whereas Proposition 5 in fact holds for any h < π/t, the proof of Proposition 5 is essentially based on the choice of h = π/t since the key point is to be able to write E(q, p, t E(q, p, t as the general term of an alternating series The impressive convergence rate of E(q, p, t definitely relies on the choice of this particular discretisation step For a general step h, it is much more difficult to study the convergence rate and one cannot give an explicit upper-bound Remark 53 For αt = 84 and q = p = 5, the global error is bounded by S 8 t f ( αf( As one can see, the method we use to invert Laplace transforms provides a very good accuracy with few computations Remark 54 Considering the case of call options in Theorem 5 is sufficient since put prices are computed using parity relations and their accuracy is hung up to the one of call prices Theorem 5 also holds for single barrier Parisian options

13 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 3 6 Numerical examples In this section, we present some results obtained using the numerical inversion developed in Section 5 We have implemented our method in C and used the function erfc from the Octave library to compute the function N at a complex point In the examples, we choose p = 5, q = 5 and α = 84/T Hence, when the spot is of order the accuracy of our method is ensured up to 6 In Table, we compare the prices of a double barrier Parisian out call with S = K =, L = 9, L =, r = 95, δ = and T = obtained with our method and the corrected Monte Carlo method of Baldi et al(baldi, Caramellino, and Iovino with samples For the results obtained by the corrected Monte Carlo method, we precise the width of the confidence interval at level 95% The accuracy showed by this approach decreases as the delay of the option increases Our method is far more accurate and incredibly faster For instance, if we consider the option described above with = and 5 time steps for the Monte Carlo, our algorithm takes 5 ms (CPU time whereas the corrected Monte Carlo algorithm runs in sec (CPU time elay MC Price Price CI Laplace Table Comparison corrected Monte Carlo and Laplace Transform Figure shows the evolution of the price of a double Parisian knock out call wrt the delay when using the Laplace transform method or the corrected Monte Carlo one We can see that the price given by the Laplace transform method is in the confidence interval given by the corrected Monte Carlo method Figures 3 and 4 show the evolution of the price and the delta of a double barrier Parisian in call with respect to the spot and the strike The delta is computed using a finite difference scheme

14 4 C LABART AN J LELONG Price Monte Carlo Laplace Transform Inf Price Monte Carlo Sup Price Monte Carlo 35 Price elay Figure Comparison of corrected Monte Carlo and Laplace Transform

15 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 5 Price Strike Spot 5 Figure 3 Price of a ouble barrier Parisian In Call (σ =, r =, δ =, L = 8, U =

16 6 C LABART AN J LELONG elta Strike Spot 5 5 Figure 4 elta of a ouble barrier Parisian In Call

17 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 7 Appendix A Regularity of option prices Proposition A Let f(t be the star price of a double barrier Parisian option of maturity t If b < and b >, f is of class C and for all k, f (k (t = O (e (mσ t when t goes to infinity For the sake of clearness, we will only prove Proposition A for single barrier Parisian options as the scheme of the proof is still valid for double barrier Parisian options Once again, we can restrict to calls Let f(t = PIC(x, t; K, L; r, δ f(t = E e mzt (S t K {T b <t} Let W t denote Z tt Z b T Relying on the strong Markov property, b (A f(t = E {T Let ν denote the density of Z T b for its expression and µ the density of T b Z T b and T b f(t = b <t}e (xe σ(w tτ z K e m(w tτ z z=z T b, τ=t b (see Chesney et al(997chesney, Jeanblanc-Picqué, and Yor (see Proposition B for a proof of existence Since are independent, Equation (A can be written t dτ dz dw (xe σ(w tτz K e m(w tτz p(wν(zµ(τ where p(w = π e w A change of variable on τ gives t f(t = dτ dz dw (xe σ(w τz K e m(w τz p(wν(zµ(t τ Since µ is of class C and all its derivatives are null at and bounded on any interval, T (see Appendix B, one can easily prove that f is of class C and that for all k f (k (t = t dτ dz dw (xe σ(w τz K e m(w τz p(wν(zµ (k (t τ This proves the first part of Proposition A From Proposition B, we know that µ and all its derivatives are bounded Then, we can bound f (k f (k (t t dτ dz dw xe (mσ(w τz µ (k p(wν(z, µ (k t e (mσ τ dτ, xe (mσz ν(zdz e (mσ t x µ (k (m σ e (mσz ν(zdz Relying on one more use of the strong Markov property, the same kind of computations can be reproduced for double barrier Parisian options

18 8 C LABART AN J LELONG In this section, we assume b < Appendix B Regularity of the density of T b Proposition B The rv T b has a density µ wrt to Lebesgue s measure µ is of class C and for all k, µ (k ( = µ (k ( = To prove this proposition, we need the two following lemmas Lemma B Let N be the analytic prolongation of the cumulative normal distribution function on the complex plane The following equivalent holds N (r( i when r Lemma B For b <, we have for u R ( ( E e iut b = O e b u when u Proof of Proposition B We recall that (B E (e λ T b = e λb ψ(λ We define O = {z C; π 4 < arg(z < π 4 } One can easily prove that the function z ( E e z T b is holomorphic on the open set O and hence analytic Moreover, z ezb ψ(z is also analytic on O except perhaps in a countable number of isolated points These two functions coincide on R, so they are equal on O Consequently, we can derive the following equality For all z C with positive real part, we have ( (B E e zt b = e zb ψ( z We use the following convention: for any z C with positive real part, z is the only complex number z O such that z = z z Thanks to the continuity of both terms in (B, the equality also holds for pure imaginary numbers Hence, by setting z = iu for u R in Equation (B, we obtain the Fourier transform of T b ( E e iut b = e uib ψ( iu From Lemma B, we know that the Fourier transform of T b has a density µ wrt the Lebesgue measure given by T b is integrable on R, thus the rv Moreover, thanks to Lemma B, u u k µ(t = e uib π ψ( iu eiut du e uib ψ( iu is integrable and continuous Hence, µ is of class C Since µ(t = for t <, for all k, µ (k ( = Lemma B3 yields that for all k, lim t µ (k (t =

19 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 9 Proof of Lemma B N (x iy = π x e (viy dv It is easy to check that x N (x iy y N (x iy = and this definition coincides with the cumulative normal distribution function on the real axis, so it is the unique analytic prolongation We write N (x iy = N (x y yn (x iy, to get y x N (x iy = N (x i (v iu e (viu dvdu, π Taking x iy = r( i gives (B3 y = N (x i π r N (r( i = N (r i π e (xiu du e (riu du, = N (r i e r (t e itr rdt π For t,, e r (t r tends to when r goes to infinity The function r e r (t r is maximum for r = t, hence the following upper bound holds e r (t r t e (t for all t, The upper bound is integrable on,, so by using the bounded convergence theorem, we can assert that the integral on the right hand side of (B3 tends to when r goes to infinity Proof of Lemma B We only do the proof for u > For r >, ψ(r( i = r( i π e ri N (r( i Using the equivalent of N (r( i when r goes to infinity (see Lemma B enables to establish that ψ(r( i r π when r goes to infinity Noticing that iu = u ( i ends the proof Here is a quite obvious lemma we used in the proof of Proposition B Lemma B3 Let g be an integrable function on R, then lim t g(u e iut du = Appendix C Formulae of Â, Â Let us recall the definitions of A, A A =E {T <T } E b {T T <T } emz T (xe σz T K F b b T, b A =E {T <T } E b {T T <T } emz T (xe σz T K F b b T b

20 C LABART AN J LELONG C Formula of  Case L K  = Ke(b b θ θψ (θ ψ (θ e (mθk e(b b θ ψ (θ ψ(θ e mb m θ m θ σ Kψ(m m θ L ψ((m σ (m σ θ Case L < K  = K eb θ θψ(θ ψ(θ e (mθk m θ m σ θ K e(b b θ ψ (θ ψ(θ πe λ e mk e θ(bk N (θ b ( k m θ e θ(kb N (θ b k m σ θ ( m θ m σ θ e(b b θ ψ (θ ψ(θ πe mb mk m θ e m N (m b k L (m σ (m σ θ e (mσ N ((m σ b k C Formula of  Instead of computing  directly, which would mean doing again the same type of computations we did to evaluate Â, we first rewrite A to use as much as possible the computations we have already done in the valuation of  As Z and Z have the same law, introducing a new Brownian motion Z = Z leads to A = E = E Let A 3 be defined as E {T ( Z<T } {T b b {T ( Z<T } {T b b ( Z T b ( Z<T } em Z T (xe σ Z T K ( Z T b ( Z<T } e(mσ Z T (x Ke σ Z T {T b <T } {T b T b <T } emz T (K xe σz T Analogously with Theorem 4, we can write, for L x L, A 3 = k dyemy (K xe σy h(t, y, where the Laplace transform of h is still given by Equation (4 Then, we compute Â3, and we get  by replacing in Â3 m by (m σ, x by K, K by x, b by b and b by b (which means we replace L by xk L and L by xk L Case K L Case K L  = xe(b b θ θψ (θ ψ (θ e (mσθk m θ m σ θ

21 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS Â = xeb θ θψ(θ ψ(θ e (mσθk m θ m σ θ xe(b b θ ψ (θ ψ(θ πe λ e (mσk e (bkθ N (θ k b ( m θ m σ θ xe(b b θ L ψ (θ ψ(θ e (mσb K m θ L (m σ θ e (kb θ N (θ k b ( m θ m σ θ ( ψ(m m πe m N (m k b ( ψ((m σ (m σ πe (mσ N ((m σ k b Appendix Laplace transforms of single barrier Parisian option prices In this section, we only recall the prices of single barrier Parisian options that are required to compute the double barrier Parisian option prices In the following, d denotes bk Standard call option BSC(x, λ; K; r, δ = ( K θ e(mθk mθ mσθ K x Ke(mθk m θ (mσ( θ θ mθ mσθ for K x, for K x Parisian down in call PIC(x, λ; K, L; r, δ = ψ(θ e bθ ( θψ(θ Ke (mθk m θ, m σ θ for K > L and x L PIC(x, λ; K, L = e(mθb ψ(θ L (m σ θ Ke(mθk θψ(θ ( K ψ( m θ m πe m mn (d m ψ( (m σ πe (mσ (m σn ( d (m σ ( m θ ψ(θ θe λ πn (d θ m σ θ ( eλ π ψ(θ Kebθ e (mθk N (d θ m σ θ m θ for K L x

22 C LABART AN J LELONG 3 Parisian up in call PUIC(x, π λ; K, L; r, δ = e (mθb ψ(θ K m θ e m mn (d m L (m σ θ e (mσ (m σn (d (m σ ebθ ψ(θ Ke(mθk e λ πn (d θ ( m σ θ m θ ( e(mθk θψ(θ K m θ ( ψ(θ θ πe λ N (d θ m σ θ for x L K PUIC(x, λ; K, L; r, δ = e(mθb ψ(θ K m θ ψ( L m (m σ θ ψ( (m σ ebθ ψ(θ ( θψ(θ K e (mθk m θ m θ σ for K L and x L References Abate et al(999abate, Choudhury, and Whitt J Abate, L Choudhury, and G Whitt An introduction to numerical transform inversion and its application to probability models Computing Probability, pages 57 33, 999, Andersen and Brotherton-Ratcliffe(996 L Andersen and R Brotherton-Ratcliffe Exact exotics Risk, 9 (:85 89, 996 Avellaneda and Wu(999 M Avellaneda and L Wu Pricing parisian-style options with a lattice method International Journal of Theoretical and Applied Finance, (: 6, 999 Baldi et al(baldi, Caramellino, and Iovino P Baldi, L Caramellino, and M G Iovino Pricing complex barrier options with general features using sharp large deviation estimates In Monte Carlo and quasi- Monte Carlo methods 998 (Claremont, CA, pages 49 6 Springer, Berlin,, 3 Chesney et al(997chesney, Jeanblanc-Picqué, and Yor M Chesney, M Jeanblanc-Picqué, and M Yor Brownian excursions and Parisian barrier options Adv in Appl Probab, 9(:65 84, 997 ISSN -8678, 7, 8, 7 Haber et al(999haber, Schonbucher, and PWilmott R Haber, P Schonbucher, and PWilmott An american in paris OFRC Working Papers Series 999mf4, Oxford Financial Research Centre, 999 available at Hartley( P Hartley Pricing parisian options by laplace inversion ecisions in Economics & Finance, Labart and Lelong(5 C Labart and J Lelong Pricing parisian options Technical report, ENPC, ecember 5, 4 Revuz and Yor(999 Revuz and M Yor Continuous martingales and Brownian motion, volume 93 of Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences Springer-Verlag, Berlin, third edition, 999 ISBN Schröder(3 M Schröder Brownian excursions and Parisian barrier options: a note J Appl Probab, 4 (4: , 3 ISSN -9 Widder(94 V Widder The Laplace Transform Princeton Mathematical Series, v 6 Princeton University Press, Princeton, N J, 94 9 Wilmott(998 P Wilmott erivatives University Edition, 998

23 PRICING OUBLE BARRIER PARISIAN OPTIONS USING LAPLACE TRANSFORMS 3 CMAP, ECOLE POLYTECHNIQUE, CNRS, Route de Saclay 98 Palaiseau, FRANCE address: labart@cmappolytechniquefr CERMICS, Ecole des Ponts, ParisTech, 6-8 avenue Blaise Pascal, Champs sur Marne Marne La Vallée, FRANCE address: lelong@cermicsenpcfr