Pricing double Parisian options using numerical inversion of Laplace transforms

Transcription

1 Pricing double Parisian options using numerical inversion of Laplace transforms Jérôme Lelong (joint work with C. Labart) Conference on Numerical Methods in Finance (Udine) Thursday 26 June 2008 J. Lelong (MathFi INRIA) June 26, / 26

2 Plan 1 Presentation of the Parisian options Definition The different options Mathematical setting 2 Several approaches Link between single and double Parisian options 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) June 26, / 26

3 Presentation of the Parisian options Definition Definition barrier options counting the time spent in a row above (resp. below) a fixed level (the barrier). If this time is longer than a fixed value (the window width), the option is activated ( In ) or canceled ( Out ). J. Lelong (MathFi INRIA) June 26, / 26

4 Presentation of the Parisian options Definition Definition D b -1.0 D FIG.: single barrier Parisian option J. Lelong (MathFi INRIA) June 26, / 26

5 Presentation of the Parisian options Definition Definition D b FIG.: single barrier Parisian option J. Lelong (MathFi INRIA) June 26, / 26

6 Presentation of the Parisian options Definition Definition (double barrier) bup 0.0 blow -0.5 D -1.0 D FIG.: double barrier Parisian option J. Lelong (MathFi INRIA) June 26, / 26

7 Presentation of the Parisian options Definition Definition (double barrier) bup D 0.0 blow -0.5 D -1.0 D FIG.: double barrier Parisian option J. Lelong (MathFi INRIA) June 26, / 26

8 Presentation of the Parisian options The different options A few Payoffs Single Parisian Down In Call, φ(s) = (S T K ) + 1 { 0 t 1 <t 2 T t 2 t 1 D, s.t. } u [t. 1,t 2 ] S u L L is the barrier and D the option window. Double Parisian Out Call φ(s) = (S T K ) + 1 { 0 t 1 <t 2 T t 2 t 1 D, u [t 1,t 2 ] s.t. S u <L up and S u >L low }. J. Lelong (MathFi INRIA) June 26, / 26

9 Presentation of the Parisian options Mathematical setting Mathematical setting Let W = (W t,0 t T) be a B.M on (Ω,F,Q), with F = σ(w ). Assume that σ2 (r δ S t = x e 2 )t+σw t. We set m = r δ σ2 2 σ. We can introduce P Q s.t. e rt m2 (r+ E Q (φ(s t,t T)) = e 2 )T E P (e mz T φ(x e Z t,t T)) where Z is P-B.M. The price of a Parisian Down In Call (PDIC) is given by m2 (r+ f (T) = e ) t 2 t 1 D, s.t. } u [t, 1,t 2 ] Z u b }{{} "star" price 2 )T E P (e mz T (x e σz T K ) + 1 { 0 t 1 <t 2 T where b = 1 σ log( L x ) and Z is a P B.M. J. Lelong (MathFi INRIA) June 26, / 26

10 Presentation of the Parisian options Mathematical setting Brownian Excursions I D b Tb T b FIG.: Brownian excursions J. Lelong (MathFi INRIA) June 26, / 26

11 Presentation of the Parisian options Mathematical setting Brownian Excursions II Single Parisian Down In Call, φ(s) = (S T K ) + 1 { 0 t 1 <t 2 T t 2 t 1 D, s.t. u [t. 1,t 2 ] S u L }{{} =1 {T <T} b } J. Lelong (MathFi INRIA) June 26, / 26

12 Presentation of the Parisian options Mathematical setting Brownian Excursions II Single Parisian Down In Call, φ(s) = (S T K ) + 1 { 0 t 1 <t 2 T t 2 t 1 D, s.t. u [t. 1,t 2 ] S u L }{{} =1 {T <T} b Double Parisian Out Call, φ(s) = (S T K ) + 1 { } 0 t 1 <t 2 T t 2 t 1 D, u [t. 1,t 2 ] s.t. S u <L up and S u >L low }{{} =1 {T blow >T}1 {T + >T} bup } J. Lelong (MathFi INRIA) June 26, / 26

13 Plan 1 Presentation of the Parisian options Definition The different options Mathematical setting 2 Several approaches Link between single and double Parisian options 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) June 26, / 26

14 Several approaches Several approaches Crude Monte Carlo simulations perform badly because of the time discretization. Improvement by Baldi, Caramellino and Iovino (2000) using sharp large deviations. 2-dimensional PDE (Haber, Schonbucker and Willmott (1999)) : a second state variable counts the length of the excursion of interest. Chesney, Jeanblanc and Yor (1997) have shown that it is possible to compute the Laplace transforms (w.r.t. maturity time) of the single Parisian option prices. There is no explicit formula for the law of T : we only know its Laplace b transform. We know that the r.v. T b and Z T are independent and we know the b density of Z T. b J. Lelong (MathFi INRIA) June 26, / 26

15 Several approaches Laplace transform approach Use Laplace transforms as suggested by Chesney, Jeanblanc and Yor 1. Few numerical computations but not straightforward to implement. We have managed to find closed formulae for the Laplace transforms of the Parisian (single and double barrier) option prices. 1 [Chesney et al., 1997] J. Lelong (MathFi INRIA) June 26, / 26

16 Link between single and double Parisian options Link between single and double barrier Parisian options Consider a Double Parisian Out Call (DPOC) [ ] m2 ( DPOC(x,T;K,L low,l up ;r,δ) = e 2 +r)t E e mz T (x e σz T K ) + 1 {T >T}1 b {T + low bup >T}. Rewrite the two indicators 1 {T >T}1 b {T + low bup >T} = }{{} 1 1 {T <T} 1 b {T + low bup <T} Call }{{} PDIC(L low ) }{{} PUIC(L up ) +1 {T b low <T}1 {T + bup <T} } {{ } A = new term. J. Lelong (MathFi INRIA) June 26, / 26

17 Link between single and double Parisian options Double Parisian option prices Theorem 1 DPOC (x,λ;l low,l up ) =ŜC (x,λ) PDIC (x,λ;l low ) PUIC (x,λ;l up ) + Â(x,λ;L low,l up ) where  is the Laplace transform of A w.r.t. maturity time given by [ ] [ ] Â(x,λ;L low,l up ) = E e λt 2λZT b low 1 {T <T + b low bup } E e blow PUIC x<lup (x,λ;l up ) [ ] [ + E e λt + bup 1 {T + bup <T } E e ] 2λZT + bup PDIC b x>llow (x,λ;l low ), low where PUIC x<lup (resp. PDIC x>llow ) means that we use the definition of PUIC (resp. PDIC ) in the case x < L up (resp. x > L low ). J. Lelong (MathFi INRIA) June 26, / 26

18 Plan 1 Presentation of the Parisian options Definition The different options Mathematical setting 2 Several approaches Link between single and double Parisian options 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) June 26, / 26

19 Inversion formula Fourier series representation Fact Let f be a continuous function defined on R + and α a positive number. Assume that the function f (t)e αt is integrable. Then, given the Laplace transform f, f can be recovered from the contour integral f (t) = 1 2πi α+i α i e st f (s)ds, t > 0. Problem : the Laplace transforms have been computed for real values of the parameter λ. = Prove that they are analytic in a complex half plane = Find their abscissa of convergence. J. Lelong (MathFi INRIA) June 26, / 26

20 Analytical prolongation Analytical prolongation Proposition 1 (abscissa of convergence) The abscissa of convergence of the Laplace transforms of the star prices of Parisian options is smaller than (m+σ)2 2. All these Laplace transforms are analytic on the complex half plane {z C : Re(z) > (m+σ)2 2 }. Lemma 2 (Analytical prolongation of N ) The unique analytic prolongation of the normal cumulative distribution function on the complex plane is defined by N (x + iy) = 1 2π x e (v+iy)2 2 dv. J. Lelong (MathFi INRIA) June 26, / 26

21 Euler summation Trapezoidal rule I f (t) = 1 2πi α+i α i A trapezoidal discretization with step h = π t f π t eαt e (t) = f αt (α) + 2t t k=1 e st f (s)ds. leads to ( ( 1) k Re f ( α + i kπ t )). Proposition 2 (adapted from [Abate et al., 1999]) If f is a continuous bounded function satisfying f (t) = 0 for t < 0, we have e π t (t) = f (t) f π t (t) f e 2αt 1 e 2αt. J. Lelong (MathFi INRIA) June 26, / 26

22 Euler summation Trapezoidal rule II We want to compute numerically Truncation of the series f π (t) = eαt e f αt ( (α) + ( 1) k Re f t 2t t s p (t) = eαt e f αt (α) + 2t t k=1 p k=1 ( ( 1) k Re f ( α + i kπ t )). ( α + i πk )). t very slow convergence of s p (t) = need of an acceleration technique. J. Lelong (MathFi INRIA) June 26, / 26

23 Euler summation Euler summation For p,q > 0, we set q E(q,p,t) = Cq k 2 q s p+k (t). k=0 Proposition 3 Let p,q > 0 and f C q+4 such that there exists ɛ > 0 s.t. k q + 4, f (k) (s) = O (e (α ɛ)s ), where α is the abscissa of convergence of f. Then, f π t (t) E(q,p,t) te αt when p goes to infinity. f (0) αf (0) π 2 p! (q + 1)! 2 q (p + q + 2)! + O ( 1 p q+3 ), J. Lelong (MathFi INRIA) June 26, / 26

24 Regularity of the Parisian option prices Regularity of the Parisian option prices Theorem 3 (Regularity of double Parisian option prices) Let f (t) be the star price of a double barrier Parisian option with maturity time t. If b low < 0 and ( b up > ) 0, f is of class C and for all k 0, f (k) (t) = O e (m+σ)2 2 t when t goes to infinity. If b low > 0 or b up < 0, f is discontinuous in t = D. If b low = 0 or b up = 0, f is continuous. Moreover, if b low = 0 (resp. b up = 0), call prices (resp. put prices) are C 1 if x K (resp. if x K ). The price of a single barrier Parisian option, when the Parisian time of interest has a density function µ, can be written f (t) = t 0 φ(t u)µ(u)du with φ of class C. J. Lelong (MathFi INRIA) June 26, / 26

25 Regularity of the Parisian option prices Density of the Parisian times Theorem 4 The following assertions hold For b < 0 (resp. b > 0), the r.v. T b (resp. T + ) has a density µ w.r.t b Lebesgue s measure. µ is of class C and for all k 0, µ (k) (0) = µ (k) ( ) = 0. For b > 0 (resp. b < 0), the r.v. T b (resp. T + ) is not absolutely b continuous w.r.t Lebesgue s measure and P(T = D) > 0 (resp. b P(T + = D) > 0). b T0 has a density which tends to infinity in D+ and equals 0 in D. Nonetheless, the jump in D is integrable. J. Lelong (MathFi INRIA) June 26, / 26

26 Regularity of the Parisian option prices Density of the Parisian times FIG.: Density function of T 0 FIG.: Cumulative distribution function of T 0. J. Lelong (MathFi INRIA) June 26, / 26

27 Practical implementation Practical implementation For 2α/t = 18.4, p = q = 15, f (t) E(q,p,t) S t f (0) αf (0) Very few terms are needed to achieve a very good accuracy. The computation of E(q,p,t) only requires the computation of p + q terms. J. Lelong (MathFi INRIA) June 26, / 26

28 Practical implementation Numerical convergence for a PUOC PUOC with S 0 = 110, r = 0.1, σ = 0.2, K = 100, T = 1, L = 110, D = 0.1 year FIG.: Convergence of the standard summation w.r.t. p J. Lelong (MathFi INRIA) June 26, / 26

29 Practical implementation Regularity of a Double Out Call option S 0 = 100, r = 0.1, σ = 0.2, K = 90, L low = 100, L up = 120, D = 0.1 year FIG.: Regularity w.r.t maturity time J. Lelong (MathFi INRIA) June 26, / 26

30 Practical implementation Conclusion Closed formulae for double Parisian option prices Regularity of the Parisian option prices Existence and regularity of the density of Parisian times Accuracy of the inversion algorithm J. Lelong (MathFi INRIA) June 26, / 26

31 Practical implementation Abate, J., Choudhury, L., and Whitt, G. (1999). An introduction to numerical transform inversion and its application to probability models. Computing Probability, pages Chesney, M., Jeanblanc-Picqué, M., and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. in Appl. Probab., 29(1): J. Lelong (MathFi INRIA) June 26, / 26