Pricing Parisian options using numerical inversion of Laplace transforms

Transcription

1 using numerical inversion of Laplace transforms Jérôme Lelong (joint work with C. Labart) Tuesday 23 October 2007 J. Lelong (MathFi INRIA) Tuesday 23 October / 33

2 Plan 1 Presentation of the Parisian options Definition The different options Some parity relationships 2 What is known Several approaches 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) Tuesday 23 October / 33

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 ). Parisian options are less sensitive to influential agent on the market than standard barrier options. Parisian options naturally appears in the analysis of structured products such as re-callable convertible bond: the owner wants to recall its bond if ever the underlying stock has been traded out of a given range for a while. Some attempts to use Parisian options to price credit risk derivatives. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

4 Presentation of the Parisian options Definition Definition (single barrier) D b -1.0 D FIG.: single barrier Parisian option J. Lelong (MathFi INRIA) Tuesday 23 October / 33

5 Presentation of the Parisian options Definition Definition (single barrier) D b FIG.: single barrier Parisian option J. Lelong (MathFi INRIA) Tuesday 23 October / 33

6 Presentation of the Parisian options Definition Definition (double barrier) bup D 0.0 bdown -0.5 D -1.0 D FIG.: double barrier Parisian option J. Lelong (MathFi INRIA) Tuesday 23 October / 33

7 Presentation of the Parisian options The different options Payoff Parisian Down In Call, (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. Parisian Up Out Call, Parisian Double Out Call (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 (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 down }. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

8 Presentation of the Parisian options The different options Definition 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. Let W = (W t,t 0) 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. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

9 Presentation of the Parisian options The different options Brownian Excursions I D b Tb T b FIG.: Brownian excursions J. Lelong (MathFi INRIA) Tuesday 23 October / 33

10 Presentation of the Parisian options The different options Brownian Excursions II g b t = sup{u t Z u = b}, T b = inf{t > 0 (t g b t ) 1 {Z t <b} > D}, T + b = inf{t > 0 (t g b t ) 1 {Z t >b} > D}. T : first time the B.M. Z stays below b longer than D. b Price of a PDIC : m2 (r+ f (T) = e 2 )T E P (e mz T (x e σz T K ) }{{ + 1 } {T b <T} ). }{{} Call payoff Parisian part Price of a Double Parisian Out Call : m2 (r+ f (T) = e 2 )T E P (e mz T (x e σz T K ) }{{ + } Call payoff 1 {T b low >T}1 {T + bup >T} } {{ } Parisian part ). J. Lelong (MathFi INRIA) Tuesday 23 October / 33

11 Presentation of the Parisian options Some parity relationships Some parity relationships Same kind of parity relationships as for standard barrier options. For instance, { }{ } { } { } ( D O U O 1 P P (x,t;k,l;r,δ) = xk P C U I D I x,t; 1 K, 1 ) L ;δ,r. ( ) ( PDOP(x,T;K,L;r,δ) = E e mz T (K x e σz T ) + 1 {T b >T} e By introducing the new B.M. W = Z, we can rewrite ( E e mw T (K x e σw T ) + 1 {T + xk E ) ( b >T} e r+ m2 2 ( e (m+σ)w T ( 1 ) T = x eσw T 1 K r+ m2 2 ) T. ) + ) (r+ 1 {T )e m2 + b >T} 2 T. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

12 Presentation of the Parisian options Some parity relationships Link between single and double barrier Parisian options Consider a Double Parisian Out Call (DPOC) [ ] m2 ( DPOC(x,T;K,L d,l u ;r,δ) = e 2 +r)t E e mz T (S T K ) + 1 {T >T}1 b {T + d bu >T}. Rewrite the two indicators 1 {T >T}1 b {T + d bu >T} = }{{} 1 Call 1 {T b d <T} }{{} PDIC(L=L d ) 1 {T + bu <T} }{{} PUIC(L=L u ) +1 {T <T}1 b {T + d bu <T} } {{ } new term. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

13 Plan 1 Presentation of the Parisian options Definition The different options Some parity relationships 2 What is known Several approaches 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) Tuesday 23 October / 33

14 What is known What is known about Parisian options There is no explicit formula for the law of T : we only know its b Laplace transform 1. We know that the r.v. T b and Z T are independent and we know the b density of Z T. b Chesney et al. have shown that it is possible to compute the Laplace transforms (w.r.t. maturity time) of the Parisian option prices. 1 [Chesney et al., 1997] J. Lelong (MathFi INRIA) Tuesday 23 October / 33

15 Several approaches Monte Carlo approach Crude Monte Carlo simulations perform badly because of the time discretisation. Attempt to improve the crude Monte Carlo by Baldi, Caramellino and Iovino. Recover the density of T by numerically inverting its the Laplace b transform. E P (e mz T (x e σz T K ) + 1 {T b <T} ) = where W Z. f (T) = 0 E P (e mz T b e mw T T b (x e σz T b e σw T T b K ) + 1 {T b <T} ), R e my e m2 2 (T t) C(x e σy,t t)1 {t<t} d T b Z T b (t,y), where C(z,t) is the price of a Call option with maturity t and spot z. Actually d T b Z T b = d T d Z b T. b J. Lelong (MathFi INRIA) Tuesday 23 October / 33

16 Several approaches EDP approach EDP for the price V (t,x) of a vanilla call option t V σ2 x 2 xx V + rx x V rv = 0 (t,x) [0,T) R +. The payoff is not Markovian. Need to introduce a second state variable = solve a 2-dimensional EDP 2. τ = t sup{u t : S u L}. { t V σ2 x 2 xx V + rx x V rv = 0, on [0,T) (L, ), t V σ2 x xx V + rx x V rv + τ V = 0, on [0,T) [0,L), continuity condition : V (L,t,τ) = V (L,t,0). 2 [Haber et al., 1999] J. Lelong (MathFi INRIA) Tuesday 23 October / 33

17 Several approaches Laplace transform approach Use Laplace transforms as suggested by Chesney, Jeanblanc and Yor 3. 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. 3 [Chesney et al., 1997] J. Lelong (MathFi INRIA) Tuesday 23 October / 33

18 Several approaches Example of a Laplace transform For all λ > (m+σ)2 2 and K > L, the Laplace transform of a Parisian Down In call price is given by with f (λ) = ψ( θ D)e 2bθ ( ) θψ(θ K e (m θ)k 1 D) m θ 1, m + σ θ ψ(z) = N (z) = + 0 z xe x2 2 +zx dx = 1 + z 2πe z2 2 N (z), 1 2π e u2 2 du. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

19 Plan 1 Presentation of the Parisian options Definition The different options Some parity relationships 2 What is known Several approaches 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi INRIA) Tuesday 23 October / 33

20 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 ˆf, f can be recovered from the contour integral f (t) = 1 2πi α+i α i e st ˆf (s)ds, t > 0. Problem : the Laplace transforms have been computed for real values of the parameter λ. = We have to prove that they are analytic in a complex half plane and find their abscissa of convergence. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

21 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 }. Proof : It is sufficient to notice that the star price of a Parisian option is bounded by E(e mz T (x e σw T +K )). E(e mz T (x e σw T +K )) K e m2 2 T +x e (m+σ)2 2 T = O (e (m+σ)2 2 T ). Hence, [Widder, 1941, Theorem 2.1] yields that the abscissa of convergence of the Laplace transforms of the star prices is smaller that (m+σ) 2 2. The second part of the proposition ensues from [Widder, 1941, Theorem 5.a]. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

22 Analytical prolongation Analytical prolongation Lemma 1 (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. Proof : It is sufficient to notice that the function defined above is holomorphic on the complex plane (and hence analytic) and that it coincides with the normal cumulative distribution function on the real axis. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

23 Euler summation Trapezoidal rule I f (t) = 1 2πi α+i α i A trapezoidal discretisation of step h = π t f π t eαt e (t) = f αt (α) + 2t t k=1 e st ˆf (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 Proposition 2 actually holds for h < 2 π t. e 2αt 1 e 2αt. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

24 Euler summation Trapezoidal rule II We want to compute numerically f π (t) = eαt e f αt ( (α) + ( 1) k Re f t 2t t k=1 We still need to approximate the series. s p (t) = eαt e f αt (α) + 2t t 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) Tuesday 23 October / 33

25 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 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 ˆf. Then, f π (t) E(q,p,t) te αt f (0) αf (0) ( ) p! (q + 1)! 1 t π 2 2 q (p + q + 2)! + O p q+3, when p goes to infinity. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

26 Regularity of the Parisian option prices Regularity of the Parisian option prices Proposition 4 Let f be the star price of a ( Parisian) option of maturity t. 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. Proof : [ ] f (t) = E e mz t (e σz t K ) + 1 {T b <t}. Relying on the strong Markov property, f (t) = E where W F T f (t) = t 0 dτ ( 1 {T b <t} E [ (xe σ(w t τ+z) K ) + e m(w t τ+z) ] z=z T b, τ=t b. b dz dw (xe σ(w τ+z) K ) + e m(w τ+z) p(w)ν(z)µ(t τ). J. Lelong (MathFi INRIA) Tuesday 23 October / 33 ),

27 Regularity of the Parisian option prices Density for a Parisian time Proposition 5 T b has a density µ w.r.t the Lebesgue measure. µ is of class C and for all k 0, µ (k) (0) = µ (k) ( ) = 0. Proof : ( ) E e λ2 2 T b = e λb ψ(λ D) for λ R. Both sides are analytic on O = {z C; π 4 < arg(z) < π 4 }. Continuity for ) arg(z) = ± π 4 (e = for all u R E iut b = e 2uib ψ(. 2iuD) µ(t) = 1 2π e 2uib ψ( 2iuD) e iut du. ( ) ( ) Moreover, E e iut b = O e b u when u = µ is C. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

28 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) Tuesday 23 October / 33

29 Practical implementation Numerical convergence for a PUOC I Consider a Parisian Up Out Call with S 0 = 110, r = 0.1, σ = 0.2, T = 1, L = 110, D = 0.1 year. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

30 Practical implementation Numerical convergence for a PUOC I Consider a Parisian Up Out Call with S 0 = 110, r = 0.1, σ = 0.2, T = 1, L = 110, D = 0.1 year. To achieve an acceptable accuracy without any Euler summation, p 1000 is required whereas the use of the Euler summation enables to cut down to p = q 15 (e.g. only 30 terms to be computed). J. Lelong (MathFi INRIA) Tuesday 23 October / 33

31 Practical implementation Numerical convergence for a PUOC II FIG.: Convergence of the Euler summation w.r.t. q for p = 10 J. Lelong (MathFi INRIA) Tuesday 23 October / 33

32 Practical implementation Improved Monte Carlo method for a PUOC FIG.: Convergence of the Improved Monte Carlo Method with 250 time steps. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

33 Practical implementation Convergence of a PUOC to a standard Up Out Call Barrier option J. Lelong (MathFi INRIA) Tuesday 23 October / 33

34 Practical implementation Price Monte Carlo Laplace Transform Inf Price Monte Carlo Sup Price Monte Carlo 3.5 Price Delay FIG.: Comparison of the improved MC with the Laplace method J. Lelong (MathFi INRIA) Tuesday 23 October / 33

35 Practical implementation Hedging Is the replicating portfolio generated by the delta? Probably yes, but not straightforward. How to compute the delta Differentiate the Laplace transforms of the prices to obtain the Laplace transforms of the deltas. Use some automatic differentiation tools to compute the Laplace transforms of the deltas. J. Lelong (MathFi INRIA) Tuesday 23 October / 33

36 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): Haber, R., Schonbucker, P., and Wilmott, P. (1999). Pricing parisian options. Journal of Derivatives, (6): Widder, D. V. (1941). The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J. J. Lelong (MathFi INRIA) Tuesday 23 October / 33