Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing double Parisian options using numerical inversion of Laplace transforms Jérôme Lelong (joint work with C. Labart) ❤tt♣✿✴✴❝❡r♠✐❝s✳❡♥♣❝✳❢r✴⑦❧❡❧♦♥❣ Conference on Numerical Methods in Finance (Udine) Thursday 26 June 2008 J. Lelong (MathFi – INRIA) June 26, 2008 1 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Plan Presentation of the Parisian options 1 Definition The different options Mathematical setting Pricing of Parisian options 2 Several approaches Link between single and double Parisian options Numerical evaluation 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi – INRIA) June 26, 2008 2 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation 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, 2008 3 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Definition Definition 1.5 1.0 0.5 0.0 b D -0.5 D -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F IG .: single barrier Parisian option J. Lelong (MathFi – INRIA) June 26, 2008 3 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Definition Definition 1.5 1.0 0.5 0.0 b D -0.5 -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F IG .: single barrier Parisian option J. Lelong (MathFi – INRIA) June 26, 2008 3 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Definition Definition (double barrier) 1.5 1.0 0.5 b up 0.0 b low D -0.5 D -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F IG .: double barrier Parisian option J. Lelong (MathFi – INRIA) June 26, 2008 4 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Definition Definition (double barrier) 1.5 1.0 0.5 D b up 0.0 b low D -0.5 D -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F IG .: double barrier Parisian option J. Lelong (MathFi – INRIA) June 26, 2008 4 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation The different options A few Payoffs Single Parisian Down In Call, φ ( S ) = ( S T − K ) + 1 � ∃ 0 ≤ t 1 < t 2 ≤ T � . , s.t. ∀ u ∈ [ t 1 , t 2 ] S u ≤ L t 2 − t 1 ≥ D 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 � . , ∃ u ∈ [ t 1 , t 2 ] s.t. S u < L up and S u > L low t 2 − t 1 ≥ D J. Lelong (MathFi – INRIA) June 26, 2008 5 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation 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 S t = x e ( r − δ − σ 2 2 ) t + σ W t . r − δ − σ 2 2 We set m = . We can introduce P ∼ Q s.t. σ e − rT E Q ( φ ( S t , t ≤ T )) = e − ( r + m 2 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 � � f ( T ) = e − ( r + m 2 2 ) T E P e mZ T ( x e σ Z T − K ) + 1 � ∃ 0 ≤ t 1 < t 2 ≤ T � , , s.t. ∀ u ∈ [ t 1 , t 2 ] Z u ≤ b t 2 − t 1 ≥ D � �� � "star" price � L � where b = 1 σ log and Z is a P − B.M. x J. Lelong (MathFi – INRIA) June 26, 2008 6 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Mathematical setting Brownian Excursions I 1.5 1.0 0.5 0.0 D b -0.5 -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T b T − b F IG .: Brownian excursions J. Lelong (MathFi – INRIA) June 26, 2008 7 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Mathematical setting Brownian Excursions II Single Parisian Down In Call, φ ( S ) = ( S T − K ) + 1 � ∃ 0 ≤ t 1 < t 2 ≤ T � . , s.t. ∀ u ∈ [ t 1 , t 2 ] S u ≤ L t 2 − t 1 ≥ D � �� � = 1 { T − b < T } J. Lelong (MathFi – INRIA) June 26, 2008 8 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Presentation of the Parisian options Numerical evaluation Mathematical setting Brownian Excursions II Single Parisian Down In Call, φ ( S ) = ( S T − K ) + 1 � ∃ 0 ≤ t 1 < t 2 ≤ T � . , s.t. ∀ u ∈ [ t 1 , t 2 ] S u ≤ L t 2 − t 1 ≥ D � �� � = 1 { T − b < T } Double Parisian Out Call, φ ( S ) = ( S T − K ) + 1 � ∀ 0 ≤ t 1 < t 2 ≤ T � . , ∃ u ∈ [ t 1 , t 2 ] s.t. S u < L up and S u > L low t 2 − t 1 ≥ D � �� � = 1 { T − > T } 1 { T + bup > T } blow J. Lelong (MathFi – INRIA) June 26, 2008 8 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Pricing of Parisian options Numerical evaluation Plan Presentation of the Parisian options 1 Definition The different options Mathematical setting Pricing of Parisian options 2 Several approaches Link between single and double Parisian options Numerical evaluation 3 Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation J. Lelong (MathFi – INRIA) June 26, 2008 9 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Pricing of Parisian options Numerical evaluation 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 − b : we only know its Laplace transform. We know that the r.v. T − b and Z T − b are independent and we know the density of Z T − b . J. Lelong (MathFi – INRIA) June 26, 2008 10 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Pricing of Parisian options Numerical evaluation 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, 2008 11 / 26
Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Pricing of Parisian options Numerical evaluation Link between single and double Parisian options Link between single and double barrier Parisian options Consider a Double Parisian Out Call (DPOC) � � DPOC ( x , T ; K , L low , L up ; r , δ ) = e − ( m 2 2 + r ) T E e mZ T ( x e σ Z T − K ) + 1 { T − blow > T } 1 { T + . bup > T } Rewrite the two indicators 1 { T − blow > T } 1 { T + bup > T } = 1 − 1 { T − − 1 { T + + 1 { T − blow < T } 1 { T + . blow < T } ���� bup < T } bup < T } � �� � � �� � � �� � Call PDIC ( L low ) PUIC ( L up ) A = new term J. Lelong (MathFi – INRIA) June 26, 2008 12 / 26
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