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Pricing double Parisian options using numerical inversion of Laplace - - PowerPoint PPT Presentation

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Pricing Parisian options Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing double Parisian options using numerical inversion of Laplace transforms Jrme Lelong (joint work with C. Labart)

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options

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

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options

Plan

1

Presentation of the Parisian options Definition The different options Mathematical setting

2

Pricing of Parisian options Several approaches Link between single and double Parisian options

3

Numerical evaluation Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation

  • J. Lelong (MathFi – INRIA)

June 26, 2008 2 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options 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, 2008 3 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Definition

Definition

D D b 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • 1.5
  • 1.0
  • 0.5

0.0 1.0 1.5 0.5

FIG.: single barrier Parisian option

  • J. Lelong (MathFi – INRIA)

June 26, 2008 3 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Definition

Definition

D b 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • 1.5
  • 1.0

0.0 0.5 1.0 1.5

  • 0.5

FIG.: single barrier Parisian option

  • J. Lelong (MathFi – INRIA)

June 26, 2008 3 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Definition

Definition (double barrier)

D D bup blow 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.5 1.0

FIG.: double barrier Parisian option

  • J. Lelong (MathFi – INRIA)

June 26, 2008 4 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Definition

Definition (double barrier)

D D D bup blow 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5

FIG.: double barrier Parisian option

  • J. Lelong (MathFi – INRIA)

June 26, 2008 4 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options The different options

A few Payoffs

Single Parisian Down In Call, φ(S) = (ST −K)+1 ∃ 0≤t1<t2≤T

t2−t1≥D

, s.t. ∀u∈[t1,t2]Su≤L .

L is the barrier and D the option window. Double Parisian Out Call φ(S) = (ST −K)+1 ∀ 0≤t1<t2≤T

t2−t1≥D

,∃u∈[t1,t2] s.t. Su<Lup and Su>Llow .

  • J. Lelong (MathFi – INRIA)

June 26, 2008 5 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Mathematical setting

Mathematical setting

Let W = (Wt,0 ≤ t ≤ T) be a B.M on (Ω,F,Q), with F = σ(W). Assume that St = xe(r−δ− σ2

2 )t+σWt .

We set m =

r−δ− σ2

2

σ

. We can introduce P ∼ Q s.t. e−rT EQ(φ(St,t ≤ T)) = e−(r+ m2

2 )T EP(emZT φ(xeZt,t ≤ T))

where Z is P-B.M. The price of a Parisian Down In Call (PDIC) is given by f (T) = e−(r+ m2

2 )T EP

  • emZT (xeσZT −K)+1 ∃ 0≤t1<t2≤T

t2−t1≥D

, s.t. ∀u∈[t1,t2]Zu≤b

  • "star" price

, where b = 1

σ log

L

x

  • and Z is a P−B.M.
  • J. Lelong (MathFi – INRIA)

June 26, 2008 6 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Mathematical setting

Brownian Excursions I

Tb T−

b

D b 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5

FIG.: Brownian excursions

  • J. Lelong (MathFi – INRIA)

June 26, 2008 7 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Mathematical setting

Brownian Excursions II

Single Parisian Down In Call, φ(S) = (ST −K)+ 1 ∃ 0≤t1<t2≤T

t2−t1≥D

, s.t. ∀u∈[t1,t2]Su≤L

  • =1{T−

b <T}

.

  • J. Lelong (MathFi – INRIA)

June 26, 2008 8 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Presentation of the Parisian options Mathematical setting

Brownian Excursions II

Single Parisian Down In Call, φ(S) = (ST −K)+ 1 ∃ 0≤t1<t2≤T

t2−t1≥D

, s.t. ∀u∈[t1,t2]Su≤L

  • =1{T−

b <T}

. Double Parisian Out Call, φ(S) = (ST −K)+ 1 ∀ 0≤t1<t2≤T

t2−t1≥D

,∃u∈[t1,t2] s.t. Su<Lup and Su>Llow

  • =1{T−

blow >T}1{T+ bup >T}

.

  • J. Lelong (MathFi – INRIA)

June 26, 2008 8 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Pricing of Parisian options

Plan

1

Presentation of the Parisian options Definition The different options Mathematical setting

2

Pricing of Parisian options Several approaches Link between single and double Parisian options

3

Numerical evaluation Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation

  • J. Lelong (MathFi – INRIA)

June 26, 2008 9 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Pricing of Parisian options 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 ZT−

b are independent and we know the

density of ZT−

b .

  • J. Lelong (MathFi – INRIA)

June 26, 2008 10 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Pricing of Parisian options Several approaches

Laplace transform approach

Use Laplace transforms as suggested by Chesney, Jeanblanc and Yor1. 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

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Pricing of Parisian options 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,Llow,Lup;r,δ) = e−( m2

2 +r)TE

  • emZT (xeσZT −K)+1{T−

blow >T}1{T+ bup>T}

  • .

Rewrite the two indicators 1{T−

blow >T}1{T+ bup>T} =

1

  • Call

−1{T−

blow <T}

  • PDIC(Llow)

−1{T+

bup<T}

  • PUIC(Lup)

+1{T−

blow <T}1{T+ bup<T}

  • A = new term

.

  • J. Lelong (MathFi – INRIA)

June 26, 2008 12 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Pricing of Parisian options Link between single and double Parisian options

Double Parisian option prices

Theorem 1

  • DPOC

⋆(x,λ;Llow,Lup) =

SC

⋆(x,λ)−

PDIC

⋆(x,λ;Llow)

− PUIC

⋆(x,λ;Lup)+

A(x,λ;Llow,Lup) where A is the Laplace transform of A w.r.t. maturity time given by

  • A(x,λ;Llow,Lup) = E
  • e

−λT−

blow 1{T− blow <T+ bup}

  • E
  • e
  • 2λZT−

blow

  • PUIC

⋆ |x<Lup (x,λ;Lup)

+E

  • e

−λT+

bup 1{T+ bup<T− blow }

  • E
  • e

−

  • 2λZT+

bup

  • PDIC

⋆ |x>Llow (x,λ;Llow),

where PUIC

⋆ |x<Lup (resp.

PDIC

⋆ |x>Llow ) means that we use the definition of

  • PUIC

⋆ (resp.

PDIC

⋆) in the case x < Lup (resp. x > Llow).

  • J. Lelong (MathFi – INRIA)

June 26, 2008 13 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation

Plan

1

Presentation of the Parisian options Definition The different options Mathematical setting

2

Pricing of Parisian options Several approaches Link between single and double Parisian options

3

Numerical evaluation Inversion formula Analytical prolongation Euler summation Regularity of the Parisian option prices Practical implementation

  • J. Lelong (MathFi – INRIA)

June 26, 2008 14 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation 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∞

est 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, 2008 15 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation 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, 2008 16 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Euler summation

Trapezoidal rule I

f (t) = 1 2πi α+i∞

α−i∞

est f (s)ds. A trapezoidal discretization with step h = π

t leads to

f π

t (t) = eαt

2t

  • f (α)+ eαt

t

∞

  • k=1

(−1)k Re

  • f
  • α+ikπ

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, 2008 17 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Euler summation

Trapezoidal rule II

We want to compute numerically f π

t (t)

∆

= eαt 2t

  • f (α)+ eαt

t

∞

  • k=1

(−1)k Re

  • f
  • α+ikπ

t

  • .

Truncation of the series sp(t)

∆

= eαt 2t

  • f (α)+ eαt

t

p

  • k=1

(−1)k Re

  • f
  • α+iπk

t

  • .

very slow convergence of sp(t) = ⇒ need of an acceleration technique.

  • J. Lelong (MathFi – INRIA)

June 26, 2008 18 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Euler summation

Euler summation

For p,q > 0, we set E(q,p,t) =

q

  • k=0

Ck

q2−qsp+k(t).

Proposition 3

Let p,q > 0 and f ∈ C q+4 such that there exists ǫ > 0 s.t. ∀k ≤ q+4, f (k)(s) = O(e(α−ǫ)s), where α is the abscissa of convergence of f . Then,

  • f π

t (t)−E(q,p,t)

  • ≤

teαt f ′(0)−αf (0)

  • π2

p!(q+1)! 2q (p+q+2)! +O

  • 1

pq+3

  • ,

when p goes to infinity.

  • J. Lelong (MathFi – INRIA)

June 26, 2008 19 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation 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 blow < 0 and bup > 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 blow > 0 or bup < 0, f is discontinuous in t = D. If blow = 0 or bup = 0, f is continuous. Moreover, if blow = 0 (resp. bup = 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, 2008 20 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation 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+ b ) has a density µ w.r.t

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+ b ) is not absolutely

continuous w.r.t Lebesgue’s measure and P(T−

b = D) > 0 (resp.

P(T+

b = D) > 0).

T−

0 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, 2008 21 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Regularity of the Parisian option prices

Density of the Parisian times

0.04 0.08 0.12 0.16 0.20 0.24 0.28 3 7 11 15 19 23 27 31

FIG.: Density function of T−

0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.1 0.2 0.3 0.4

FIG.: Cumulative distribution function

  • f T−

0 .

  • J. Lelong (MathFi – INRIA)

June 26, 2008 22 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Practical implementation

Practical implementation

For 2α/t = 18.4, p = q = 15,

  • f (t)−E(q,p,t)
  • ≤ S010−8 +t
  • f ′(0)−αf (0)
  • 10−11.

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, 2008 23 / 26

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Presentation of the Parisian options Pricing of Parisian options Numerical evaluation Pricing Parisian options Numerical evaluation Practical implementation

Numerical convergence for a PUOC

PUOC with S0 = 110, r = 0.1, σ = 0.2, K = 100, T = 1, L = 110, D = 0.1 year.

100 200 300 400 500 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 100 200 300 400 500 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

FIG.: Convergence of the standard summation w.r.t. p

  • J. Lelong (MathFi – INRIA)

June 26, 2008 24 / 26

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Regularity of a Double Out Call option

S0 = 100, r = 0.1, σ = 0.2, K = 90, Llow = 100, Lup = 120, D = 0.1 year.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 4 5 6 7 8 9 10

FIG.: Regularity w.r.t maturity time

  • J. Lelong (MathFi – INRIA)

June 26, 2008 25 / 26

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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, 2008 26 / 26

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Abate, J., Choudhury, L., and Whitt, G. (1999).

An introduction to numerical transform inversion and its application to probability models. Computing Probability, pages 257 – 323.

Chesney, M., Jeanblanc-Picqué, M., and Yor, M. (1997).

Brownian excursions and Parisian barrier options.

  • Adv. in Appl. Probab., 29(1):165–184.
  • J. Lelong (MathFi – INRIA)

June 26, 2008 26 / 26