pricing double parisian options using numerical inversion of laplace transforms · 2017. 2. 11. ·...

Presentation of the Parisian optionsPricing of Parisian options

Numerical evaluation

Pricing Parisian options

Pricing double Parisian options using numericalinversion of Laplace transforms

Jérôme Lelong (joint work with C. Labart)

http://cermics.enpc.fr/~lelong

Conference on Numerical Methods in Finance (Udine)

Thursday 26 June 2008

J. Lelong (MathFi – INRIA) June 26, 2008 1 / 26

http://cermics.enpc.fr/~lelong




Plan

1 Presentation of the Parisian optionsDefinitionThe different optionsMathematical setting

2 Pricing of Parisian optionsSeveral approachesLink between single and double Parisian options

3 Numerical evaluationInversion formulaAnalytical prolongationEuler summationRegularity of the Parisian option pricesPractical implementation





Presentation of the Parisian options

Definition

Definition

barrier options counting the time spent in a row above (resp. below) afixed level (the barrier). If this time is longer than a fixed value (thewindow width), the option is activated (“In”) or canceled (“Out”).






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






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






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






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






The different options

A few Payoffs

Single Parisian Down In Call,

φ(S) = (ST −K )+1 ∃ 0≤t1<t2≤Tt2−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≤Tt2−t1≥D ,∃u∈[t1,t2] s.t. Su<Lup and Su>Llow

.






Mathematical setting

Mathematical settingLet W = (Wt ,0 ≤ t ≤ T) be a B.M on (Ω,F ,Q), with F =σ(W ). Assumethat

St = x e(r−δ− σ22 )t+σWt .

We set m = r−δ− σ22

σ . We can introduce P∼Q s.t.

e−rT EQ(φ(St , t ≤ T)) = e−(r+ m22 )T EP(emZT φ(x eZt , t ≤ T))

where Z is P-B.M.

The price of a Parisian Down In Call (PDIC) is given by

f (T) = e−(r+ m22 )T EP

(emZT (x eσZT −K )+1 ∃ 0≤t1<t2≤T

t2−t1≥D , s.t. ∀u∈[t1,t2] Zu≤b)︸︷︷︸

"star" price

,

where b = 1σ log

( Lx

)and Z is a P−B.M.







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







Brownian Excursions II

Single Parisian Down In Call,

φ(S) = (ST −K )+ 1 ∃ 0≤t1<t2≤Tt2−t1≥D , s.t. ∀u∈[t1,t2] Su≤L

︸︷︷︸=1T−

b<T

.

Double Parisian Out Call,

φ(S) = (ST −K )+ 1 ∀ 0≤t1<t2≤Tt2−t1≥D ,∃u∈[t1,t2] s.t. Su<Lup and Su>Llow

︸︷︷︸=1T−

blow>T 1T+

bup>T

.





Pricing of Parisian options

Plan









Several approaches

Several approaches

Crude Monte Carlo simulations perform badly because of the timediscretization. Improvement by Baldi, Caramellino and Iovino (2000)using sharp large deviations.

2-dimensional PDE (Haber, Schonbucker and Willmott (1999)) : asecond state variable counts the length of the excursion of interest.

Chesney, Jeanblanc and Yor (1997) have shown that it is possible tocompute the Laplace transforms (w.r.t. maturity time) of the singleParisian 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 thedensity of ZT−

b.






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 Laplacetransforms of the Parisian (single and double barrier) option prices.

1[Chesney et al., 1997]






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−( m22 +r)TE

[emZT (x eσZT −K )+1T−

blow>T 1T+

bup>T

].

Rewrite the two indicators

1T−blow

>T 1T+bup

>T = 1︸︷︷︸Call

−1T−blow

<T ︸︷︷︸PDIC(Llow)

−1T+bup

<T ︸︷︷︸PUIC(Lup)

+1T−blow

<T 1T+bup

<T ︸︷︷︸A = new term

.






Link between single and double Parisian options

Double Parisian option prices

Theorem 1

àDPOC?

(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 1T−blow

<T+bup

]E

[e

p2λZT−

blow

] PUIC?|x<Lup

(x,λ;Lup)

+E[

e−λT+

bup 1T+bup

<T−blow

]E

[e−p2λZT+

bup

]PDIC?|x>Llow

(x,λ;Llow),

where PUIC?|x<Lup

(resp. PDIC?|x>Llow

) means that we use the definition ofPUIC?

(resp. PDIC?

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






Plan









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 Laplacetransform 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 ofthe parameter λ.=⇒ Prove that they are analytic in a complex half plane=⇒ Find their abscissa of convergence.






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 distributionfunction on the complex plane is defined by

N (x+ iy) = 1p2π

∫ x

−∞e−

(v+iy)2

2 dv.






Euler summation

Trapezoidal rule I

f (t) = 1

2πi

∫ α+i∞

α−i∞est f (s)ds.

A trapezoidal discretization with step h = πt leads to

f πt

(t) = eαt

2tf (α)+ eαt

t

∞∑k=1

(−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 πt

(t)∣∣∣ ∆= ∣∣∣f (t)− f π

t(t)

∣∣∣≤ ∥∥f∥∥∞ e−2αt

1−e−2αt .






Euler summation

Trapezoidal rule II

We want to compute numerically

f πt

(t)∆= eαt

2tf (α)+ eαt

t

∞∑k=1

(−1)k Re

(f

(α+ i

kπ

t

)).

Truncation of the series

sp(t)∆= eαt

2tf (α)+ eαt

t

p∑k=1

(−1)k Re

(f

(α+ i

πk

t

)).

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






Euler summation

Euler summation

For p,q > 0, we set

E(q,p, t) =q∑

k=0Ck

q 2−qsp+k(t).

Proposition 3

Let p,q > 0 and f ∈C q+4 such that there exists ε> 0 s.t.∀k ≤ q+4, f (k)(s) =O (e(α−ε)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.






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 maturitytime 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 ofinterest has a density function µ, can be written f (t) = ∫ t

0 φ(t −u)µ(u)duwith φ of class C∞.







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.tLebesgue’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 absolutelycontinuous 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.







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

0.04 0.08 0.12 0.16 0.20 0.24 0.280

0.1

0.2

0.3

0.4

FIG.: Cumulative distribution functionof T−

0 .






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+qterms.







Numerical convergence for a PUOCPUOC 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







Regularity of a Double Out Call optionS0 = 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







Conclusion

Closed formulae for double Parisian option prices


Existence and regularity of the density of Parisian times

Accuracy of the inversion algorithm







Ï Abate, J., Choudhury, L., and Whitt, G. (1999).An introduction to numerical transform inversion and its application toprobability models.Computing Probability, pages 257 – 323.

Ï Chesney, M., Jeanblanc-Picqué, M., and Yor, M. (1997).Brownian excursions and Parisian barrier options.Adv. in Appl. Probab., 29(1):165–184.


pricing double parisian options using numerical inversion of laplace transforms · 2017. 2. 11. ·...

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