pricing 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 Parisian options using numericalinversion of Laplace transforms

Jérôme Lelong (joint work with C. Labart)

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

Tuesday 23 October 2007

J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 1 / 33

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




Plan

1 Presentation of the Parisian optionsDefinitionThe different optionsSome parity relationships

2 Pricing of Parisian optionsWhat is knownSeveral approaches

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”).

Parisian options are less sensitive to influential agent on the marketthan standard barrier options.

Parisian options naturally appears in the analysis of structuredproducts such as re-callable convertible bond: the owner wants torecall its bond if ever the underlying stock has been traded out of agiven range for a while.

Some attempts to use Parisian options to price credit risk derivatives.






Definition

Definition (single barrier)

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 (single barrier)

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

D

bup

bdown

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

Payoff

Parisian Down In Call,

(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.

Parisian Up Out Call,

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

.

Parisian Double Out Call

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

.







DefinitionThe 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.

Let W = (Wt , t ≥ 0) 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.







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

gbt = supu ≤ t | Zu = b,

T−b = inft > 0 | (t −gb

t ) 1Zt<b > D,

T+b = inft > 0 | (t −gb

t ) 1Zt>b > D.

T−b : first time the B.M. Z stays below b longer than D.

Price of a PDIC :

f (T) = e−(r+ m22 )T EP(emZT (x eσZT −K )+︸︷︷︸

Call payoff

1T−b <T ︸︷︷︸

Parisian part

).

Price of a Double Parisian Out Call :

f (T) = e−(r+ m22 )T EP(emZT (x eσZT −K )+︸︷︷︸

Call payoff

1T−blow

>T 1T+bup

>T ︸︷︷︸Parisian part

).






Some parity relationships


Same kind of parity relationships as for standard barrier options. Forinstance,

P

D

U

O

I

P (x,T ;K ,L;r,δ) = xK P

U

D

O

I

C

(1

x,T ;

1

K,

1

L;δ,r

).

PDOP(x,T ;K ,L;r,δ) = E(emZT (K −x eσZT )+ 1T−

b >T

)e−

(r+ m2

2

)T

.

By introducing the new B.M. W =−Z , we can rewrite

E(e−mWT (K −x e−σWT )+ 1T+

−b>T

)e−

(r+ m2

2

)T =

xK E

(e−(m+σ)WT

(1

xeσWT − 1

K

)+1T+

−b>T

)e−

(r+ m2

2

)T

.







Link between single and double barrier Parisian options

Consider a Double Parisian Out Call (DPOC)

DPOC(x,T ;K ,Ld,Lu;r,δ) = e−( m22 +r)TE

[emZT (ST −K )+1T−

bd>T 1T+

bu>T

].

Rewrite the two indicators

1T−bd

>T 1T+bu

>T = 1︸︷︷︸Call

− 1T−bd

<T ︸︷︷︸PDIC(L=Ld)

− 1T+bu

<T ︸︷︷︸PUIC(L=Lu)

+1T−bd

<T 1T+bu

<T ︸︷︷︸new term

.





Pricing of Parisian options

Plan









What is known

What is known about Parisian options

There is no explicit formula for the law of T−b : we only know its

Laplace transform1.

We know that the r.v. T−b and ZT−

bare independent and we know the

density of ZT−b

.

Chesney et al. have shown that it is possible to compute the Laplacetransforms (w.r.t. maturity time) of the Parisian option prices.

1[Chesney et al., 1997]






Several approaches

Monte Carlo approachCrude Monte Carlo simulations perform badly because of the timediscretisation.Attempt to improve the crude Monte Carlo by Baldi, Caramellino andIovino.

Recover the density of T−b by numerically inverting its the Laplace

transform.

EP(emZT (x eσZT −K )+1T−b <T ) =

EP(emZT−

b emWT−T−

b (x eσZT−

b eσWT−T−

b −K )+1T−b <T ),

where W ⊥⊥ Z .

f ?(T) =∫ ∞

0

∫R

emy em2

2 (T−t) C(x eσy ,T − t)1t<T dT−b ⊗ZT−

b(t,y),

where C(z, t) is the price of a Call option with maturity t and spot z.Actually dT−

b ⊗ZT−b= dT−

b×dZT−

b.






Several approaches

EDP approach

EDP for the price V (t,x) of a vanilla call option

∂t V + 1

2σ2x2∂xxV + rx∂xV − rV = 0 (t,x) ∈ [0,T)×R∗

+.

The payoff is not Markovian. Need to introduce a second statevariable =⇒ solve a 2-dimensional EDP2.

τ= t − supu ≤ t : Su ≥ L.∂t V + 1

2 σ2x2∂xxV + rx∂xV − rV = 0, on [0,T)× (L,∞),

∂t V + 12 σ

2x∂xxV + rx∂xV − rV +∂τV = 0, on [0,T)× [0,L),

continuity condition : V (L, t,τ) = V (L, t,0).

2[Haber et al., 1999]






Several approaches

Laplace transform approach

Use Laplace transforms as suggested by Chesney, Jeanblanc and Yor3.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.

3[Chesney et al., 1997]






Several approaches

Example of a Laplace transform

For all λ> (m+σ)2

2 and K > L, the Laplace transform of a Parisian Down Incall price is given by

f ?(λ) = ψ(−θpD)e2bθ

θψ(θp

D)K e(m−θ)k

(1

m−θ − 1

m+σ−θ)

,

with

ψ(z)∆=

∫ +∞

0xe−

x22 +zxdx = 1+z

p2πe

z22 N (z),

N (z)∆=

∫ z

−∞1p2π

e−u22 du.






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.=⇒ We have to prove that they are analytic in a complex half plane andfind their abscissa of convergence.






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 isbounded by E(emZT (x eσWT +K )).

E(emZT (x eσWT +K )) ≤ K em2

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 ofconvergence 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]. N








Lemma 1 (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.

Proof : It is sufficient to notice that the function defined above isholomorphic on the complex plane (and hence analytic) and that itcoincides with the normal cumulative distribution function on the realaxis. N






Euler summation

Trapezoidal rule I

f (t) = 1

2πi

∫ α+i∞

α−i∞est f (s)ds.

A trapezoidal discretisation of 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 .

Proposition 2 actually holds for h < 2πt .






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

)).

We still need to approximate 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 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 πt

(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


Proposition 4Let 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[

emZt (eσZt −K )+1T−b <t

].

Relying on the strong Markov property,

f (t) = E(

1T−b <tE

[(xeσ(Wt−τ+z) −K )+em(Wt−τ+z)]

|z=ZT−b

,τ=T−b

),

where W ⊥⊥FT−b

.

f (t) =∫ t

0dτ

∫ ∞

−∞dz

∫ ∞

−∞dw (xeσ(w

pτ+z) −K )+em(w

pτ+z)p(w)ν(z)µ(t −τ).

NJ. Lelong (MathFi – INRIA) Tuesday 23 October 2007 25 / 33






Density for a Parisian time

Proposition 5T−

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−

λ22 T−

b

)= eλb

ψ(λp

D)for λ ∈R.

Both sides are analytic on O = z ∈C;−π4 < arg(z) < π

4 . Continuity for

arg(z) =±π4 =⇒ for all u ∈R E

(e−iuT−

b

)= e

p2uib

ψ(p

2iuD).

µ(t) = 1

2π

∫ ∞

−∞ep

2uib

ψ(p

2iuD)e−iut du.

Moreover, E(e−iuT−

b

)=O

(e−|b|

p|u|)

when |u|→∞=⇒ µ is C ∞. N






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 PUOC I

Consider a Parisian Up Out Call withS0 = 110, r = 0.1, σ= 0.2, T = 1, L = 110, D = 0.1 year.







Numerical convergence for a PUOC I

Consider a Parisian Up Out Call withS0 = 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 tocut down to p = q ≈ 15 (e.g. only 30 terms to be computed).







Numerical convergence for a PUOC II

FIG.: Convergence of the Euler summation w.r.t. q for p = 10







Improved Monte Carlo method for a PUOC

FIG.: Convergence of the Improved Monte Carlo Method with 250 time steps.







Convergence of a PUOC to a standard Up Out Call Barrieroption







0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Pric

e

Delay

Price Monte Carlo

Laplace Transform

Inf Price Monte Carlo

Sup Price Monte Carlo

FIG.: Comparison of the improved MC with the Laplace method







Hedging

Is the replicating portfolio generated by the delta? Probably yes, butnot straightforward.

How to compute the deltaDifferentiate the Laplace transforms of the prices to obtain the Laplacetransforms of the deltas.Use some automatic differentiation tools to compute the Laplacetransforms of the deltas.







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.

Ï Haber, R., Schonbucker, P., and Wilmott, P. (1999).Pricing parisian options.Journal of Derivatives, (6):71–79.

Ï Widder, D. V. (1941).The Laplace Transform.Princeton Mathematical Series, v. 6. Princeton University Press,Princeton, N. J.


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

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