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
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 2 / 33
Presentation of the Parisian optionsPricing 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) 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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 3 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 4 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 4 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 5 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
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
.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 6 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
The different options
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 7 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
The different options
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) Tuesday 23 October 2007 8 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
The different options
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
).
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 9 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
Some parity relationships
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
.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 10 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
Some parity relationships
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
.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 11 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Pricing of Parisian options
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 12 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Pricing of Parisian options
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]
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 13 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Pricing of Parisian options
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 14 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Pricing of Parisian options
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]
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 15 / 33
Presentation of the Parisian optionsPricing 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 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]
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 16 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Pricing of Parisian options
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 17 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 18 / 33
Presentation of the Parisian optionsPricing 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 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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 19 / 33
Presentation of the Parisian optionsPricing 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 .
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 20 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Analytical prolongation
Analytical prolongation
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 21 / 33
Presentation of the Parisian optionsPricing 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 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 .
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 22 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 23 / 33
Presentation of the Parisian optionsPricing 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=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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 24 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Regularity of the Parisian option prices
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
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Regularity of the Parisian option prices
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 26 / 33
Presentation of the Parisian optionsPricing 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+qterms.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 27 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 28 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
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).
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 28 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
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 2007 29 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
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 2007 30 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
Convergence of a PUOC to a standard Up Out Call Barrieroption
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 31 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
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
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 32 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 33 / 33
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
Practical implementation
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.
J. Lelong (MathFi – INRIA) Tuesday 23 October 2007 33 / 33
Top Related