Post on 14-Sep-2020
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
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
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
J. Lelong (MathFi – INRIA) June 26, 2008 2 / 26
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”).
J. Lelong (MathFi – INRIA) June 26, 2008 3 / 26
Presentation of the Parisian optionsPricing 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
Presentation of the Parisian optionsPricing of Parisian options
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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
Presentation of the Parisian optionsPricing of Parisian options
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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
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
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
Presentation of the Parisian optionsPricing 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≤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
.
J. Lelong (MathFi – INRIA) June 26, 2008 5 / 26
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Presentation of the Parisian options
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.
J. Lelong (MathFi – INRIA) June 26, 2008 6 / 26
Presentation of the Parisian optionsPricing of Parisian options
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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
Presentation of the Parisian optionsPricing 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≤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
.
J. Lelong (MathFi – INRIA) June 26, 2008 8 / 26
Presentation of the Parisian optionsPricing 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≤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
.
J. Lelong (MathFi – INRIA) June 26, 2008 8 / 26
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 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
J. Lelong (MathFi – INRIA) June 26, 2008 9 / 26
Presentation of the Parisian optionsPricing 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 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.
J. Lelong (MathFi – INRIA) June 26, 2008 10 / 26
Presentation of the Parisian optionsPricing of Parisian options
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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 Laplacetransforms of the Parisian (single and double barrier) option prices.
1[Chesney et al., 1997]
J. Lelong (MathFi – INRIA) June 26, 2008 11 / 26
Presentation of the Parisian optionsPricing 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−( 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
.
J. Lelong (MathFi – INRIA) June 26, 2008 12 / 26
Presentation of the Parisian optionsPricing of Parisian options
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Pricing Parisian options
Pricing of Parisian options
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).
J. Lelong (MathFi – INRIA) June 26, 2008 13 / 26
Presentation of the Parisian optionsPricing of Parisian options
Numerical evaluation
Pricing Parisian options
Numerical evaluation
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
J. Lelong (MathFi – INRIA) June 26, 2008 14 / 26
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.=⇒ Prove that they are analytic in a complex half plane=⇒ Find their abscissa of convergence.
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Presentation of the Parisian optionsPricing of Parisian options
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Pricing Parisian options
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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.
J. Lelong (MathFi – INRIA) June 26, 2008 16 / 26
Presentation of the Parisian optionsPricing of Parisian options
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Pricing Parisian options
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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 .
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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.
J. Lelong (MathFi – INRIA) June 26, 2008 18 / 26
Presentation of the Parisian optionsPricing of Parisian options
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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 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.
J. Lelong (MathFi – INRIA) June 26, 2008 19 / 26
Presentation of the Parisian optionsPricing of Parisian options
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Pricing Parisian options
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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 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∞.
J. Lelong (MathFi – INRIA) June 26, 2008 20 / 26
Presentation of the Parisian optionsPricing 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.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.
J. Lelong (MathFi – INRIA) June 26, 2008 21 / 26
Presentation of the Parisian optionsPricing 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
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 .
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Presentation of the Parisian optionsPricing of Parisian options
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Pricing Parisian options
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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.
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Practical implementation
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
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Presentation of the Parisian optionsPricing of Parisian options
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Practical implementation
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
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Presentation of the Parisian optionsPricing of Parisian options
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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
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.
J. Lelong (MathFi – INRIA) June 26, 2008 26 / 26