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1/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte Carlo Simulation of the Law of the Maximum of aLevy Process
A. E. Kyprianou, J.C.Pardo and K. van Schaik
Department of Mathematical Sciences, University of Bath
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
2/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Motivation
Levy process. A (one dimensional) process with stationary andindependent increments which has paths which are right continuous withleft limits and therefore includes Brownian motion with drift, compoundPoisson processes, stable processes amongst many others).
A popular model in mathematical finance for the evolution of a risky assetis
St := eXt , t ≥ 0
where {Xt : t ≥ 0} is a Levy process.
Barrier options: The value of up-and-out barrier option with expiry date Tand barrier bis typically priced as
Es(f (ST )1{ST≤b})
where ST = supu≤T Su = exp{supu≤T Xu}, f is some nice function.
Other motivations from queuing theory, population models etc.
One is fundamentally interested in the joint distribution
P(Xt ∈ dx , X t ∈ dy)
for any Levy process (X ,P).
3/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Fourier methods
Theoretically, things are already difficult enough if considering P(X t ∈ dy)or P(Xt ∈ dx), especially in the former case.
For the case of P(Xt ∈ dx) one is sometimes lucky and knows this inexplicit form. But usually one only knows something about
Ψ(θ) := −1
tlog E(e iθXt )
= aiθ +1
2σ2θ2 +
ZR(1− e iθx + iθx1{|x |≤1})Π(dx)
where a ∈ R, σ ∈ R and Π is a measure concentrated on R\{0} satisfyingRR(1 ∧ x2)Π(dx) < ∞...
...in which case there are fast-Fourier methods for inverting exp{−Ψ(θ)t}to give P(Xt ∈ dx).
For the case of P(X t ∈ dx), recent methods have concentrated on Fourierinversion of the, so called, Wiener-Hopf factors.
3/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Fourier methods
Theoretically, things are already difficult enough if considering P(X t ∈ dy)or P(Xt ∈ dx), especially in the former case.
For the case of P(Xt ∈ dx) one is sometimes lucky and knows this inexplicit form. But usually one only knows something about
Ψ(θ) := −1
tlog E(e iθXt )
= aiθ +1
2σ2θ2 +
ZR(1− e iθx + iθx1{|x |≤1})Π(dx)
where a ∈ R, σ ∈ R and Π is a measure concentrated on R\{0} satisfyingRR(1 ∧ x2)Π(dx) < ∞...
...in which case there are fast-Fourier methods for inverting exp{−Ψ(θ)t}to give P(Xt ∈ dx).
For the case of P(X t ∈ dx), recent methods have concentrated on Fourierinversion of the, so called, Wiener-Hopf factors.
3/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Fourier methods
Theoretically, things are already difficult enough if considering P(X t ∈ dy)or P(Xt ∈ dx), especially in the former case.
For the case of P(Xt ∈ dx) one is sometimes lucky and knows this inexplicit form. But usually one only knows something about
Ψ(θ) := −1
tlog E(e iθXt )
= aiθ +1
2σ2θ2 +
ZR(1− e iθx + iθx1{|x |≤1})Π(dx)
where a ∈ R, σ ∈ R and Π is a measure concentrated on R\{0} satisfyingRR(1 ∧ x2)Π(dx) < ∞...
...in which case there are fast-Fourier methods for inverting exp{−Ψ(θ)t}to give P(Xt ∈ dx).
For the case of P(X t ∈ dx), recent methods have concentrated on Fourierinversion of the, so called, Wiener-Hopf factors.
3/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Fourier methods
Theoretically, things are already difficult enough if considering P(X t ∈ dy)or P(Xt ∈ dx), especially in the former case.
For the case of P(Xt ∈ dx) one is sometimes lucky and knows this inexplicit form. But usually one only knows something about
Ψ(θ) := −1
tlog E(e iθXt )
= aiθ +1
2σ2θ2 +
ZR(1− e iθx + iθx1{|x |≤1})Π(dx)
where a ∈ R, σ ∈ R and Π is a measure concentrated on R\{0} satisfyingRR(1 ∧ x2)Π(dx) < ∞...
...in which case there are fast-Fourier methods for inverting exp{−Ψ(θ)t}to give P(Xt ∈ dx).
For the case of P(X t ∈ dx), recent methods have concentrated on Fourierinversion of the, so called, Wiener-Hopf factors.
3/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Fourier methods
Theoretically, things are already difficult enough if considering P(X t ∈ dy)or P(Xt ∈ dx), especially in the former case.
For the case of P(Xt ∈ dx) one is sometimes lucky and knows this inexplicit form. But usually one only knows something about
Ψ(θ) := −1
tlog E(e iθXt )
= aiθ +1
2σ2θ2 +
ZR(1− e iθx + iθx1{|x |≤1})Π(dx)
where a ∈ R, σ ∈ R and Π is a measure concentrated on R\{0} satisfyingRR(1 ∧ x2)Π(dx) < ∞...
...in which case there are fast-Fourier methods for inverting exp{−Ψ(θ)t}to give P(Xt ∈ dx).
For the case of P(X t ∈ dx), recent methods have concentrated on Fourierinversion of the, so called, Wiener-Hopf factors.
4/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Wiener-Hopf-Fourier methods
Recall that it turns out that one may always uniquely decompose
q + Ψ(θ) = κ+(q ,−iθ)× κ−(q , iθ)
such that
E(e iθXeq ) =κ+(q , 0)
κ+(q ,−iθ)and E(e
iθXeq ) =κ−(q , 0)
κ−(q , iθ)
where eq is an independent and exponentially distributed random variablewith rate q > 0 and X t = infs≤t Xs .
Fourier inversion, first in θ and then in q would be an option for accessingthe law of X t and X t . However one is rarely in possession of the factorsκ+ and κ− (even after 60 years of research into this topic), and even thenthere is the issue of the double Fourier inversion.
There are no convenient formulae which contain both Xt and X t whichcould be Fourier inverted.
4/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Wiener-Hopf-Fourier methods
Recall that it turns out that one may always uniquely decompose
q + Ψ(θ) = κ+(q ,−iθ)× κ−(q , iθ)
such that
E(e iθXeq ) =κ+(q , 0)
κ+(q ,−iθ)and E(e
iθXeq ) =κ−(q , 0)
κ−(q , iθ)
where eq is an independent and exponentially distributed random variablewith rate q > 0 and X t = infs≤t Xs .
Fourier inversion, first in θ and then in q would be an option for accessingthe law of X t and X t . However one is rarely in possession of the factorsκ+ and κ− (even after 60 years of research into this topic), and even thenthere is the issue of the double Fourier inversion.
There are no convenient formulae which contain both Xt and X t whichcould be Fourier inverted.
4/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Wiener-Hopf-Fourier methods
Recall that it turns out that one may always uniquely decompose
q + Ψ(θ) = κ+(q ,−iθ)× κ−(q , iθ)
such that
E(e iθXeq ) =κ+(q , 0)
κ+(q ,−iθ)and E(e
iθXeq ) =κ−(q , 0)
κ−(q , iθ)
where eq is an independent and exponentially distributed random variablewith rate q > 0 and X t = infs≤t Xs .
Fourier inversion, first in θ and then in q would be an option for accessingthe law of X t and X t . However one is rarely in possession of the factorsκ+ and κ− (even after 60 years of research into this topic), and even thenthere is the issue of the double Fourier inversion.
There are no convenient formulae which contain both Xt and X t whichcould be Fourier inverted.
4/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Wiener-Hopf-Fourier methods
Recall that it turns out that one may always uniquely decompose
q + Ψ(θ) = κ+(q ,−iθ)× κ−(q , iθ)
such that
E(e iθXeq ) =κ+(q , 0)
κ+(q ,−iθ)and E(e
iθXeq ) =κ−(q , 0)
κ−(q , iθ)
where eq is an independent and exponentially distributed random variablewith rate q > 0 and X t = infs≤t Xs .
Fourier inversion, first in θ and then in q would be an option for accessingthe law of X t and X t . However one is rarely in possession of the factorsκ+ and κ− (even after 60 years of research into this topic), and even thenthere is the issue of the double Fourier inversion.
There are no convenient formulae which contain both Xt and X t whichcould be Fourier inverted.
5/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Suppose that e(1), e(2), · · · are a sequence of i.i.d unit mean exponentiallydistributed random variables.
Note that
g(n, λ) :=
nXi=1
1
λe(i)
is a Gamma (Erlang) distribution with parameters n and λ and by thestrong law of Large numbers, for t > 0,
g(n,n/t) =
nXi=1
t
ne(i) → t
almost surely.
Hence for a suitably large n, we have in distribution
(Xg(n,n/t),X g(n,n/t)) ' (Xt ,X t).
Indeed since t is not a jump time with probability 1, we have that(Xg(n,n/t),X g(n,n/t)) → (Xt ,X t) almost surely.
5/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Suppose that e(1), e(2), · · · are a sequence of i.i.d unit mean exponentiallydistributed random variables.
Note that
g(n, λ) :=
nXi=1
1
λe(i)
is a Gamma (Erlang) distribution with parameters n and λ and by thestrong law of Large numbers, for t > 0,
g(n,n/t) =
nXi=1
t
ne(i) → t
almost surely.
Hence for a suitably large n, we have in distribution
(Xg(n,n/t),X g(n,n/t)) ' (Xt ,X t).
Indeed since t is not a jump time with probability 1, we have that(Xg(n,n/t),X g(n,n/t)) → (Xt ,X t) almost surely.
5/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Suppose that e(1), e(2), · · · are a sequence of i.i.d unit mean exponentiallydistributed random variables.
Note that
g(n, λ) :=
nXi=1
1
λe(i)
is a Gamma (Erlang) distribution with parameters n and λ and by thestrong law of Large numbers, for t > 0,
g(n,n/t) =
nXi=1
t
ne(i) → t
almost surely.
Hence for a suitably large n, we have in distribution
(Xg(n,n/t),X g(n,n/t)) ' (Xt ,X t).
Indeed since t is not a jump time with probability 1, we have that(Xg(n,n/t),X g(n,n/t)) → (Xt ,X t) almost surely.
5/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Suppose that e(1), e(2), · · · are a sequence of i.i.d unit mean exponentiallydistributed random variables.
Note that
g(n, λ) :=
nXi=1
1
λe(i)
is a Gamma (Erlang) distribution with parameters n and λ and by thestrong law of Large numbers, for t > 0,
g(n,n/t) =
nXi=1
t
ne(i) → t
almost surely.
Hence for a suitably large n, we have in distribution
(Xg(n,n/t),X g(n,n/t)) ' (Xt ,X t).
Indeed since t is not a jump time with probability 1, we have that(Xg(n,n/t),X g(n,n/t)) → (Xt ,X t) almost surely.
6/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
A reformulation of the Wiener-Hopf factorization states that
Xeq
d= Sq + Iq
where Sq is independent of Iq and they are respectively equal indistribution to X eq and X eq
.
Taking advantage of the above, the fact that X has stationary andindependent increments and the fact that, as a time, g(n,n/t) can beseen as n subsequent independent exponential time periods we have thefollowing:
6/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
A reformulation of the Wiener-Hopf factorization states that
Xeq
d= Sq + Iq
where Sq is independent of Iq and they are respectively equal indistribution to X eq and X eq
.
Taking advantage of the above, the fact that X has stationary andindependent increments and the fact that, as a time, g(n,n/t) can beseen as n subsequent independent exponential time periods we have thefollowing:
6/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
A reformulation of the Wiener-Hopf factorization states that
Xeq
d= Sq + Iq
where Sq is independent of Iq and they are respectively equal indistribution to X eq and X eq
.
Taking advantage of the above, the fact that X has stationary andindependent increments and the fact that, as a time, g(n,n/t) can beseen as n subsequent independent exponential time periods we have thefollowing:
7/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Theorem. For all n ∈ {1, 2, · · · } and λ > 0,
(Xg(n,λ),X g(n,λ))d= (V (n, λ), J (n, λ))
where
V (n, λ) =
nXj=1
{S (j)λ +I
(j)λ } and J (n, λ) :=
n−1_i=0
iX
j=1
{S (j)λ + I
(j)λ }+ S
(i+1)λ
!.
Here, S(0)λ = I
(0)λ = 0, {S (j)
λ : j ≥ 1} are an i.i.d. sequence of randomvariables with common distribution equal to that of X eλ and
{I (j)λ : j ≥ 1} are another i.i.d. sequence of random variable with common
distribution equal to that of X eλ.
Moreover, we have the following obvious:
Corollary. We have as n ↑ ∞
(V (n,n/t), J (n,n/t)) → (Xt ,X t)
where the convergence is understood in the distributional sense.
7/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Theorem. For all n ∈ {1, 2, · · · } and λ > 0,
(Xg(n,λ),X g(n,λ))d= (V (n, λ), J (n, λ))
where
V (n, λ) =
nXj=1
{S (j)λ +I
(j)λ } and J (n, λ) :=
n−1_i=0
iX
j=1
{S (j)λ + I
(j)λ }+ S
(i+1)λ
!.
Here, S(0)λ = I
(0)λ = 0, {S (j)
λ : j ≥ 1} are an i.i.d. sequence of randomvariables with common distribution equal to that of X eλ and
{I (j)λ : j ≥ 1} are another i.i.d. sequence of random variable with common
distribution equal to that of X eλ.
Moreover, we have the following obvious:
Corollary. We have as n ↑ ∞
(V (n,n/t), J (n,n/t)) → (Xt ,X t)
where the convergence is understood in the distributional sense.
7/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Some new ideas
Theorem. For all n ∈ {1, 2, · · · } and λ > 0,
(Xg(n,λ),X g(n,λ))d= (V (n, λ), J (n, λ))
where
V (n, λ) =
nXj=1
{S (j)λ +I
(j)λ } and J (n, λ) :=
n−1_i=0
iX
j=1
{S (j)λ + I
(j)λ }+ S
(i+1)λ
!.
Here, S(0)λ = I
(0)λ = 0, {S (j)
λ : j ≥ 1} are an i.i.d. sequence of randomvariables with common distribution equal to that of X eλ and
{I (j)λ : j ≥ 1} are another i.i.d. sequence of random variable with common
distribution equal to that of X eλ.
Moreover, we have the following obvious:
Corollary. We have as n ↑ ∞
(V (n,n/t), J (n,n/t)) → (Xt ,X t)
where the convergence is understood in the distributional sense.
8/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte-Carlo simulation
The previous results suggest that to simulate, for example, E(g(Xt ,X t))one should follow the following algorithm:
Sample independently from the distribution X en/tand X en/t
n ×m-times
and then construct m independent versions of the variables V (n,n/t) andJ (n,n/t), say
{V (i)(n,n/t) : i = 1, · · · ,m} and {J (i)(n,n/t) : i = 1, · · · ,m}.
Then approximate
E(g(Xt ,X t)) '1
m
mXi=1
g(V (i)(n,n/t), J (i)(n,n/t)).
In effect this still leaves the problem of simulating from the unknowndistribution X en/t
and X en/t.
8/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte-Carlo simulation
The previous results suggest that to simulate, for example, E(g(Xt ,X t))one should follow the following algorithm:
Sample independently from the distribution X en/tand X en/t
n ×m-times
and then construct m independent versions of the variables V (n,n/t) andJ (n,n/t), say
{V (i)(n,n/t) : i = 1, · · · ,m} and {J (i)(n,n/t) : i = 1, · · · ,m}.
Then approximate
E(g(Xt ,X t)) '1
m
mXi=1
g(V (i)(n,n/t), J (i)(n,n/t)).
In effect this still leaves the problem of simulating from the unknowndistribution X en/t
and X en/t.
8/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte-Carlo simulation
The previous results suggest that to simulate, for example, E(g(Xt ,X t))one should follow the following algorithm:
Sample independently from the distribution X en/tand X en/t
n ×m-times
and then construct m independent versions of the variables V (n,n/t) andJ (n,n/t), say
{V (i)(n,n/t) : i = 1, · · · ,m} and {J (i)(n,n/t) : i = 1, · · · ,m}.
Then approximate
E(g(Xt ,X t)) '1
m
mXi=1
g(V (i)(n,n/t), J (i)(n,n/t)).
In effect this still leaves the problem of simulating from the unknowndistribution X en/t
and X en/t.
8/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte-Carlo simulation
The previous results suggest that to simulate, for example, E(g(Xt ,X t))one should follow the following algorithm:
Sample independently from the distribution X en/tand X en/t
n ×m-times
and then construct m independent versions of the variables V (n,n/t) andJ (n,n/t), say
{V (i)(n,n/t) : i = 1, · · · ,m} and {J (i)(n,n/t) : i = 1, · · · ,m}.
Then approximate
E(g(Xt ,X t)) '1
m
mXi=1
g(V (i)(n,n/t), J (i)(n,n/t)).
In effect this still leaves the problem of simulating from the unknowndistribution X en/t
and X en/t.
8/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Monte-Carlo simulation
The previous results suggest that to simulate, for example, E(g(Xt ,X t))one should follow the following algorithm:
Sample independently from the distribution X en/tand X en/t
n ×m-times
and then construct m independent versions of the variables V (n,n/t) andJ (n,n/t), say
{V (i)(n,n/t) : i = 1, · · · ,m} and {J (i)(n,n/t) : i = 1, · · · ,m}.
Then approximate
E(g(Xt ,X t)) '1
m
mXi=1
g(V (i)(n,n/t), J (i)(n,n/t)).
In effect this still leaves the problem of simulating from the unknowndistribution X en/t
and X en/t.
9/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Recent advances in Wiener-Hopf theory
The suggested Monte-Carlo method would not make sense were it not forrecent advances in the theory of Wiener-Hopf factorization.Specifically there are many new examples of (two-sided) jumping Levyprocesses emerging for which sufficient analytical structure is in place inorder to sample from the two distributions X en/t
and X en/t. (β-Levy
processes, Lamperti-stable processes, Hypergeometric Levy processes, · · · )Case in point: The β-class of Levy processes (Kuznetsov (2009)).
Ψ(θ) = iaθ+1
2σ2θ2− c1
β1B
„α1 −
iθ
β1, 1− λ1
«− c2
β2B
„α2 −
iθ
β2, 1− λ2
«+γ
Where γ is a special constant and the underlying Levy measure Π hasdensity π given by
π(x) = c1e−α1β1x
(1− e−β1x )λ11{x>0} + c2
eα2β2x
(1− eβ2x )λ21{x<0}.
Note that the β-class of Levy process has exponential moments (needed towork with risk neutral measures), there is asymmetry in the jump structureand locally jumps are stable-like (similarly to eg CGMY processes)Moreover we can have infinite or finite activity, bounded or unboundedpath variation.
9/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Recent advances in Wiener-Hopf theory
The suggested Monte-Carlo method would not make sense were it not forrecent advances in the theory of Wiener-Hopf factorization.
Specifically there are many new examples of (two-sided) jumping Levyprocesses emerging for which sufficient analytical structure is in place inorder to sample from the two distributions X en/t
and X en/t. (β-Levy
processes, Lamperti-stable processes, Hypergeometric Levy processes, · · · )Case in point: The β-class of Levy processes (Kuznetsov (2009)).
Ψ(θ) = iaθ+1
2σ2θ2− c1
β1B
„α1 −
iθ
β1, 1− λ1
«− c2
β2B
„α2 −
iθ
β2, 1− λ2
«+γ
Where γ is a special constant and the underlying Levy measure Π hasdensity π given by
π(x) = c1e−α1β1x
(1− e−β1x )λ11{x>0} + c2
eα2β2x
(1− eβ2x )λ21{x<0}.
Note that the β-class of Levy process has exponential moments (needed towork with risk neutral measures), there is asymmetry in the jump structureand locally jumps are stable-like (similarly to eg CGMY processes)Moreover we can have infinite or finite activity, bounded or unboundedpath variation.
9/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Recent advances in Wiener-Hopf theory
The suggested Monte-Carlo method would not make sense were it not forrecent advances in the theory of Wiener-Hopf factorization.Specifically there are many new examples of (two-sided) jumping Levyprocesses emerging for which sufficient analytical structure is in place inorder to sample from the two distributions X en/t
and X en/t. (β-Levy
processes, Lamperti-stable processes, Hypergeometric Levy processes, · · · )
Case in point: The β-class of Levy processes (Kuznetsov (2009)).
Ψ(θ) = iaθ+1
2σ2θ2− c1
β1B
„α1 −
iθ
β1, 1− λ1
«− c2
β2B
„α2 −
iθ
β2, 1− λ2
«+γ
Where γ is a special constant and the underlying Levy measure Π hasdensity π given by
π(x) = c1e−α1β1x
(1− e−β1x )λ11{x>0} + c2
eα2β2x
(1− eβ2x )λ21{x<0}.
Note that the β-class of Levy process has exponential moments (needed towork with risk neutral measures), there is asymmetry in the jump structureand locally jumps are stable-like (similarly to eg CGMY processes)Moreover we can have infinite or finite activity, bounded or unboundedpath variation.
9/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Recent advances in Wiener-Hopf theory
The suggested Monte-Carlo method would not make sense were it not forrecent advances in the theory of Wiener-Hopf factorization.Specifically there are many new examples of (two-sided) jumping Levyprocesses emerging for which sufficient analytical structure is in place inorder to sample from the two distributions X en/t
and X en/t. (β-Levy
processes, Lamperti-stable processes, Hypergeometric Levy processes, · · · )Case in point: The β-class of Levy processes (Kuznetsov (2009)).
Ψ(θ) = iaθ+1
2σ2θ2− c1
β1B
„α1 −
iθ
β1, 1− λ1
«− c2
β2B
„α2 −
iθ
β2, 1− λ2
«+γ
Where γ is a special constant and the underlying Levy measure Π hasdensity π given by
π(x) = c1e−α1β1x
(1− e−β1x )λ11{x>0} + c2
eα2β2x
(1− eβ2x )λ21{x<0}.
Note that the β-class of Levy process has exponential moments (needed towork with risk neutral measures), there is asymmetry in the jump structureand locally jumps are stable-like (similarly to eg CGMY processes)Moreover we can have infinite or finite activity, bounded or unboundedpath variation.
9/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Recent advances in Wiener-Hopf theory
The suggested Monte-Carlo method would not make sense were it not forrecent advances in the theory of Wiener-Hopf factorization.Specifically there are many new examples of (two-sided) jumping Levyprocesses emerging for which sufficient analytical structure is in place inorder to sample from the two distributions X en/t
and X en/t. (β-Levy
processes, Lamperti-stable processes, Hypergeometric Levy processes, · · · )Case in point: The β-class of Levy processes (Kuznetsov (2009)).
Ψ(θ) = iaθ+1
2σ2θ2− c1
β1B
„α1 −
iθ
β1, 1− λ1
«− c2
β2B
„α2 −
iθ
β2, 1− λ2
«+γ
Where γ is a special constant and the underlying Levy measure Π hasdensity π given by
π(x) = c1e−α1β1x
(1− e−β1x )λ11{x>0} + c2
eα2β2x
(1− eβ2x )λ21{x<0}.
Note that the β-class of Levy process has exponential moments (needed towork with risk neutral measures), there is asymmetry in the jump structureand locally jumps are stable-like (similarly to eg CGMY processes)Moreover we can have infinite or finite activity, bounded or unboundedpath variation.
10/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Case in point: The β-class of Levy processes
Kuznetsov shows that all the roots of the equation q + Ψ(θ) are onimaginary axis and those on the positive imaginary axis are all the roots onκ−(q , iθ) and those on the negative imaginary axis are all the roots ofκ+(q ,−iθ).
Analytic considerations allow him to conclude that eg E(e iθXeq ) is aninfinite product of rational functions.
The latter can be inverted giving the form of the density
P(X eq ∈ dx) =
0@Xn≤0
knζneζnx
1A dx
and he gives small intervals O(1) in length indicating the location of theroots iζn .
10/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Case in point: The β-class of Levy processes
Kuznetsov shows that all the roots of the equation q + Ψ(θ) are onimaginary axis and those on the positive imaginary axis are all the roots onκ−(q , iθ) and those on the negative imaginary axis are all the roots ofκ+(q ,−iθ).
Analytic considerations allow him to conclude that eg E(e iθXeq ) is aninfinite product of rational functions.
The latter can be inverted giving the form of the density
P(X eq ∈ dx) =
0@Xn≤0
knζneζnx
1A dx
and he gives small intervals O(1) in length indicating the location of theroots iζn .
10/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Case in point: The β-class of Levy processes
Kuznetsov shows that all the roots of the equation q + Ψ(θ) are onimaginary axis and those on the positive imaginary axis are all the roots onκ−(q , iθ) and those on the negative imaginary axis are all the roots ofκ+(q ,−iθ).
Analytic considerations allow him to conclude that eg E(e iθXeq ) is aninfinite product of rational functions.
The latter can be inverted giving the form of the density
P(X eq ∈ dx) =
0@Xn≤0
knζneζnx
1A dx
and he gives small intervals O(1) in length indicating the location of theroots iζn .
10/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Case in point: The β-class of Levy processes
Kuznetsov shows that all the roots of the equation q + Ψ(θ) are onimaginary axis and those on the positive imaginary axis are all the roots onκ−(q , iθ) and those on the negative imaginary axis are all the roots ofκ+(q ,−iθ).
Analytic considerations allow him to conclude that eg E(e iθXeq ) is aninfinite product of rational functions.
The latter can be inverted giving the form of the density
P(X eq ∈ dx) =
0@Xn≤0
knζneζnx
1A dx
and he gives small intervals O(1) in length indicating the location of theroots iζn .
11/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Test case: Brownian motion (simulated and known density) t = 1
12/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
β-class t = 1,n = 10, a = 0, σ = 3/10, α1 = 10, β1 = 1/4, λ1 =1/2, c1 = 1/100, α2 = 10, β2 = 1/4, λ2 = 1/2, c2 = 1/100)
13/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Simulated value function of European up-and-out call option with Xfrom β-family
2 4 6 8 10
0.005
0.01
0.015
0.02
0.025
0.03
0.035
With Gaussian part (strike=5, barrier=10, T=1)
14/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Simulated value function of European up-and-out call option with Xfrom β-family
2 4 6 8 10
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Irregular upwards (strike=5, barrier=10, T=1)
15/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
Advantages over standard random walk approach
Standard random walk:- in general law of Xt not known, needs to be obtained by numericalFourier inversion.- always produces an atom at 0 when simulating X t (W-H MC methodproduces atom iff it is really present, i.e. iff X is irregular upwards).- well known bad performance when simulating X t (misses excursionsbetween grid points), W-H MC method performs significantly better inBrownian motion test case
16/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
W-H MC method vs. random walk: P(X 1 ≤ z ) where X is BM; n =number of time steps (for r.w. 2n time steps)
17/ 17
Monte Carlo Simulation of the Law of the Maximum of a Levy Process
W-H MC method vs. random walk: P(X1 ≤ z1, X 1 ≥ z2) where X isBM; 1000 time steps (for r.w. 2000 time steps)