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Duality and Derivative Pricing with Time-Changed Lévy Processes

Jos´ e Fajardo Ernesto Mordecki

IBMEC Business School Universidad de La Republica del Uruguay Fourth Bachelier Finance Society Congress. Tokyo, August 20, 2006

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Outline

• Motivation

– p. 2/3

Outline

• Motivation
• Time-Changed Lévy processes

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Outline

• Motivation
• Time-Changed Lévy processes
• Bidimensional Derivative pricing

– p. 2/3

Outline

• Motivation
• Time-Changed Lévy processes
• Bidimensional Derivative pricing
• Duality and Symmetry

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Outline

• Motivation
• Time-Changed Lévy processes
• Bidimensional Derivative pricing
• Duality and Symmetry
• Conclusions

– p. 2/3

Motivation

BiDimensional Derivative Pricing: European type American Perpetual type BM Margrabe (1978) Gerber and Shiu (1996) LP Fajardo and Mordecki (2006) Fajardo and Mordecki (2006) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ?

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Motivation

BiDimensional Derivative Pricing: European type American Perpetual type BM Margrabe (1978) Gerber and Shiu (1996) LP Fajardo and Mordecki (2006) Fajardo and Mordecki (2006) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ? Duality Put-Call Exotic Derivatives BM Carr and Chesney (1996) Henderson and Wojakowski (2002) LP Fajardo and Mordecki (2006) Eberlein and Papantaleon (2005a) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ? SM Schroder (1999) ?

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Lévy Processes

We say that X = {Xt}t≥0 is a Lévy Process, or a process with stationary and independent increments, if:

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Lévy Processes

We say that X = {Xt}t≥0 is a Lévy Process, or a process with stationary and independent increments, if:

• X has paths RCLL

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Lévy Processes

We say that X = {Xt}t≥0 is a Lévy Process, or a process with stationary and independent increments, if:

• X has paths RCLL
• X0 = 0, and has independent increments, given

0 < t1 < t2 < ... < tn, the r.v. Xt1, Xt2 − Xt1, · · · , Xtn − Xtn−1

are independents.

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Lévy Processes

We say that X = {Xt}t≥0 is a Lévy Process, or a process with stationary and independent increments, if:

• X has paths RCLL
• X0 = 0, and has independent increments, given

0 < t1 < t2 < ... < tn, the r.v. Xt1, Xt2 − Xt1, · · · , Xtn − Xtn−1

are independents.

• The distribution of the increment Xt − Xs is

homogenous in time, that is, depends just on the difference t − s.

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Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of Xt as:

E(ezXt) = etψ(z)

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Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of Xt as:

E(ezXt) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

– p. 5/3

Lévy-Khintchine Formula

A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of Xt as:

E(ezXt) = etψ(z)

Where ψ is called characteristic exponent, and is given by:

ψ(z) = bz + 1 2σ2z2 +

• I

R

(ezy − 1 − zy1{|y|<1})Π(dy),

where b and σ ≥ 0 are real constants, and Π is a positive measure in I

R − {0} such that

• (1 ∧ y2)Π(dy) < ∞, and is called Lévy measure,

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Remarks

• Due to the independence of increments we can

not model range dependence.

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Remarks

• Due to the independence of increments we can

not model range dependence.

• Due to the homogeneity in time we can not have

flexible representations for the term structure of volatility.

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This Paper

• Time-inhomogenous Processes

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This Paper

• Time-inhomogenous Processes
• Derivative Pricing and Duality Relationship.

– p. 7/3

Carr and Wu (2004)

• Monroe (1978)

– p. 8/3

Carr and Wu (2004)

• Monroe (1978)
• Stochastic Time change on Lévy Processes→

stochastic volatility

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Carr and Wu (2004)

• Monroe (1978)
• Stochastic Time change on Lévy Processes→

stochastic volatility

• Correlation between Lévy processes and random

clock→ leverage effect.

– p. 8/3

Carr and Wu (2004)

• Monroe (1978)
• Stochastic Time change on Lévy Processes→

stochastic volatility

• Correlation between Lévy processes and random

clock→ leverage effect.

• Original clock as calendar time and new random

clock as business time

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Time-Change

Now let t → Tt, , t ≥ 0, be an increasing RCLL process, such that for each fixed t, Tt is a stopping time with respect to F.

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Time-Change

Now let t → Tt, , t ≥ 0, be an increasing RCLL process, such that for each fixed t, Tt is a stopping time with respect to F. Furthermore, suppose Tt is finite P − a.s., ∀t ≥ 0 and

Tt → ∞ as t → ∞.

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Time-Change

Now let t → Tt, , t ≥ 0, be an increasing RCLL process, such that for each fixed t, Tt is a stopping time with respect to F. Furthermore, suppose Tt is finite P − a.s., ∀t ≥ 0 and

Tt → ∞ as t → ∞.

Then {Tt} defines a random change on time, we can also impose ETt = t.

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Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt, t ≥ 0,

using different triplets for X and different time changes

Tt, we can obtain a good candidate for the underlying

asset return process. .

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Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt, t ≥ 0,

using different triplets for X and different time changes

Tt, we can obtain a good candidate for the underlying

asset return process. We know that if Tt is an independent subordinator, then Y is a Lévy process.

– p. 10/3

Time-Changed Lévy processes

Then, consider the process Yt defined by:

Yt ≡ XTt, t ≥ 0,

using different triplets for X and different time changes

Tt, we can obtain a good candidate for the underlying

asset return process. We know that if Tt is an independent subordinator, then Y is a Lévy process. A more general situation is when Tt is modeled by a non-decreasing semimartingale.

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Time-Changed Lévy Processes

In that case

Tt = bt + t ∞ yµ(dy, ds)

where b is a drift and µ is the counting measure of jumps of the time change and we can take µ = 0 and just take locally deterministic time changes, so we need to specify the local intensity ν:

Tt = t ν(s−)ds

(1)

where ν is the instantaneous activity rate, observe that

ν must be non-negative.

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Time-Changed Lévy Processes

In that case

Tt = bt + t ∞ yµ(dy, ds)

where b is a drift and µ is the counting measure of jumps of the time change and we can take µ = 0 and just take locally deterministic time changes, so we need to specify the local intensity ν:

Tt = t ν(s−)ds

(2)

where ν is the instantaneous activity rate, observe that

ν must be non-negative.

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Remarks

• When Xt is the Brownian motion , ν is proportional

to the instantaneous variance rate of the Brownian motion

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Remarks

• When Xt is the Brownian motion , ν is proportional

to the instantaneous variance rate of the Brownian motion

• When Xt is a pure jump Lévy process, ν is

proportional to the Lévy intensity of jumps.

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Time-Changed Lévy Processes

We can obtain the characteristic function of Yt:

φYt(z) = E(ez′XTt) = E

• E
• ez′Xu/Tt = u
• .

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Time-Changed Lévy Processes

We can obtain the characteristic function of Yt:

φYt(z) = E(ez′XTt) = E

• E
• ez′Xu/Tt = u
• .

If Tt and Xt were independent, then:

φYt(z) = LTt(ψ(z))

where LTt is the Laplace transform of Tt.

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Time-Changed Lévy Processes

So if the Laplace transform of T and the characteristic exponent of X have closed forms, we can obtain a closed form for φYt. Using equation (1) we have:

LTt(λ) = E(e−λ

R t

0 ν(s−)ds)

(3)

From bond pricing literature Carr and Wu (2004)

• btained a closed forms for φYt.

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Time-Changed Lévy Processes

• When T is a Lévy processes the characteristics of

Y are known explicitly.

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Time-Changed Lévy Processes

• When T is a Lévy processes the characteristics of

Y are known explicitly.

• When T has independent increments we can
• btain a deterministic characteristics for Y .

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Time-Changed Lévy Processes

• When T is a Lévy processes the characteristics of

Y are known explicitly.

• When T has independent increments we can
• btain a deterministic characteristics for Y .
• When T is just a processes that turns Y in to a

general semimartingale we can obtain a predictable version of the characteristics.

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Model

Consider a market model with three assets (S1, S2, S3) given by

S1

t = eY 1

t ,

S2

t = S2 0eY 2

t ,

S3

t = S3 0eY 3

t

(4)

where (Y 1, Y 2, Y 3) is a three dimensional Time-Changed Lévy process.

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Model

Consider a market model with three assets (S1, S2, S3) given by

S1

t = eY 1

t ,

S2

t = S2 0eY 2

t ,

S3

t = S3 0eY 3

t

(5)

where (Y 1, Y 2, Y 3) is a three dimensional Time-Changed Lévy process. Consider a function:

f : (0, ∞) × (0, ∞) → I R

homogenous of an arbitrary degree α; We introduce a derivative contract with payoff given by

Φt = f(S2

t , S3 t )

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Problem

We want to price that derivative contract. In the European case, the problem reduces to the computation of

ET = E(S2

0, S3 0, T) = E

• e−Y 1

T f(S2

0eY 2

T , S3

0eY 3

T )

• (6)

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Problem

We want to price that derivative contract. In the European case, the problem reduces to the computation of

ET = E(S2

0, S3 0, T) = E

• e−Y 1

T f(S2

0eY 2

T , S3

0eY 3

T )

• (7)

In the American case, if MT denotes the class of stopping times up to time T, i.e:

MT = {τ : 0 ≤ τ ≤ T, τ stopping time}

for the finite horizon case, putting T = ∞ for the perpetual case. Hence we need to find the value function AT and an optimal stopping time τ∗ in MT.

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Dual market Method

such that

AT = A(S2

0, S3 0, T) = sup τ∈MT

E

• e−Y 1

τ f(S2

0eY 2

τ , S3

0eY 3

τ3)

• = E
• e−Y 1

τ∗f(S2

0eY 2

τ∗, S3

0eY 3

τ∗)

• .

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Dual market Method

Observe that

e−Y 1

t f(S2

0eY 2

t , S3

0eY 3

t ) = e−Y 1 t +αY 3 t f(S2

0eY 2

t −Y 3 t , S3

0).

Let ρ = − log Ee−Y 1

1 +αY 3 1 , that we assume finite. The

process

Zt = e−Y 1

t +αY 3 t +ρTt

(8)

is a density process (i.e. a positive martingale starting at Z0 = 1) that allow us to introduce a new measure ˜

P

by its restrictions to each Ft by the formula

d ˜ Pt dPt = Zt.

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Dual market Method

Denote now by Yt = Y 2

t − Y 3 t , and St = S2

• 0eYt. Finally, let

F(x) = f(x, S3

0).

With the introduced notations, under the change of measure we obtain

ET = ˜ E

• e−ρTT F(ST)
• AT = sup

τ∈MT

˜ E

• e−ρTτF(Sτ)
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Example

Let S2

T and S3 T be two risky assets, a contract with

payoff Φt = (S2

T − S3 T )+ can be priced using The Dual

Market Method:

D = E

• e−rT(S2

T − S3 T )+

. =

• A

e−rT(S2

0eY 2

T − S3

0eY 3

T )dP

Assuming for simplicity S2

0 = S3 0 = 1,

Then A = {ω ∈ Ω : Y 2

T (ω) > Y 3 T (ω)}, we apply the

method:

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Example

D =

• A

e−rT(eY 2

T − eY 3 T )dP =

• {ST >1}

e−rTeY 3

T (ST − 1)dP

where ST = eYT and Y = Y 2 − Y 3.

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Example

D =

• A

e−rT(eY 2

T − eY 3 T )dP =

• {ST >1}

e−rTeY 3

T (ST − 1)dP

where ST = eYT and Y = Y 2 − Y 3. Now the dual measure: ρ = − log Ee−r+Y 3

1 = r − log EeY 3 1 , then:

d P = er(t−Tt)+Y 3

T

(EeY 3

1 )TT dP.

– p. 22/3

Example

D =

• A

e−rT(eY 2

T − eY 3 T )dP =

• {ST >1}

e−rTeY 3

T (ST − 1)dP

where ST = eYT and Y = Y 2 − Y 3. Now the dual measure: ρ = − log Ee−r+Y 3

1 = r − log EeY 3 1 , then:

d P = er(t−Tt)+Y 3

T

(EeY 3

1 )TT dP.

To simplify assume r = 0, then:

D =

• {ST >1}

e−ρTT ST d P −

• {ST >1}

e−ρTT d P

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Duality

Consider a market with two assets given by

S1

t = eYt, and S2 t = S2 0ert

where (Y ) is a one dimensional Time-Changed Lévy process, and for simplicity, and without loss of generality we take S1

0 = 1.

– p. 23/3

Duality

Assume that T has independent increments. Let

Ψ = (B, C, ν) be the characteristic triplet of Y .

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Duality

Assume that T has independent increments. Let

Ψ = (B, C, ν) be the characteristic triplet of Y . Then, as

we assume that we are under the risk neutral measure, B is completely determined by the other characteristics:

Bt = (r − δ)t − 1 2 t csds − t

• I

R

(ex − 1 − x)ν(ds, dx),

– p. 24/3

Duality

Assume that T has independent increments. Let

Ψ = (B, C, ν) be the characteristic triplet of Y . Then, as

we assume that we are under the risk neutral measure, B is completely determined by the other characteristics:

Bt = (r − δ)t − 1 2 t csds − t

• I

R

(ex − 1 − x)ν(ds, dx),

where,

Bt = t bsds and Ct = t csds,

.

– p. 24/3

Duality

Let us assume that τ is a stopping time with respect to the given filtration F, that is τ : Ω → [0, ∞] belongs to Ft for all t ≥ 0; and introduce the notation

C(S0, K, r, δ, τ, Ψ) = Ee−rτ(Sτ − K)+

(9)

P(S0, K, r, δ, τ, Ψ) = Ee−rτ(K − Sτ)+

(10)

– p. 25/3

Duality

Let us assume that τ is a stopping time with respect to the given filtration F, that is τ : Ω → [0, ∞] belongs to Ft for all t ≥ 0; and introduce the notation

C(S0, K, r, δ, τ, Ψ) = Ee−rτ(Sτ − K)+

(11)

P(S0, K, r, δ, τ, Ψ) = Ee−rτ(K − Sτ)+

(12)

If τ = T, where T is a fixed constant time, then formulas (9) and (10) give the price of the European call and put options respectively.

– p. 25/3

Put-Call Duality

Proposition 0.1 Consider a Time-changed Lévy market with driving process Y with characteristic triplet

Ψ = (B, C, ν). Then, for the expectations introduced in

(9) and (10) we have

C(S0, K, r, δ, τ, Ψ) = P(K, S0, δ, r, τ, ˜ Ψ),

(13)

where ˜

Ψ(z) = ( ˜ B, ˜ C, ˜ ν) is the characteristic triplet (of a

certain semimartingale) that satisfies:

     ˜ Bt = (δ − r)t − 1

2

t

0 csds −

t ex − 1 − x1{|x|≤1}

• ˜

ν(ds, dx), ˜ C = C, ˜ ν(dy) = e−yν(−dy)

– p. 26/3

Remark

When Yt has not independent increments, the characteristic triplet is predictable.

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Remark

When Yt has not independent increments, the characteristic triplet is predictable. Since P is the risk neutral probability, the process B is also determined by the other characteristics.

Bt = (r − δ)t − 1 2Ct − t

• I

R

(ex − 1 − h(x))ν(ds, dx),

– p. 27/3

Remark

When Yt has not independent increments, the characteristic triplet is predictable. Since P is the risk neutral probability, the process B is also determined by the other characteristics.

Bt = (r − δ)t − 1 2Ct − t

• I

R

(ex − 1 − h(x))ν(ds, dx),

where (B, C, ν) is the triplet characteristics of Y and h is a truncation function.

– p. 27/3

Remark

When Yt has not independent increments, the characteristic triplet is predictable. Since P is the risk neutral probability, the process B is also determined by the other characteristics.

Bt = (r − δ)t − 1 2Ct − t

• I

R

(ex − 1 − h(x))ν(ds, dx),

where (B, C, ν) is the triplet characteristics of Y and h is a truncation function. Gyrsanov’s Theorem for semimartingales:

˜ Bt = −Bt − Ct − h(x)(ex − 1)ν(ds, dx)

(17)

– p. 27/3

Symmetric markets

Lets define symmetric markets by

L

• e−(r−δ)t+Yt | P
• = L
• e−(δ−r)t−Yt | ˜

P

• ,

(18)

In view of (0.1), we have

ν(dy) = e−yν(−dy).

(19)

This ensures ˜

ν = ν, and from this follows B − (r − δ) = ˜ B − (δ − r)

, giving (18), as we always have ˜

C = C.

– p. 28/3

Bates’s x%-Rule

If the call and put options have strike prices x%

• ut-of-the money relative to the forward price, then the

call should be priced x% higher than the put.

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Bates’s x%-Rule

If the call and put options have strike prices x%

• ut-of-the money relative to the forward price, then the

call should be priced x% higher than the put. If r = δ, we can take the future price F as the underlying asset in Proposition 1.

– p. 29/3

Bates’s x%-Rule

If the call and put options have strike prices x%

• ut-of-the money relative to the forward price, then the

call should be priced x% higher than the put. If r = δ, we can take the future price F as the underlying asset in Proposition 1. Corollary 0.3 Take r = δ and assume (19) holds, we have

C(F0, Kc, r, τ, Ψ) = (1 + x) P(F0, Kp, r, τ, Ψ),

(22)

where Kc = (1 + x)F0 and Kp = F0/(1 + x), with x > 0.

– p. 29/3

Conclusions

Time-Changed Lévy Processes:

• Bidimensional derivatives

– p. 30/3

Conclusions

Time-Changed Lévy Processes:

• Bidimensional derivatives
• Duality and Symmetry

– p. 30/3

Conclusions

Time-Changed Lévy Processes:

• Bidimensional derivatives
• Duality and Symmetry
• Bates’s x% Rule.

– p. 30/3

Conclusions

Time-Changed Lévy Processes:

• Bidimensional derivatives
• Duality and Symmetry
• Bates’s x% Rule.

Extensions:

• Duality with Exotic Derivatives
• Skewness Premium
• Perpetual American Option with TCLP

– p. 30/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

– p. 31/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

• Fajardo and Mordecki (2006) Put-Call Duality and Symmetry with L´

evy Processes.

Quantitative Finance 6, 3, 219–227.

– p. 31/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

• Fajardo and Mordecki (2006) Put-Call Duality and Symmetry with L´

evy Processes.

Quantitative Finance 6, 3, 219–227.

• Fajardo and Mordecki (2005) A Note on Pricing, Duality and Symmetry for Two Dimensional

L´ evy Markets. “From Stochastic Analysis to Mathematical Finance - Festschrift for

A.N. Shiryaev”. Eds. Y. Kabanov, R. Lipster and J. Stoyanov, Springer Verlag, New York.

– p. 31/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

• Fajardo and Mordecki (2006) Put-Call Duality and Symmetry with L´

evy Processes.

Quantitative Finance 6, 3, 219–227.

• Fajardo and Mordecki (2005) A Note on Pricing, Duality and Symmetry for Two Dimensional

L´ evy Markets. “From Stochastic Analysis to Mathematical Finance - Festschrift for

A.N. Shiryaev”. Eds. Y. Kabanov, R. Lipster and J. Stoyanov, Springer Verlag, New York.

• Fajardo and Mordecki (2003). Duality and Derivative Pricing with L´

evy Processes. Preprint

CMAT- Uruguay.

– p. 31/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

• Fajardo and Mordecki (2006) Put-Call Duality and Symmetry with L´

evy Processes.

Quantitative Finance 6, 3, 219–227.

• Fajardo and Mordecki (2005) A Note on Pricing, Duality and Symmetry for Two Dimensional

L´ evy Markets. “From Stochastic Analysis to Mathematical Finance - Festschrift for

A.N. Shiryaev”. Eds. Y. Kabanov, R. Lipster and J. Stoyanov, Springer Verlag, New York.

• Fajardo and Mordecki (2003). Duality and Derivative Pricing with L´

evy Processes. Preprint

CMAT- Uruguay.

• Eberlein and Papapantoleon (2005a). Equivalence of Floating and Fixed Strike Asian and

Lookback Options. Stochastic Processes and Their Applications, 115, 31-40

– p. 31/3

References

• Fajardo and Mordecki (2006) Pricing Derivatives on Two-dimensional L´

evy Processes. Int.

Journal of Theoretical and Applied Finance. 9, 2, 185–197

• Fajardo and Mordecki (2006) Put-Call Duality and Symmetry with L´

evy Processes.

Quantitative Finance 6, 3, 219–227.

• Fajardo and Mordecki (2005) A Note on Pricing, Duality and Symmetry for Two Dimensional

L´ evy Markets. “From Stochastic Analysis to Mathematical Finance - Festschrift for

A.N. Shiryaev”. Eds. Y. Kabanov, R. Lipster and J. Stoyanov, Springer Verlag, New York.

• Fajardo and Mordecki (2003). Duality and Derivative Pricing with L´

evy Processes. Preprint

CMAT- Uruguay.

• Eberlein and Papapantoleon (2005a). Equivalence of Floating and Fixed Strike Asian and

Lookback Options. Stochastic Processes and Their Applications, 115, 31-40

• Eberlein and Papapantoleon (2005b). Symmetries and pricing of exotic options in L´

evy

• models. “Exotic Option Pricing and Advanced Lévy Models”. A. Kyprianou, W.

Schoutens, P . Wilmott (Eds.), Wiley.

– p. 31/3

References

• Huang, J. and Wu, L.,(2004). Specification Analysis of Option Pricing Models Based on

Time-Changed L´ evy Processes, Journal of Finance, vol. LIX, no. 3, 1405–1440.

– p. 32/3

References

• Huang, J. and Wu, L.,(2004). Specification Analysis of Option Pricing Models Based on

Time-Changed L´ evy Processes, Journal of Finance, vol. LIX, no. 3, 1405–1440.

• Carr, P

. and Wu, L. (2004), Time-changed L´

evy Processes and option pricing. Journal of

Financial Economics 71, 113–141.

– p. 32/3

References

• Huang, J. and Wu, L.,(2004). Specification Analysis of Option Pricing Models Based on

Time-Changed L´ evy Processes, Journal of Finance, vol. LIX, no. 3, 1405–1440.

• Carr, P

. and Wu, L. (2004), Time-changed L´

evy Processes and option pricing. Journal of

Financial Economics 71, 113–141.

• Cherny, A. S. and Shiryaev, A. N. (2002). Change of time and measure for L´

evy processes.

University of Aarhus. Centre for Mathematical Physics and Stochastics, Lecture Notes 13.

– p. 32/3

Triplet for Semimartingales

When Yt has indep. increments. and it’s distribution is inf. divisible: E(eiuYt) = eψt(u), with ψt(u) = iubt − u2 2 ct +

• (eiux − 1 − iuh(x))Ft(dx)

bt ∈ I R, ct ∈ I R+ and Ft a positive measure which integrates x2 ∧ 1. h is abounded Borel function with compact support which behaves like “x” near the origin, the independence of increments imply: eiuYt eψt(u) , is a martingale If Y is a semimartingale, we can find a unique (B, C, ν) such that if we use this triplet instead if (b, c, F) to define ψt, we still have the martingale property.

– p. 33/3

Gyrsanov for Semimartingales

Lemma 0.1 Let Y be a d-dimensional semimartingale with finite variation with triplet (B, C, ν) under P, let u, v be vectors in d and v ∈ [−M, M]d. Moreover let ˜ P ∼ P, with density d ˜

P dP = e<v,YT > E[e<v,YT >]. Then the process Y ∗ :=< u, Y > is a ˜

P−semimartingale with triplet: b∗

s

= < u, bs > + 1 2(< u, csv > + < v, csu >) + Z

d < u, x > (e<v,x> − 1)λs(dx)

c∗

s

= < u, csu > λ∗

s

= Λ(κs) where Λ is a mapping λ : I Rd → I R such that x → Λ(x) =< u, x > and κs is a measure defined by κs(A) = R

A e<v,x>λs(dx).

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Stationarity and Independence

Lemma 0.2 Let {Xt} be a Lévy process and {τt}t≤T be an independent increasing cádlág process with stationary increments. Then {Xτt} has stationary increments.

– p. 35/3

Stationarity and Independence

Lemma 0.3 Let {Xt} be a Lévy process and {τt}t≤T be an independent increasing cádlág process with stationary increments. Then {Xτt} has stationary increments. Lemma 0.4 Let {Xt} be a Lévy process such that , for any t ≥ 0, EX2

t < ∞ and

EXt = 0. Let {τt}t≤T be and independent cádlág process such that , for any t ≥ 0, Eτt < ∞. Then, for any t ≥ 0, EX2

τt < ∞ and EXτt = 0. Moreover, the

increments of Xτt over disjoint intervals are not correlated. See Cherny and Shiryaev (2002). Ch. 5, Lemma 5.2 and 5.3.

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