Lévy Finance *[0.5Cm] Models and Results

Home , Jump diffusion, Variance gamma process

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Stochastic Calculus for Lévy Processes Lévy-Process Driven Financial Market Models Jump-Diffusion Models Merton-Model Kou-Model General Lévy Models Variance-Gamma model CGMY model GH models Variance-mean mixtures European Style Options Equivalent Martingale Measure Jump-Diffusion Models Variance-Gamma Model NIG Model

Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Stochastic Integral for LévyProcesses

Let (Xt ) be a Lévy process with Lévy-Khintchine triplet (α, σ, ν(dx)). By the Lévy-Itódecomposition we know

X = X (1) + X (2) + X (3),

where the X (i) are independent L´evyprocesses. X (1) is a Brownian motion with drift, X (2) is a compound Poisson process with jump (3) distributed concentrated on R/(−1, 1) and X is a square-integrable martingale (which can be viewed as a limit of compensated compound Poisson processes with small jumps). We know how to deﬁne the stochastic integral with respect to any of these processes!

From the L´evy-It´odecomposition we deduce the canonical decomposition (useful for applying the general semi-martingale theory) Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx), 0 R where Z t Z X xµX (ds, dx) = ∆X (s) 0 R 0

d˜ Assume ∼ ˜ and [ P |FT ] = Z(T ). Then there exists a P P E dP deterministic process β and a measurable non-negative deterministic process Y , satisfying

Z t Z |x(Y (s, x) − 1)| ν(dx)ds < ∞ 0 R and Z t (σβ(s))2ds < ∞. 0 β and Y can be expressed in terms of X and Z.

Conversely, if " # dP˜ Z(t) = E |Ft dP Z t 1 Z t = exp β(s)σdW (s) − β2(s)σ2ds 0 2 0 Z t Z + (Y (s, x) − 1) µX − νX (ds, dx) 0 R Z t Z − (Y (s, x) − 1 − log Y (s, x)) µX (ds, dx) 0 R

then it deﬁnes a probability measure P˜ ∼ P.

In both cases Z t W˜ (t) = W (t) − β(s)σds 0 X X is a P˜ Brownian motion,ν ˜ (ds, dx) = Y (s, x)ν (ds, dx) is the P˜ compensator of µX and X has the following canonical decomposition under P˜ Z t Z X (t) =α ˜t + σW˜ (t) + x µX − ν˜X (ds, dx), 0 R where Z t Z t Z α˜t = αt + β(s)σds + x(Y (s, x) − 1)νX (ds, dx) 0 0 R

Professor Dr. R¨udigerKiesel L´evyFinance I if (β, Y ) are deterministic and dependent on time, then X becomes a process with independent (but not stationary) increments; often called an additive process;

I in the general case X is a semi-martingale.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Girsanov theorem

X is not necessary a L´evyprocess under P˜ I if (β, Y ) are deterministic and independent of time, then X remains a L´evyprocess with triplet (˜α, σ, Y · ν);

Professor Dr. R¨udigerKiesel L´evyFinance I in the general case X is a semi-martingale.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Girsanov theorem

X is not necessary a L´evyprocess under P˜ I if (β, Y ) are deterministic and independent of time, then X remains a L´evyprocess with triplet (˜α, σ, Y · ν);

I if (β, Y ) are deterministic and dependent on time, then X becomes a process with independent (but not stationary) increments; often called an additive process;

X is not necessary a L´evyprocess under P˜ I if (β, Y ) are deterministic and independent of time, then X remains a L´evyprocess with triplet (˜α, σ, Y · ν);

I if (β, Y ) are deterministic and dependent on time, then X becomes a process with independent (but not stationary) increments; often called an additive process;

I in the general case X is a semi-martingale.

Let (Xt ) be a L´evy process with L´evy-Khintchine triplet (α, σ, ν) and f ∈ C 2. Then Z t 0 f (X (t)) = f (x0) + f (X (u−))dX (u) 0 σ2 Z t + f 00(X (u))du 2 0 X + f (X (s)) − f (X (s−)) − ∆X (s)f 0(X (s−)) . 0

The stochastic exponential of a L´evy process X is the solution Z of the SDE dZ(t) = Z(t−)dX (t), Z(0) = 1 which is

σ2t Y Z(t) = exp X (t) − (1 + ∆X (s)) e−∆X (s). 2 0

Professor Dr. R¨udigerKiesel L´evyFinance I The solution of the SDE is the stochastic exponential

σ2t Y S(t) = S(0) exp X (t) − (1 + ∆X (s)) e−∆X (s) = E (X ) . 2 t 0

I Problems:

I the asset price can take negative values unless jumps are restricted to be larger than −1

I the distribution of log returns is not known

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Dynamic Specification

I We assume the asset price process dynamics to be

dS(t) = S(t−)dX (t)

with X a suitable driving L´evyprocess with triplet (α, σ, ν).

Professor Dr. R¨udigerKiesel L´evyFinance I Problems:

I the asset price can take negative values unless jumps are restricted to be larger than −1

I the distribution of log returns is not known

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Dynamic Specification

I We assume the asset price process dynamics to be

dS(t) = S(t−)dX (t)

with X a suitable driving L´evyprocess with triplet (α, σ, ν).

I The solution of the SDE is the stochastic exponential

σ2t Y S(t) = S(0) exp X (t) − (1 + ∆X (s)) e−∆X (s) = E (X ) . 2 t 0

I We assume the asset price process dynamics to be

dS(t) = S(t−)dX (t)

with X a suitable driving L´evyprocess with triplet (α, σ, ν).

I The solution of the SDE is the stochastic exponential

σ2t Y S(t) = S(0) exp X (t) − (1 + ∆X (s)) e−∆X (s) = E (X ) . 2 t 0

I Problems:

I the asset price can take negative values unless jumps are restricted to be larger than −1

I the distribution of log returns is not known

Professor Dr. R¨udigerKiesel L´evyFinance I Also, the risk-free bank account (discount) factor is

B(t) = ert , r ≥ 0

I The SDE for S(t) is

1 dS(t) = S(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Exponential Specification

I In the following, we will assume that the stock price process is

S(t) = S(0) exp{X (t)}.

Professor Dr. R¨udigerKiesel L´evyFinance I The SDE for S(t) is

1 dS(t) = S(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Exponential Specification

I In the following, we will assume that the stock price process is

S(t) = S(0) exp{X (t)}.

I Also, the risk-free bank account (discount) factor is

B(t) = ert , r ≥ 0

I In the following, we will assume that the stock price process is

S(t) = S(0) exp{X (t)}.

I Also, the risk-free bank account (discount) factor is

B(t) = ert , r ≥ 0

I The SDE for S(t) is

1 dS(t) = S(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

Professor Dr. RüdigerKiesel LévyFinance I Using the product formula and Itô’sformula dS˜(t) = −rS˜(t−)dt

1 +S˜(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

I Now use the canonical decomposition of X Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx). 0 R

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Existence of EMM - calculations

I We want S˜(t) = S(0) exp{X (t) − rt} to be a martingale.

Professor Dr. R¨udigerKiesel L´evyFinance I Now use the canonical decomposition of X Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx). 0 R

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Existence of EMM - calculations

I We want S˜(t) = S(0) exp{X (t) − rt} to be a martingale. I Using the product formula and Itˆo’sformula dS˜(t) = −rS˜(t−)dt

1 +S˜(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

I We want S˜(t) = S(0) exp{X (t) − rt} to be a martingale. I Using the product formula and Itˆo’sformula dS˜(t) = −rS˜(t−)dt

1 +S˜(t−) dX (t) + σ2dt + e∆X (t) − 1 − ∆X (t) 2

I Now use the canonical decomposition of X Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx). 0 R

Professor Dr. R¨udigerKiesel L´evyFinance I Now we change the measure according to the Girsanov theorem and obtain Z t W (t) = W ∗(t) + σβ(u)du 0 ν∗(du, dx) = Y (u, x)νX (du, dx)

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Existence of EMM - calculations

I So we get the integral form Z t 1 Z t S˜(t) = S˜(0) + S˜(u−) σ2 − r + α du + S˜(u−)σdW (u) 0 2 0 Z t Z + S˜(u−)(ex − 1)µX (du, dx) 0 R Z t Z − S˜(u−)xνX (du, dx) 0 R

I So we get the integral form Z t 1 Z t S˜(t) = S˜(0) + S˜(u−) σ2 − r + α du + S˜(u−)σdW (u) 0 2 0 Z t Z + S˜(u−)(ex − 1)µX (du, dx) 0 R Z t Z − S˜(u−)xνX (du, dx) 0 R I Now we change the measure according to the Girsanov theorem and obtain Z t W (t) = W ∗(t) + σβ(u)du 0 ν∗(du, dx) = Y (u, x)νX (du, dx) Professor Dr. R¨udigerKiesel L´evyFinance ˜ ∗ I We want S to be a P martingale, so the drift must vanish.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Existence of EMM - calculations

I Now the dynamics are (we already use the L´evy measure ν)

1 dS˜(t) = S˜(t−) σ2 − r + α + σβ(t) dt 2 +S˜(t−)σdW ∗(t)

Z + S˜(t−)(ex − 1) µX − ν∗ (dt, dx) R Z + S˜(t−) [(ex − 1)Y (t, x) − x] ν(dx)dt R

I Now the dynamics are (we already use the L´evy measure ν)

1 dS˜(t) = S˜(t−) σ2 − r + α + σβ(t) dt 2 +S˜(t−)σdW ∗(t)

Z + S˜(t−)(ex − 1) µX − ν∗ (dt, dx) R Z + S˜(t−) [(ex − 1)Y (t, x) − x] ν(dx)dt R ˜ ∗ I We want S to be a P martingale, so the drift must vanish.

Professor Dr. R¨udigerKiesel L´evyFinance I In terms of the triple after the measure change 1 Z σ2 − r +α ˜ + [(ex − 1 − x)Y (t, x)])ν(dx) = 0 2 R

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Drift Conditions

I In terms of the original triple 1 Z σ2 − r + α + σβ(t) + [(ex − 1)Y (t, x) − x] ν(dx) = 0 2 R

I In terms of the triple after the measure change 1 Z σ2 − r +α ˜ + [(ex − 1 − x)Y (t, x)])ν(dx) = 0 2 R

Consider the exponential L´evymodel

S(t) = S(0)eX (t), t ≥ 0.

Assume that the moment-generating function

h hX (t)i M(h, t) = E e

of X (t) exists; then

M(h, t) = [M(h, 1)]t.

The process n o ehX (t)M(h, 1)−t t≥0 is a positive martingale and can be used to define a change of probability measure Q, which is called the Esscher measure of parameter h. The risk-neutral Esscher measure is the Esscher measure of parameter h = h∗ such that the process −rt e S(t) t≥0 is a martingale. The condition −rt ∗ E e S(t); h = S(0) yields rt h X (t) ∗i e Professor= E Dr.e RüdigerKiesel; h LévyFinance

" ∗ # eX (t)+h X (t) M(1 + h∗, 1)t = = E M(h∗, 1)t M(h∗, 1) or M(1 + h∗, 1) er = . M(h∗, 1) This equation then uniquely determines the parameter h∗. Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Option Pricing

We apply the above technique to the valuation of a European call with maturity T and strike K on the underlying with price dynamics S(t). By the risk-neutral valuation principle, we have to calculate

h −rT + ∗i E e (S(T ) − K) ; h

h −rT ∗i = E e (S(T ) − K)1{S(T )>K}; h

∗ −rT ∗ = S(0)E 1{S(T )>K}; h + 1 − Ke E 1{S(T )>K}; h .

Professor Dr. R¨udigerKiesel L´evyFinance I The tuple that characterizes the change of measure is (β, Y ) with Y (x) = eδx , which is deterministic and independent of time.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Esscher Transform via Girsanov

∗ I Assume P ∼ P with Z t Z Z(t) = exp βσW (t) + δx(µX − νX )(ds, dx) 0 R σ2β2 Z − + (eδx − 1 − δx)ν(dx) t 2 R

where β ∈ R+ and δ ∈ R.

∗ I Assume P ∼ P with Z t Z Z(t) = exp βσW (t) + δx(µX − νX )(ds, dx) 0 R σ2β2 Z − + (eδx − 1 − δx)ν(dx) t 2 R

where β ∈ R+ and δ ∈ R. I The tuple that characterizes the change of measure is (β, Y ) with Y (x) = eδx , which is deterministic and independent of time.

Professor Dr. R¨udigerKiesel L´evyFinance I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Mean-correcting EMM

I Assume S(t) = S(0)eX (t) .

Professor Dr. R¨udigerKiesel L´evyFinance I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Jump-Diffusion Models General LévyModels European Style Options Mean-correcting EMM

I Assume S(t) = S(0)eX (t) .

I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Assume S(t) = S(0)eX (t) .

I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Merton-Model Jump-Diffusion Models Kou-Model General LévyModels European Style Options Classical Construction

The classical formulation is dS(t) = S(t−)dX (t) with driving process N Xt Xt = µt + σ1Wt + i , i=1 where I µ, σ1 > 0 are a constant; I Wt is a standard Brownian motion; I Nt is a Poisson process with intensity λ; I (i ) is a family of independent random variables with distribution F , such that Vi = 1 + i are log-normally distributed and k = R xF (dx).

So S(t) = E (Xt ) which is here

N(t) σ2 Y S(t) = S(0) exp σ W (t) + µ − 1 t V 1 2 i i=1

If 1 + is log-normally distributed with parameters

δ2 log(1 + ) = γ − , Var log(1 + ) = δ2, = k = eγ − 1 E 2 E

under the historical measure P, one can ﬁnd an equivalent ∗ martingale measure P such that the intensity of N is λ˜ > 0 and the (1 + i ) are log-normally distributed with parameters

δ2 ∗ ∗ log(1 + ) =γ ˜ − , Var log(1 + ) = δ2, ∗ = k˜ = eγ˜ − 1 E 2 E ∗ under P .

We ﬁnd for the price of a European call

∞ X (λ0T )n C(S, 1, T , λ,˜ σ˜) = exp{−λ0T }C (S, 1, ˜r , T , σ˜ ), n! BS n n n=0

with CBS the Black-Scholes call price and parameters

nγ˜ 1 nδ2 λ0 = λ˜(1 + k˜), ˜r = − λ˜k˜, σ˜2 = σ2T + . n T n T 2

Professor Dr. R¨udigerKiesel L´evyFinance I The characteristic function of X is 2 2 σ u iµ u−σ2 u2/2 φ (u) = exp iµu − + λ e Y Y − 1 , X 2

and the L´evy triplet is (µ, σ, λfY ).

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Merton-Model Jump-Diffusion Models Kou-Model General LévyModels European Style Options Merton in exponential Lévyframework

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

2 where Yk ∼ N(µY , σY ) with normal density fY for k = 1, 2 ....

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

2 where Yk ∼ N(µY , σY ) with normal density fY for k = 1, 2 ....

I The characteristic function of X is 2 2 σ u iµ u−σ2 u2/2 φ (u) = exp iµu − + λ e Y Y − 1 , X 2

and the L´evy triplet is (µ, σ, λfY ).

The classical formulation is dS(t) = S(t−)dX (t) with driving process N Xt Xt = µt + σWt + (Vi − 1), i=1 where I µ, σ > 0 are a constant; I Wt is a standard Brownian motion; I Nt is a Poisson process with intensity λ; I (Vi ) is a family of i.i.d. random variables, such that Yi = log Vi have an asymmetric double exponential distribution DbExpo(p, η1, η2). Professor Dr. R¨udigerKiesel L´evyFinance I In distribution  +  ξ , with probability p log V = −  −ξ , with probability q

+ − where ξ , ξ are exponential rvs with mean 1/η1 resp. 1/η2

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Merton-Model Jump-Diffusion Models Kou-Model General LévyModels European Style Options Kou’s Construction

I The density of a double exponential distribution is

−η1y −η2y fY (x) = pη1e 1{y≥0} + qη2e 1{y<0}

where p, q ≥ 0; p + q = 1 represent the probabilities of upward and downward jumps.

I The density of a double exponential distribution is

−η1y −η2y fY (x) = pη1e 1{y≥0} + qη2e 1{y<0}

where p, q ≥ 0; p + q = 1 represent the probabilities of upward and downward jumps.

I In distribution  +  ξ , with probability p log V = −  −ξ , with probability q

+ − where ξ , ξ are exponential rvs with mean 1/η1 resp. 1/η2

Professor Dr. R¨udigerKiesel L´evyFinance I N(t) σ2 Y S(t) = S(0) exp σW (t) + µ − t V 2 i i=1 I p q 1 1 2 p q E(Y ) = − ; Var(Y ) = pq + + 2 + 2 ; η1 η2 η1 η2 η1 η2 Y η2 η1 E(V ) = E(e ) = q + p η2 + 1 η1 − 1 where we need η1 > 1 to ensure existence of relevant moments (it implies that the average upward jump does not exceed 100 percent)

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Merton-Model Jump-Diffusion Models Kou-Model General LévyModels European Style Options Kou’s Construction

I So S(t) = E (Xt )

Professor Dr. R¨udigerKiesel L´evyFinance I p q 1 1 2 p q E(Y ) = − ; Var(Y ) = pq + + 2 + 2 ; η1 η2 η1 η2 η1 η2 Y η2 η1 E(V ) = E(e ) = q + p η2 + 1 η1 − 1 where we need η1 > 1 to ensure existence of relevant moments (it implies that the average upward jump does not exceed 100 percent)

Stochastic Calculus for LévyProcesses Lévy-Process Driven Financial Market Models Merton-Model Jump-Diffusion Models Kou-Model General LévyModels European Style Options Kou’s Construction

I So S(t) = E (Xt )

I N(t) σ2 Y S(t) = S(0) exp σW (t) + µ − t V 2 i i=1

I So S(t) = E (Xt )

I N(t) σ2 Y S(t) = S(0) exp σW (t) + µ − t V 2 i i=1 I p q 1 1 2 p q E(Y ) = − ; Var(Y ) = pq + + 2 + 2 ; η1 η2 η1 η2 η1 η2 Y η2 η1 E(V ) = E(e ) = q + p η2 + 1 η1 − 1 where we need η1 > 1 to ensure existence of relevant moments (it implies that the average upward jump does not exceed 100 percent) Professor Dr. R¨udigerKiesel L´evyFinance I The characteristic function of X is 2 2 σ u pη1 qη2 φX (u) = exp iµu − + λ − − 1 , 2 η1 − iu η2 + iu

and the L´evy triplet is (µ, σ, λfY ).

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

where Yk ∼ DbExpo(p, η1, η2) with DbExpo-density fY for k = 1, 2 ....

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

where Yk ∼ DbExpo(p, η1, η2) with DbExpo-density fY for k = 1, 2 ....

I The characteristic function of X is 2 2 σ u pη1 qη2 φX (u) = exp iµu − + λ − − 1 , 2 η1 − iu η2 + iu

and the L´evy triplet is (µ, σ, λfY ).

Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Examples of Lévy-type models

I Variance-Gamma model (Madan and Seneta 1990, Carr, Chang, and Madan 1998)

I CGMY model (Carr, Geman, Madan, and Yor 2002)

I hyperbolic distributions (Eberlein and Keller 1995, Eberlein, Keller, and Prause 1998)

I normal inverse Gaussian distributions (Barndorﬀ-Nielsen 1998)

I generalized hyperbolic distributions (Eberlein 2001)

Professor Dr. RüdigerKiesel LévyFinance I The Lévy-Khintchine triplet is 1 a(1 − e−b)/b, 0, ae−bx 1 dx x {x>0}

I A Gamma process is an increasing pure-jump process and has inﬁnitely many jumps in each interval

Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Gamma Process

G I A Gamma process X with parameters a, b > 0 has X G (t + s) − X G (t) ∼ Γ(as, b), i.e. has density

e−x/b f (x; as, b) = xas−1 bas−1Γ(as)

Professor Dr. RüdigerKiesel LévyFinance I A Gamma process is an increasing pure-jump process and has infinitely many jumps in each interval

G I A Gamma process X with parameters a, b > 0 has X G (t + s) − X G (t) ∼ Γ(as, b), i.e. has density

e−x/b f (x; as, b) = xas−1 bas−1Γ(as)

I The L´evy-Khintchine triplet is 1 a(1 − e−b)/b, 0, ae−bx 1 dx x {x>0}

G I A Gamma process X with parameters a, b > 0 has X G (t + s) − X G (t) ∼ Γ(as, b), i.e. has density

e−x/b f (x; as, b) = xas−1 bas−1Γ(as)

I The L´evy-Khintchine triplet is 1 a(1 − e−b)/b, 0, ae−bx 1 dx x {x>0}

I A Gamma process is an increasing pure-jump process and has inﬁnitely many jumps in each interval

Professor Dr. R¨udigerKiesel L´evyFinance Advanced Financial Mathematics and Structured Derivatives

Extended Black-Scholes Model Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy ProcessLévy-Process Driven FinancialDriven Market ModelsModels CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures • Examples: LévyEuropean processes Style Options Gamma Process – Gamma process: Density of a Gamma(10,20)-distributed random variable: Density of a Gamma(10,20)-distributed random variable

www.executiveacademy.at  Prof. Dr. Rudi Zagst, Dr. Matthias Scherer, Stephan Höcht 139 Professor Dr. R¨udigerKiesel L´evyFinance Advanced Financial Mathematics and Structured Derivati ves

Extended Black-Scholes ModelStochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Lévy Process Driven Models Variance-mean mixtures • Examples: Lévy processes European Style Options – Gamma process:Gamma Process • Sample pathSample of a Gamma path of process a Gamma(10,20)-process with parameters a=1 0 and b=20.

www.executiveacademy.at  Prof. Dr. Rudi Zagst, Dr. Matthias Scherer, Stephan Höcht 141

Professor Dr. R¨udigerKiesel L´evyFinance VG G G I Also, X (t) = X (t; C, M) − X (t; C, G) with parameters C = 1/ν > 0

!−1 r1 1 1 G = θ2ν2 + σ2ν − θν > 0 4 2 2 !−1 r1 1 1 M = θ2ν2 + σ2ν + θν > 0 4 2 2

VG I A Variance Gamma process X can be deﬁned as a time-changed Brownian motion with drift X (t)VG = θX G (t) + σW (X G (t)).

VG G G I Also, X (t) = X (t; C, M) − X (t; C, G) with parameters C = 1/ν > 0

!−1 r1 1 1 G = θ2ν2 + σ2ν − θν > 0 4 2 2 !−1 r1 1 1 M = θ2ν2 + σ2ν + θν > 0 4 2 2

I The characteristic function of a Variance Gamma process X VG with parameters σ, ν, θ is 1 −u/ν φ (u) = 1 − iuθ + σ2νu2 . VG 2

I The L´evy-Khintchine triplet is (γ, 0, νVG (dx)) with −C G e−M − 1 − M e−G − 1 γ = MG The L´evymeasure of the VG-distribution is ( CeGx |x|−1 , x < 0 νVG (dx) = Ce−Mx x−1, x < 0

I The characteristic function of a Variance Gamma process X VG with parameters σ, ν, θ is 1 −u/ν φ (u) = 1 − iuθ + σ2νu2 . VG 2

Professor Dr. R¨udigerKiesel L´evyFinance I A VG process is a pure jump process I The parameter can be interpreted as follows

I θ is a skewness parameter, I σ is the variance parameter, I ν is the kurtosis parameter.

I A VG process has inﬁnitely many jumps but is of ﬁnite variation

Professor Dr. R¨udigerKiesel L´evyFinance I The parameter can be interpreted as follows

I θ is a skewness parameter, I σ is the variance parameter, I ν is the kurtosis parameter.

I A VG process has inﬁnitely many jumps but is of ﬁnite variation

I A VG process is a pure jump process

I A VG process has inﬁnitely many jumps but is of ﬁnite variation

I A VG process is a pure jump process I The parameter can be interpreted as follows

I θ is a skewness parameter, I σ is the variance parameter, I ν is the kurtosis parameter.

Professor Dr. R¨udigerKiesel L´evyFinance Advanced Financial Mathematics and Structured Derivati ves

Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models Extended Black-Scholes Model CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures Lévy Process Driven Models European Style Options • Examples: LévyVariance processes Gamma Process

– Variance gamma Sampleprocess path (continued): of a VG process with parameters • Sample pathC of= a 20 VG, G process= 40, M =with 50 parameters C=20, G=40, and M=50.

www.executiveacademy.at  Prof. Dr. RudiProfessor Zagst, Dr. Dr. RüdigerKieselMatthias Scherer, StephanLévyFinance Höcht 150 Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Variance Gamma Fit – Deutsche Bank

Professor Dr. RüdigerKiesel LévyFinance Figure: Fitted VG, GH, Normal and Empirical Densities for Deutsche Bank Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Variance Gamma Fit – SAP

Professor Dr. RüdigerKiesel LévyFinance Figure: Fitted VG, GH, Normal and Empirical Densities for SAP Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Variance Gamma Fit – QQ-Plot

Professor Dr. RüdigerKiesel LévyFinance Figure: QQ Plot for the fitted VG, GH and Normal Distribution for Deutsche Bank Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Variance Gamma Fit – QQ-Plot

Professor Dr. RüdigerKiesel LévyFinance Figure: QQ Plot for the fitted VG, GH and Normal Distribution for SAP Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options CGMY Distribution

This is the class of infinitely divisible distributions containing the Variance-Gamma distributions as subclass. The CGMY class is defined by its Lévy-Khintchine triplet (α, σ2, ν(dx)) with respect to a truncation function h(x)

Z ! C −|x| α = h(x)kCGMY (x) − x1{|x|≤1} e dx, |x|1+Y

σ = 0, ν(dx) = kCGMY (x)dx, with the four-parameter Lévy density  exp{−G |x|}  C for x < 0  1+Y k (x) = |x| CGMY exp{−M |x|}  C for x > 0.  1+Y Professor Dr. RüdigerKiesel |x|LévyFinance Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Generalized Hyperbolic Distributions

For these distributions the densities are given by:

dGH (x; λ, α, β, δ, µ) = a(λ, α, β, δ, µ) (1) 2 2 (λ− 1 )/2 ×(δ + (x − µ) ) 2 q 2 2 ×K 1 (α δ + (x − µ) ) λ− 2 × exp{β(x − µ)} where (α2 − β2)λ/2 a(λ, α, β, δ, µ) = √ λ− 1 λ p 2 2 2πα 2 δ Kλ(δ α − β )

and Kν denotes the modified Bessel function of the third kind Z ∞ 1 ν−1 1 −1 Kν(z) = y exp − z(y + y ) dy 2 0 2 Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Generalized Hyperbolic Distributions

Interpretation of parameters:

I α > 0 determines the shape;

I 0 < |β| < α is a skewness parameter; I µ ∈ R determines the location; I δ is a scaling parameter comparable to σ; I λ ∈ R characterises subclasses. Scale and location-invariant (no change under aﬃne transformations) parameterizations are

p β ζ = δ α2 − β2, ρ = , α

− 1 ξ = (1 + ζ) 2 , χ = ξρ.

Since 0 ≤ |χ| < ξ < 1 the distribution parameterised by χ and ξ can be represented by points of a triangle, the so-called shape triangle. 1 ξ 6L EE @ @ @ @ @ @ @ @ @ @ @@ - N -1 0 1 χ Professor Dr. RüdigerKiesel LévyFinance Figure: Shape Triangle Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Tail behaviour of GH-distribution

Professor Dr. RüdigerKiesel LévyFinance Figure: Tails of a GH and a Normal Distribution having the same Mean and Variance Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options Tail behaviour of GH-distribution

This is the case λ = 1. Since

1 −z K 1 (z) = (π/2z) 2 e 2 the density is now

pα2 − β2 dH (x) = × p 2 2 2αδK1(δ α − β )

q exp −α δ2 + (x − µ)2 + β(x − µ) .

Taking the logarithm of dH we get a hyperbola from the term pδ2 + (x − µ)2 (instead of the parabola which results from the normal distribution).

This is the case λ = −1/2. The density is α n p o d (x) = exp δ α2 − β2 + β(x − µ) NIG π

q x−µ 2 K1 αδ 1 + δ × q . x−µ 2 1 + δ The NIG distribution class is closed under convolution

NIG(α, β, δ1, µ1) ? NIG(α, β, δ2, µ2)

= NIG(α, β, δ1 + δ2, µ1 + µ2).

Professor Dr. RüdigerKiesel LévyFinance Figure: Realisations of NIG Lévy Motions Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options NIG LévyMotions

Professor Dr. RüdigerKiesel LévyFinance Figure: Fitted GH, HYP, NIG, Normal and Empirical Densities for Deutsche Bank Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options NIG Fit

Professor Dr. RüdigerKiesel LévyFinance Figure: Fitted GH, HYP, NIG, Normal and Empirical Densities for SAP Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options QQ-Plot

Professor Dr. RüdigerKiesel LévyFinance Figure: QQ Plot for the fitted GH, HYP, NIG and Normal Distribution for Deutsche Bank Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options QQ-Plot

Professor Dr. RüdigerKiesel LévyFinance Figure: QQ Plot for the fitted GH, HYP, NIG and Normal Distribution for SAP Returns Stochastic Calculus for LévyProcesses Variance-Gamma model Lévy-Process Driven Financial Market Models CGMY model Jump-Diffusion Models GH models General LévyModels Variance-mean mixtures European Style Options GH-distributions in Power Markets

I NIG (λ = −1/2, successful in power markets)

I Hyperbolic (λ = 1, successful in power, gas and oil markets)

I t-distribution (λ = −1/2ν, χ = ν, ψ = 0, γ = 0, ν > 0)

Densities for August 06 Power Future

Empirical Density Normal Density NIG Density 010203040

-0.0832 -0.0310 0.0 0.0211 0.0702 Log-Returns

Log-Densities for August 06 Power Future

Empirical Log-Density Normal Log-Density NIG Log-Density -1012345

-0.0832 -0.0310 0.0 0.0211 0.0702 Log-Returns

All the examples above are cases of variance-mean mixtures of Normal distributions: Let U be a random variable on [0, ∞) with law F , and

X|U=u ∼ Nr (µ + uβ, uΣ), where Σ is a symmetric positive deﬁnite r × r matrix with determinant one, µ, β are r-vectors. The distribution of X is called a normal variance-mean mixture with position µ, drift β, structure matrix Σ and mixing distribution F . If β = 0, X is a normal variance mixture.

The density is given by

n T −1 o fX(x) = exp (x − µ) Σ β

Z ∞ − 1 r × (2πu) 2 exp {−Qu(x; µ, β, Σ)} dF (u) 0 where 1 Q (x; µ, β, Σ) = (x − µ)T (uΣ)−1(x − µ) + uβT Σ−1β u 2

The characteristic function is

n o 1 ψ (t) = exp itT µ φ tT Σt − itT β , X 2 with φ(s) = E exp{−sU} the Laplace-Stieltjes transform of U. ψ is inﬁnitely divisible iﬀ φ is.

Let dGIG denote the density of a generalized inverse Gaussian distribution with parameters λ, δ and γ:

dGIG (x; λ, δ, γ)

γ λ 1 1 δ2 = xλ−1 exp − + γ2x . δ 2Kλ(δγ) 2 x Then the GH-distributions have a representation as

dGH (x; λ, α, β, δ, µ) Z ∞ p 2 2 = dN(µ+βu,u)(x)dGIG (u; λ, δ, α − β )du. 0

The characteristic function ψGH(λ,α,β,δ,µ)(t) =

λ p 2 2 δ α − β K (δpα2 − (β + it)2 eiµt λ . p 2 2 λ Kλ(δ α − β ) δpα2 − (β + it)2

Let dΓ denote the density of a Gamma distribution with parameters µ and ν:

2 µ2 µ µ ν −1 µ ν x exp − ν x dΓ(x; µ, ν) = , x > 0. ν µ2 Γ ν

Then the VG-distributions have a representation as

dVG (x; σ, β, ν) Z ∞ = dN(βu,σu)(x)dΓ(u; 1, ν)du. 0

Professor Dr. R¨udigerKiesel L´evyFinance I In terms of the triple after the measure change 1 Z σ2 − r +α ˜ + [(ex − 1 − x)Y (t, x)])ν(dx) = 0 2 R

Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options Drift Conditions

I In terms of the original triple 1 Z σ2 − r + α + σβ(t) + [(ex − 1)Y (t, x) − x] ν(dx) = 0 2 R

I In terms of the triple after the measure change 1 Z σ2 − r +α ˜ + [(ex − 1 − x)Y (t, x)])ν(dx) = 0 2 R

Professor Dr. R¨udigerKiesel L´evyFinance I The tuple that characterizes the change of measure is (β, Y ) with Y (x) = eδx , which is deterministic and independent of time.

∗ I Assume P ∼ P with Z t Z Z(t) = exp βσW (t) + δx(µX − νX )(ds, dx) 0 R σ2β2 Z − + (eδx − 1 − δx)ν(dx) t 2 R

where β ∈ R+ and δ ∈ R.

∗ I Assume P ∼ P with Z t Z Z(t) = exp βσW (t) + δx(µX − νX )(ds, dx) 0 R σ2β2 Z − + (eδx − 1 − δx)ν(dx) t 2 R

where β ∈ R+ and δ ∈ R. I The tuple that characterizes the change of measure is (β, Y ) with Y (x) = eδx , which is deterministic and independent of time.

Professor Dr. R¨udigerKiesel L´evyFinance I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

I Assume S(t) = S(0)eX (t) .

Professor Dr. R¨udigerKiesel L´evyFinance I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

I Assume S(t) = S(0)eX (t) .

I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Assume S(t) = S(0)eX (t) .

I Then S(t)e− log ϕ(−i)t = e−rt S(t)ert−log ϕ(−i)t , where ϕ(u) is the characteristic function of X (t), is a martingale.

I Observe that − log ϕ(−1) = ψ(1) the characteristic exponent.

Professor Dr. R¨udigerKiesel L´evyFinance I The characteristic function of X is 2 2 σ u iµ u−σ2 u2/2 φ (u) = exp iµu − + λ e Y Y − 1 , X 2

and the L´evy triplet is (µ, σ, λfY ).

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

2 where Yk ∼ N(µY , σY ) with normal density fY for k = 1, 2 ....

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

2 where Yk ∼ N(µY , σY ) with normal density fY for k = 1, 2 ....

I The characteristic function of X is 2 2 σ u iµ u−σ2 u2/2 φ (u) = exp iµu − + λ e Y Y − 1 , X 2

and the L´evy triplet is (µ, σ, λfY ).

Professor Dr. R¨udigerKiesel L´evyFinance I The characteristic function of X is 2 2 σ u pη1 qη2 φX (u) = exp iµu − + λ − − 1 , 2 η1 − iu η2 + iu

and the L´evy triplet is (µ, σ, λfY ).

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

where Yk ∼ DbExpo(p, η1, η2) with DbExpo-density fY for k = 1, 2 ....

I The driving process is

N(t) X X (t) = µt + σW (t) + Yk k=1

where Yk ∼ DbExpo(p, η1, η2) with DbExpo-density fY for k = 1, 2 ....

I The characteristic function of X is 2 2 σ u pη1 qη2 φX (u) = exp iµu − + λ − − 1 , 2 η1 − iu η2 + iu

and the L´evy triplet is (µ, σ, λfY ).

Use the classical formulation

dS(t) = (µ − kλ)S(t)dt + σdW (t) + S(t−)dQ(t)

where

I µ, σ > 0 are a constant; I Wt is a standard Brownian motion; I N Xt Qt = Yi , i=1

I Nt is a Poisson process with intensity λ; I (Yi ) is a family of i.i.d. random variables with density fY , such that k = E(Yi ).

N(t) σ2 Y S(t) = S(0) exp σW (t) + µ − kλ − 1 t (Y + 1) 2 i i=1 The drift condition then reads

µ − kλ = r + σβ − k˜λ˜

Many Choices possible!!

If 1 + Yi are log-normally distributed with parameters

δ2 log(1 + Y ) = γ − , Var log(1 + Y ) = δ2, Y = k = eγ − 1 E i 2 i E i

under the historical measure P, one can ﬁnd an equivalent ∗ martingale measure P such that the intensity of N is λ˜ > 0 and the (1 + Yi ) are log-normally distributed with parameters

δ2 ∗ ∗ log(1+Y ) =γ ˜ − , Var log(1+Y ) = δ2, ∗ = k˜ = eγ˜ −1 E i 2 i E ∗ under P .

We ﬁnd for the price of a European call

∞ X (λ0T )n C(S, 1, T , λ,˜ σ˜) = exp{−λ0T }C (S, 1, ˜r , T , σ˜ ), n! BS n n n=0

with CBS the Black-Scholes call price and parameters

nγ˜ 1 nδ2 λ0 = λ˜(1 + k˜), ˜r = − λ˜k˜, σ˜2 = σ2T + . n T n T 2

Professor Dr. R¨udigerKiesel L´evyFinance VG G G I Also, X (t) = X (t; C, M) − X (t; C, G) with parameters C = 1/ν > 0

!−1 r1 1 1 G = θ2ν2 + σ2ν − θν > 0 4 2 2 !−1 r1 1 1 M = θ2ν2 + σ2ν + θν > 0 4 2 2

VG I A Variance Gamma process X can be deﬁned as a time-changed Brownian motion with drift X (t)VG = θX G (t) + σW (X G (t)).

VG G G I Also, X (t) = X (t; C, M) − X (t; C, G) with parameters C = 1/ν > 0

!−1 r1 1 1 G = θ2ν2 + σ2ν − θν > 0 4 2 2 !−1 r1 1 1 M = θ2ν2 + σ2ν + θν > 0 4 2 2

Professor Dr. RüdigerKiesel LévyFinance I The Lévy-Khintchine triplet is (γ, 0, νVG (dx)) with −C G e−M − 1 − M e−G − 1 γ = MG The Lévymeasure of the VG-distribution is ( CeGx |x|−1 , x < 0 νVG (dx) = Ce−Mx x−1, x < 0

I The characteristic function of a Variance Gamma process X VG with parameters σ, ν, θ is 1 −u/ν φ (u) = 1 − iuθ + σ2νu2 . VG 2

Professor Dr. R¨udigerKiesel L´evyFinance I Distributional characteristics VG(C,G,M) VG(C,G,G) mean C(G − M/(MG)) 0 variance C(G 2 + M2)/(MG)2 2CG −2 −1 3 3 2 2 3 skewness 2C 2(G − M )/(G + M ) 2 0 kurtosis 3(1 + 2C −1(G 4 + M4)/(M2 + G 2)2) 3(1 + C −1 )

VG I The density function of a Variance Gamma process X is (CMC ) (C − M)x f (x) = √ exp VG πΓ(C) 2 C−1/2 |x| G+M KC−1/2((G + M) |x| /2)

where Kν(x) denotes the modiﬁed Bessel function of the third kind with index ν.

VG I The density function of a Variance Gamma process X is (CMC ) (C − M)x f (x) = √ exp VG πΓ(C) 2 C−1/2 |x| G+M KC−1/2((G + M) |x| /2)

where Kν(x) denotes the modified Bessel function of the third kind with index ν. I Distributional characteristics VG(C,G,M) VG(C,G,G) mean C(G − M/(MG)) 0 variance C(G 2 + M2)/(MG)2 2CG −2 −1 3 3 2 2 3 skewness 2C 2(G − M )/(G + M ) 2 0 kurtosis 3(1 + 2C −1(G 4 + M4)/(M2 + G 2)2) 3(1 + C −1 ) Professor Dr. RüdigerKiesel LévyFinance VG(σ,ν,θ) VG(σ, ν, 0) mean θ 0 variance σ2 + νθ2 σ2 skewness θν(3σ2 + 2νθ2)/(σ2 + νθ2)3/2 0 kurtosis 3(1 + 2ν νσ4(σ2 + νθ2)−2) 3(1 + ν) − Table 5: VG distribution characteristics in the (σ,ν,θ) parametrization.

Going the other way around one can use:

ν = 1/C σ2 = 2C/(MG) θ = C(G M)/(MG). − Its density function is given by

(GM)C (G M)x f (x; C,G,M)(x) = exp − VG √π Γ(C) 2 x C−1/2 Stochastic Calculus for LévyProcesses | | Equivalent MartingaleKC−1/2 ( MeasureG + M) x /2 , Lévy-Process Driven Financial Market Models × G + MJump-Diffusion Models | | Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model where Kν (x) denotesEuropean the modified Style Options Bessel function of the third kind with index ν and Γ(x) denotes the gamma function. VarianceAs Gamma shown in Figures Density 14, 15 and 16 one can see that the distribution is very flexible.

VG density − C=1.3574; G=5.8704; M=14.2699 3

2.5

1.5

0.5

0 −1 −0.5 0 0.5 1

Figure 14: The VG density Professor Dr. R¨udigerKiesel L´evyFinance

Some distribution characteristics are summarized in the Tables 5 and 6

28 VG(C,G,M) VG(C, G, G) mean C(G M)/(MG) 0 variance C(G2 +−M 2)/(MG)2 2CG−2 skewness 2C−1/2(G3 M 3)/(G2 + M 2)3/2 0 kurtosis 3(1 + 2C−1(G−4 + M 4)/(M 2 + G2)2) 3(1 + C−1) Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Table 6: VG distributionJump-Diffusion characteristics Models in the (C,G,M) parametrization. Variance-Gamma Model General LévyModels NIG Model European Style Options When θ = 0 the distribution is symmetric. Negative values of θ result Variancein negative Gamma skewness; Density positive θ’s give positive skewness. The parameter ν primarily controls the kurtosis.

VG density − σ=0.2; ν=0.5; θ=−0.25, 0.00, 0.25 3 θ=0 θ=−0.25 2.5 θ=0.25

1.5

0.5

0 −1 −0.5 0 0.5 1

Professor Dr. R¨udigerKieselFigure 15: TheL´evyFinance VG density

In terms of the (C,G,M)-parameters this reads as follows: Under this setting, G = M gives the symmetric case, GM give rise to positive skewness. The parameter C controls the kurtosis. If we fit the VG density to the Kernel density, we obtain a very good fit (compare with Normal Fit). In Figure 17, one sees a fit on a data set of daily logreturns of the SP500 over more than 30 years. Statistical χ2-tests confirm the goodness of fit.

3.2 The VG Process Recall the deﬁnition of a standard Brownian Motion W = W ,t 0 { t ≥ } W starts at zero: W = 0. • 0 W has independent increments: the distribution of increments over non- • overlapping time intervals are stochastically independent.

29 Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options Variance Gamma Density

VG density − σ=0.2; ν=0.1, 0.5, 0.8 ; θ=0 3.5 ν=0.1 ν=0.5 3 ν=0.8

2.5

1.5

0.5

0 −1 −0.5 0 0.5 1

Figure 16: The VG density Professor Dr. RüdigerKiesel LévyFinance W has stationary increments: the distribution of an increment over a • time-interval depends only on the length of the interval; not on the exact location. W W Normal(0,s): increments are Normally distributed. • s+t − t ∼ One can define in a similar way a stochastic process based on the VG distribution. (For mathematical details and other examples see [157]). A stochastic process X = Xt,t 0 is a Variance-Gamma Process with parameters C,G,M if { ≥ } X starts at zero: X = 0. • 0 X has independent increments. • X has stationary increments. • Furthermore we have that Xs+t Xt VG(Cs,G,M), i.e. increments • are VG distributed; − ∼ It will turn out (see again [157]) that a VG process is a pure jump process. Sample paths have no diffusion component in contrast with a Brownian motion (see Figure ??.

3.3 The VG Stock Price Model Instead of modeling the stock price process as an exponential of a Brownian Motion (with drift):

S = S exp((µ σ2/2)t + σW ), S > 0, t 0 − t 0

30 Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options Variance Gamma Process

A stochastic process X is a Variance-Gamma process with parameters C, G, M if

I X starts at zero: X0 = 0,

I X has independent increments,

I X has stationary increments,

I X (t + s) − X (t) ∼ VG(Cs, G, M).

Standard Brownian Motion 1

0.8

0.6

0.4

0.2

−0.2

−0.4

−0.6

−0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

VG Process (C=20; G=40; M=50) 0.1

0.05

−0.05

−0.1

−0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 18: Brownian Motion and VG paths

Professor Dr. R¨udigerKiesel L´evyFinance Note that, most of the time we immediately will work under a risk-neutral setting (after calibrating the model to market data) and we do not have to worry about the measure change.

4 Pricing Vanillas using FFT

In this Chapter, we describe how one can price very fast and eﬃciently vanilla options using the theory of characteristic functions and Fast Fourier Trans- forms. Our aim is to develop a solid understanding of the current frameworks for pricing of vanilla derivatives using these techniques and to give readers the mathematical and practical background necessary to apply and implement the

32 I One period log-Returns are VG(CGM) distributed I Mean-correcting Martingale Measure Q

S(t) = S(0) exp{(r + ω)t + X (t)}

with 1 ω = ν−1 log 1 − σ2ν − θν . 2

I Let X = (X (t)) we a VG-Process, then

S(t) = S(0) exp{X (t)}

Professor Dr. R¨udigerKiesel L´evyFinance I Mean-correcting Martingale Measure Q

S(t) = S(0) exp{(r + ω)t + X (t)}

with 1 ω = ν−1 log 1 − σ2ν − θν . 2

I Let X = (X (t)) we a VG-Process, then

S(t) = S(0) exp{X (t)}

I One period log-Returns are VG(CGM) distributed

I Let X = (X (t)) we a VG-Process, then

S(t) = S(0) exp{X (t)}

I One period log-Returns are VG(CGM) distributed I Mean-correcting Martingale Measure Q

S(t) = S(0) exp{(r + ω)t + X (t)}

with 1 ω = ν−1 log 1 − σ2ν − θν . 2

Professor Dr. R¨udigerKiesel L´evyFinance I We know the density

Z ∞ + −rT (r−ω)T +x C(K, T ) = e fVG (x; CT , G, M) S(0)e − K dx −∞

I Diﬃcult to evaluate (Bessel function).

I The risk-neutral valuation principle implies for a European call with maturity T and strike K

+ C(K, T ) = exp(−rT )EQ[(S(T ) − K) ]

Professor Dr. RüdigerKiesel LévyFinance I Difficult to evaluate (Bessel function).

I The risk-neutral valuation principle implies for a European call with maturity T and strike K

+ C(K, T ) = exp(−rT )EQ[(S(T ) − K) ]

I We know the density

Z ∞ + −rT (r−ω)T +x C(K, T ) = e fVG (x; CT , G, M) S(0)e − K dx −∞

I The risk-neutral valuation principle implies for a European call with maturity T and strike K

+ C(K, T ) = exp(−rT )EQ[(S(T ) − K) ]

I We know the density

Z ∞ + −rT (r−ω)T +x C(K, T ) = e fVG (x; CT , G, M) S(0)e − K dx −∞

I Diﬃcult to evaluate (Bessel function).

Professor Dr. R¨udigerKiesel L´evyFinance I Suppose we know the characteristic function of sT = log ST

φ(u; T ) = EQ [exp{iusT }] = EQ [exp{iu log ST }]

I Let k = log K

I Let α be a positive integer such that the αth moment of the stock price exists

Professor Dr. R¨udigerKiesel L´evyFinance I Let k = log K

I Let α be a positive integer such that the αth moment of the stock price exists

I Suppose we know the characteristic function of sT = log ST

φ(u; T ) = EQ [exp{iusT }] = EQ [exp{iu log ST }]

I Let α be a positive integer such that the αth moment of the stock price exists

I Suppose we know the characteristic function of sT = log ST

φ(u; T ) = EQ [exp{iusT }] = EQ [exp{iu log ST }]

I Let k = log K

Professor Dr. R¨udigerKiesel L´evyFinance I We know k + C(k, T ) = exp(−rT )EQ[(S(T ) − e ) ] Z ∞ = exp(−rT ) ex − ek q(x, T )dx k

I As C(k, T ) → exp(−rT )S(0) as k → −∞ it is not square-integrable and Fourier theory may not be applied!

I Let q(x, T ) be the density function of sT = log ST , then Z ∞ φ(u; T ) = exp{iux}q(x, T )dx −∞

Professor Dr. R¨udigerKiesel L´evyFinance I As C(k, T ) → exp(−rT )S(0) as k → −∞ it is not square-integrable and Fourier theory may not be applied!

I Let q(x, T ) be the density function of sT = log ST , then Z ∞ φ(u; T ) = exp{iux}q(x, T )dx −∞

I We know k + C(k, T ) = exp(−rT )EQ[(S(T ) − e ) ] Z ∞ = exp(−rT ) ex − ek q(x, T )dx k

I Let q(x, T ) be the density function of sT = log ST , then Z ∞ φ(u; T ) = exp{iux}q(x, T )dx −∞

I We know k + C(k, T ) = exp(−rT )EQ[(S(T ) − e ) ] Z ∞ = exp(−rT ) ex − ek q(x, T )dx k

I As C(k, T ) → exp(−rT )S(0) as k → −∞ it is not square-integrable and Fourier theory may not be applied!

Professor Dr. R¨udigerKiesel L´evyFinance I So the Fourier transform ρ(v) of c(k, T ) exists.

I For some α > 0 the modiﬁed call

c(k, T ) = exp(αk)C(k, T )

is square-integrable.

I For some α > 0 the modiﬁed call

c(k, T ) = exp(αk)C(k, T )

is square-integrable.

I So the Fourier transform ρ(v) of c(k, T ) exists.

Z ∞ exp(ivk)c(k, T )dk −∞ Z ∞ Z ∞ = exp(ivk) exp(αk) exp(−rT ) ex − ek dxdk −∞ k Z ∞ Z x = e−rT q(x, T ) exp(ivk) exp(αk) ex − ek dkdx −∞ −∞ Z ∞ −rT exp((α + 1 + iv)x) = e q(x, T ) 2 2 −∞ α + α − v + i(2α + 1)v exp(−rT )φ(v − (α + 1)i, T ) = = ρ(v) α2 + α − v 2 + i(2α + 1)v

Using the inverse transform we have

C(k, T ) = exp(−αk)c(k, T )

1 Z ∞ = exp(−αk) exp(−ivk)ρ(v)dv 2π −∞ 1 Z ∞ = exp(−αk) exp(−ivk)ρ(v)dv π 0 where we used that C(k, T ) is real, so ρ(v) is odd.

Professor Dr. R¨udigerKiesel L´evyFinance I Valuation can be done fast and accurate with the Fast Fourier Transform.

I So we obtain the Carr-Madan Formula exp(−α log K) Z ∞ C(K, T ) = exp(−iv log K)ρ(v)dv π 0

I Valuation can be done fast and accurate with the Fast Fourier Transform.

Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels 4.4 Calibration Results NIG Model European Style Options Doing a global calibration for the VG model introduced in Chapter ??, we a serious improvement over the Black-Scholes case. However observe still a significant difference with real market prices as can be seen in Figure 22. The VG Optioninitial parameters Pricing we – started Calibration with where (C Results= 1, G = 5, M = 5); the final optimal parameters coming out of the calibration procedure are given by: (C = 1.3574, G = 5.8703, M = 14.2699)

SP−500 / 18−04−2002 / VG / ape = 4.67 %

180

160

140

120

100

option price 80

0 1000 1100 1200 1300 1400 1500 strike

Figure 22: Global Calibration of VG

Fitting theProfessor smile at Dr. one R¨udigerKiesel maturity is doneL´evyFinance very accurately as can be seen in Figure 23. The parameters coming out of the calibration procedure for each maturity are given in Table 7. We see a quite typical term-structure for the σ and ν parameter.

5 Monte Carlo Simulation

Next, we look at possible simulation techniques for the processes encountered so far. A L´evy process can be in general simulated based on a compound Poisson

42 Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options VG Option Pricing – Calibration Results

SP500 / 18−04−2002 / VG (T=0.184) / ape =0.24034% SP500 / 18−04−2002 / VG (T=0.436) / ape =0.21534% SP500 / 18−04−2002 / VG (T=0.692) / ape =0.34172% 90 180 180

80 160 160

140 140 70

120 120 60 100 100 50 80 80 Option price Option price Option price 40 60 60

30 40 40

20 20 20

10 0 0 1040 1060 1080 1100 1120 1140 1160 1180 950 1000 1050 1100 1150 1200 1250 950 1000 1050 1100 1150 1200 1250 1300 Strike Strike Strike

SP500 / 18−04−2002 / VG (T=0.936) / ape =0.43269% SP500 / 18−04−2002 / VG (T=1.192) / ape =0.54975% SP500 / 18−04−2002 / VG (T=1.708) / ape =0.48827% 150 200 180

180 160

160 140

140 100 120 120 100 100 80

Option price Option price 80 Option price 50 60 60

40 40

20 20

0 0 0 1000 1050 1100 1150 1200 1250 1300 1350 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 Strike Strike Strike

Figure 23: Calibration of VG maturity per maturity Professor Dr. RüdigerKiesel LévyFinance approximation. However, typically, for very specific processes like the VG other much faster techniques are available. We assume we have random number generators at hand which can provide us Normal(0, 1) and Gamma(a, b) random numbers. Throughout we v allways { n} denotes a Normal(0, 1) random number; gn a Gamma random numbers. A good reference book is [139]. { }

5.1 Brownian Motion

Recall that standard Brownian motion W = Wt,t 0 has Normal distributed independent increments. We discretize time{ by taking≥ } time steps of size ∆t, which we assume to be very small. We simulate (by the Euler scheme) the value of the Brownian motion at the time points n∆t, n = 0, 1,... : { }

W0 = 0, Wn∆t = W(n−1)∆t + √∆tvn.

This leads to the following typical picture of standard Brownian motion The Matlab code for this is very simple: T=1; N=250; dt=T/N; tt=[0:dt:T];

43 I Input to the calibration algorithm

I characteristic function of the underlying stochastic dynamics, I initial guess of the parameters, I market prices

I By calling many times the pricing algorithm the optimisation procedure searches the parameter space and minimizes the error between model prices and market prices.

I Typically a nonlinear unconstrained optimisation problem. Often the Nelder-Mead Simplex algorithm is used (implemented in Matlab).

I A calibration procedure looks for the optimal parameter set such that the model prices match as best as possible the observed market prices

Professor Dr. R¨udigerKiesel L´evyFinance I By calling many times the pricing algorithm the optimisation procedure searches the parameter space and minimizes the error between model prices and market prices.

I Typically a nonlinear unconstrained optimisation problem. Often the Nelder-Mead Simplex algorithm is used (implemented in Matlab).

I A calibration procedure looks for the optimal parameter set such that the model prices match as best as possible the observed market prices I Input to the calibration algorithm

I characteristic function of the underlying stochastic dynamics, I initial guess of the parameters, I market prices

Professor Dr. R¨udigerKiesel L´evyFinance I Typically a nonlinear unconstrained optimisation problem. Often the Nelder-Mead Simplex algorithm is used (implemented in Matlab).

I A calibration procedure looks for the optimal parameter set such that the model prices match as best as possible the observed market prices I Input to the calibration algorithm

I characteristic function of the underlying stochastic dynamics, I initial guess of the parameters, I market prices

I By calling many times the pricing algorithm the optimisation procedure searches the parameter space and minimizes the error between model prices and market prices.

I A calibration procedure looks for the optimal parameter set such that the model prices match as best as possible the observed market prices I Input to the calibration algorithm

I characteristic function of the underlying stochastic dynamics, I initial guess of the parameters, I market prices

I By calling many times the pricing algorithm the optimisation procedure searches the parameter space and minimizes the error between model prices and market prices.

I Typically a nonlinear unconstrained optimisation problem. Often the Nelder-Mead Simplex algorithm is used (implemented in Matlab).

This is the case λ = −1/2 of the generalized hyperbolic distribution. The density is α n p o d (x) = exp δ α2 − β2 + β(x − µ) NIG π

q x−µ 2 K1 αδ 1 + δ × q . x−µ 2 1 + δ

Here, 1 Z ∞ 1 1 K1(z) = exp − z x + dx, z > 0, 2 0 2 x is the modiﬁed Bessel function of the third kind with index 1. The parameter α relates to the steepness, β to the asymmetry, δ to the scale and µ to the location of the density. For β = 0, the NIG-distribution is symmetric.

The characteristic function is q p 2 2 2 2 φNIG(u) = exp δ α − β − δ α − (β + iu) + iuµ

We will use the corresponding NIG L´evyprocess X in our asset model and assume that annual log returns are NIG(α, β, δ, µ) distributed. Then, Xt is NIG(α, β, δt, µt) distributed.

The density of a Meixner(α, β, δ, µ) distribution is

2δ 2 (2 cos(β/2)) i(x − µ) d (x) = exp(β(x − µ)/α) Γ δ + Meix 2απΓ(2δ) α (2) where α > 0, −π < β < π, δ > 0. For β = 0, the Meixner-distribution is symmetric with mode at µ. Its characteristic function is

cos(β/2) 2δ φ (u) = · exp{iuµ} Meixner cosh((αu − iβ)/2)

Meixner(α, β, δ, µ) NIG(α, β, δ, µ) δβ E(X ) µ + αδ tan(β/2) µ + pα2 − β2 α2δ α2δ Var(X ) 2 cos(β/2) (α2 − β2)3/2 E[(X −E(X ))3] 3β sin(β/2)p2/δ Var(X )3/2 q α δpα2 − β2 2 2 E[(X −E(X ))4] 2 − cos(β) α + 4β 2 3 + 3 + 3 Var(X ) δ δα2pα2 − β2 Table: Mean, Variance, Skewness and Kurtosis for Meixner and NIG.

The following assumptions are necessary: I X1 possesses a moment-generating function

M(z, 1) = E(exp(zX1)) = φX1 (−iz) on some open interval (a, b) with b − a > 1. ∗ I There exists a real number θ ∈ (a, b − 1) such that M(θ∗, 1) = M(θ∗ + 1, 1).

We call a change of a measure P to a locally equivalent measure Q an Esscher transform if the measures are related by

dQ θ exp(θXt ) = Z = . d t M(θ, t) P Ft For a properly chosen θ, the discounted asset price process −rt e S(t) is a martingale under Q. Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options European Call Pricing Formula - NIG

Now suppose that under the real-world measure the annual returns

ξt = Xt − Xt−1

follow a NIG(α, β, δ, m) distribution. If (r − µ)2 < δ2(2α − 1), we can perform an Esscher transform with parameter s 1 r − µ 4α2 1 θ∗ = −b − + − . 2 2 (r − µ)2 + δ2 δ2

Under the Esscher measure, annual returns ξi are NIG(α, β + θ∗, δ, m) distributed.

Assume ξt is Meixner(α, β, δ, µ) under the real-world measure. ∗ Under the Esscher measure, ξt is Meixner(α, β + αθ , δ, µ) with

α ((µ−r)/(2δ)) !! ∗ 1 − cos( 2 ) + e θ = − · β + 2 arctan α . α sin( 2 )

Parameter Fit to S&P500 option prices.

Meixner NIG BM α 0.3977 6.1882 – β -1.4940 -3.8941 – δ 0.3462 0.1622 – std deviation 0.2255 0.2363 0.1812 skewness -1.6331 -2.1374 0 kurtosis 8.5554 12.9374 3 Table: Parameters according to Schoutens (2003) and corresponding risk-neutral moments.

Meixner densities: risk neutral Std = 0.226, Skew = −1.633, Kurt = 8.555 NIG densities: risk neutral Std = 0.236, Skew = −2.137, Kurt = 12.937 3.5 risk neutral Meixner Esscher transformed 3 symmetric Meixner risk neutral NIG Esscher transformed 2.5 symmetric NIG risk neutral BM Std = 0.1812 2

1.5

0.5

0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 logreturn − r

Figure: Densities of calibrated distributions.

Professor Dr. RüdigerKiesel LévyFinance Stochastic Calculus for LévyProcesses Equivalent Martingale Measure Lévy-Process Driven Financial Market Models Jump-Diffusion Models Jump-Diffusion Models Variance-Gamma Model General LévyModels NIG Model European Style Options Barndorff-Nielsen, O.E., 1998, Processes of normal inverse Gaussian type, Finance and Stochastics 2, 41–68. Carr, P., E. Chang, and D.B. Madan, 1998, The Variance-Gamma process and option pricing, European Finance Review 2, 79–105. Carr, P., H. Geman, D.B. Madan, and M. Yor, 2002, The fine structure of asset returns: An empirical investigation., Journal of Business 75, 305–332. Eberlein, E., 2001, Applications of generalized hyperbolic Lévy motions to finance, in O.E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds.: Lévyprocesses: Theory and Applications (Birkhäuser Verlag, Boston, ). Eberlein, E., and U. Keller, 1995, Hyperbolic distributions in finance, Bernoulli 1, 281–299.

Professor Dr. R¨udigerKiesel L´evyFinance