Theory of the Jump-Diffusion Processes

Home , Additive process, Jump diffusion

OUTLINEOFTHEWHOLELECTURES.

PART 1 : JUMP-DIFFUSION PROCESSES.

I Theory and simulation.

I Stochastic calculus with jump-diffusion processes.

I Applications in ﬁnance.

PART 2 : GENERAL LÉVYPROCESSES.

I Theory.

I Simulation.

I Stochastic calculus with semimartingales.

I Applications in ﬁnance.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 1 / 34 INTRODUCTION.

CONTINUOUSTIMEMODELSWITHCONTINUOUSTRAJECTORIES. Black-Scholes model. Scale invariance of the Brownian motion. Local volatility models. Possible perfect hedging. Stochastic volatility models. Difﬁculty to obtain heavy tails, no large sudden moves.

CONTINUOUSTIMEMODELSWITHDISCONTINUOUSTRAJECTORIES.

I Market crash.

I Credit risk.

I High-frequency trading. Aït-Sahalia & Jacod, High-frequency Financial Econometrics.

I Insurance and ruin theory.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 2 / 34 INTRODUCTION.

CONTINUOUSTIMEMODELSWITHDISCONTINUOUSTRAJECTORIES.

I Market crash.

I Credit risk.

I High-frequency trading. Aït-Sahalia & Jacod, High-frequency Financial Econometrics.

I Insurance and ruin theory.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 2 / 34 TWO DIFFERENT CLASSES.

1 JUMP-DIFFUSION MODELS. Prices : diffusion process, with jumps at randon times. Jumps : rare events −→ cracks or large losses.

1 Advantages:

I price structure : easy to understand, to describe and to simulate ; I then efﬁcient Monte Carlo methods to compute path-depend prices. I Very performant to interpolate the implicit volatility smiles.

2 Inconvenients:

I unknown closed formula for the densities, I statistical estimation or moments/quantiles computations : difﬁcult to realize.

2 INFINITE ACTIVITY MODELS (general Lévy processes).

A. Popier (ENSTA) Jump-diffusion processes. January 2020 3 / 34 TWO DIFFERENT CLASSES.

1 JUMP-DIFFUSION MODELS. 2 INFINITE ACTIVITY MODELS (general Lévy processes). Models with a inﬁnite number of jumps during any time period. Unnecessary Brownian component.

1 Advantages:

I give a more realistic description of the prices at different time scales. I often obtained as subordinator of a Brownian motion (time change), I hence closed formulas or more tractable than for the jump-diffusion models.

2 Inconvenients:

I often more complicated to simulate. I Price structure less intuitive.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 3 / 34 DEFINITIONS.

DEFINITION d A stochastic process (Xt )t≥0 deﬁned on (Ω, F, P), with values in R , is a Lévy process if

1 X0 = 0 a.s. 2 its increments are independent : for every increasing sequence

t0,..., tn, the r.v. Xt0 , Xt1 − Xt0 ,..., Xtn − Xtn−1 are independent ; 3 its increments are stationnary : the law of Xt+h − Xt does not depend on t ; 4 X satisﬁes the property called stochastic continuity : for any ε > 0, lim P(|Xt+h − Xt | ≥ ε) = 0 h→0 5 there exists a subset Ω0 s.t. P(Ω0) = 1 and for every ω ∈ Ω0, t 7→ Xt (ω) is RCLL.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 4 / 34 DEFINITIONS.

If a ﬁltration (Ft )t≥0 is already given on (Ω, F, P)

DEFINITION d A stochastic process (Xt )t≥0 deﬁned on (Ω, F, P), with values in R , is a Lévy process if

1 X0 = 0 a.s.

2 its increments are independent : for any s ≤ t, the r.v. Xt − Xs is independent of Fs ; 3 its increments are stationnary; 4 X satisﬁes the property called stochastic continuity;

5 a.s. t 7→ Xt (ω) is RCLL.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 4 / 34 REMARKS.

REMARKS ON THE DEFINITIONS : X If Ft = Ft , the two definitions are equivalent. X If {Ft } is a larger filtration than (Ft ⊂ Ft ) and if Xt − Xs is independent of Fs, then {Xt ; 0 ≤ t < +∞} is a Lévy process under the large filtration.

REMARKSONTHEHYPOTHESES : If we remove Assumption 5, we speak about Lévy process in law. If we remove Assumption 3, we obtain an additive process. Dropping Assumptions 3 and 5, we have an additive process in law.

THEOREM A Lévy process (or an additive process) in law has a RCLL modiﬁcation. We can also prove that 2, 3 and 5 imply 4. A. Popier (ENSTA) Jump-diffusion processes. January 2020 5 / 34 REMARKS.

THEOREM A Lévy process (or an additive process) in law has a RCLL modiﬁcation. We can also prove that 2, 3 and 5 imply 4. A. Popier (ENSTA) Jump-diffusion processes. January 2020 5 / 34 ENLARGED FILTRATION.

For a process X = {Xt ; t ≥ 0}, we deﬁne N∞ = N the set of P-negligible events. X For any 0 ≤ t ≤ ∞, augmented ﬁltration : Ft = σ(Ft ∪ N ).

THEOREM

Let X = {Xt ; t ≥ 0} be a Lévy process. Then

the augmented ﬁltration {Ft } is right-continuous.

With respect to the enlarged ﬁltration, {Xt , t ≥ 0} is still a Lévy process.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 6 / 34 CHARACTERISTICFUNCTIONS.

PROPOSITION d Let (Xt )t≥0 be a Lévy process in R . Then there exists a function ψ : Rd → R called characteristic exponent of X s.t. :

d ihz,X i tψ(z) ∀z ∈ R , E e t = e .

A. Popier (ENSTA) Jump-diffusion processes. January 2020 7 / 34 THEORY OF THE JUMP-DIFFUSION PROCESSES.

Alexandre Popier

ENSTA, Palaiseau

January 2020

A. Popier (ENSTA) Jump-diffusion processes. January 2020 8 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 9 / 34 BROWNIANMOTION.

A Brownian motion is a Lévy process satisfying 1 for every t > 0, Xt is Gaussian with mean vector zero and covariance matrix t Id ; 2 the process X has continuous sample paths a.s. Characteristic function : ihz,B i 2 E(e t ) = exp(−t|z| /2).

A. Popier (ENSTA) Jump-diffusion processes. January 2020 10 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 11 / 34 POISSONPROCESS.

DEFINITION

A stochastic process (Xt )t≥0, with values in R, is a Poisson process with intensity λ > 0 if it is a Lévy process s.t. for every t > 0,Xt has a Poisson law with parameter λt.

PROPOSITION (CONSTRUCTION)

If (Tn)n∈N is a random walk on R s.t. for every n ≥ 1,Tn − Tn−1 is exponentially distributed with parameter λ (with T0 = 0), then the process (Xt )t≥0 deﬁned by

Xt = n ⇐⇒ Tn ≤ t < Tn+1

is a Poisson process with intensity λ.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 12 / 34 POISSONPROCESS.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 13 / 34 JUMPDISTRIBUTION.

For any interval I, we denote by N(I) the number of jumps of Xt , t ∈ I.

PROPOSITION

For 0 < s < t and n ≥ 1, the conditional law of Xs knowing Xt = n is binomial with parameters n and s/t. For 0 = t0 < t1 < . . . < tk = t and Ij =]tj−1, tj ], the conditional law of (N(I1),..., N(Ik )) knowing Xt = n is multinomial with parameters n, (t1 − t0)/t,..., (tk − tk−1)/t.

Ti : jump times of the process.

PROPOSITION

Let n ≥ 1 and t > 0. The conditional law of T1,..., Tn knowing Xt = n coincides with the law of the order statistics U(1),..., u(n) of n independent variables, uniformly distributed on [0, t].

A. Popier (ENSTA) Jump-diffusion processes. January 2020 14 / 34 COMPOUND POISSONPROCESSES.

We consider a Poisson process (Pt )t≥0 with intensity λ and jump times d Tn, and a sequence (Yn)n∈N∗ of R -valued r.v. s.t. 1 Yn are i.i.d. with distribution measure π ;

2 (Pt )t≥0 and (Yn)n∈N∗ are independent. Deﬁne P +∞ Xt X Xt = Yn = Yn1[0,t](Tn). n=1 n=1

DEFINITION

The process (Xt )t≥0 is a compound Poisson processes with intensity λ and jump distribution π.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 15 / 34 MERTON MODEL.

Compound Poisson processes with Gaussian jumps.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 16 / 34 COMPOUND POISSONPROCESSES.

PROPOSITION

The process (Xt )t≥0 is a Lévy process, with piecewise constant trajectories and characteristic function : Z d ihz,X i ihz,xi ∀z ∈ R , E e t = exp tλ (e − 1)π(dx) Rd Z = exp t (eihz,xi − 1)ν(dx) ; Rd

DEFINITION ν is a ﬁnite measure deﬁned on Rd by : ν(A) = λπ(A), A ∈ B(Rd ). ν is called the Lévy measure of the compound Poisson process. Moreover

ν(A) = E [#{t ∈ [0, 1], ∆Xt 6= 0, ∆Xt ∈ A}] .

A. Popier (ENSTA) Jump-diffusion processes. January 2020 17 / 34 COMPOUND POISSONPROCESSES.

DEFINITION

The law µ of X1 is called compound Poisson distribution and has a Z characteristic function given by : µˆ(z) = exp λ (eihz,xi − 1)π(dx) . Rd

PROPOSITION Let X be a compound Poisson process and A and B two disjointed subsets of Rd . Then : X X Yt = ∆Xs1∆Xs∈A and Zt = ∆Xs1∆Xs∈B s≤t s≤t

are two independent compound Poisson processes.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 17 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 18 / 34 DEFINITION.

DEFINITION A jump-diffusion process X is the sum of a Brownian motion and of a independent compound Poisson process. Therefore a jump-diffusion process is a Lévy process.

In other words

I a k-dimensional Brownian motion (Wt )t≥0, a d × k matrix A, I a d-dimensional vector γ,

I a Poisson process (Pt )t≥0 with intensity λ and jump times Tn, and d a sequence (Yn)n∈N∗ of R -valued r.v. such that

1 Yn are i.i.d. with distribution measure π ;

2 (Wt )t≥0, (Pt )t≥0 and (Yn)n∈N∗ are independent.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 19 / 34 DEFINITION.

In other words

I a k-dimensional Brownian motion (Wt )t≥0, a d × k matrix A, I a d-dimensional vector γ,

I a Poisson process (Pt )t≥0 with intensity λ and jump times Tn, and d a sequence (Yn)n∈N∗ of R -valued r.v. such that

1 Yn are i.i.d. with distribution measure π ;

2 (Wt )t≥0, (Pt )t≥0 and (Yn)n∈N∗ are independent.

P +∞ Xt X Xt = AWt + γt + Yn = AWt + γt + Yn1[0,t](Tn). n=1 n=1

A. Popier (ENSTA) Jump-diffusion processes. January 2020 19 / 34 DEFINITION.

X Xt = AWt + γt + ∆Xs. 0

CHARACTERISTICEXPONENT : for any z ∈ Rd : Z 1 ∗ ihz,xi ψX (z) = − hz, AA zi + ihz, γi + λ (e − 1)π(dx) 2 Rd 1 Z = − hz, AA∗zi + ihz, γi + (eihz,xi − 1)ν(dx). 2 Rd

A∗ is the transpose matrix of A.

CHARACTERISTICTRIPLE : (A, ν, γ).

A. Popier (ENSTA) Jump-diffusion processes. January 2020 19 / 34 TWOEXAMPLES.

Jump-diffusion process with Gaussian jumps (used in the Merton model).

A. Popier (ENSTA) Jump-diffusion processes. January 2020 20 / 34 TWOEXAMPLES.

Kou model where the jump sizes are given by a non symmetric Laplace distribution.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 20 / 34 SIMULATION.

For the Brownian part Xt = σWt + γt :

1 split the interval [0, t] by a grid t0 = 0 < t1 < . . . < tn = t,

2 simulate n standard Gaussian r.v. Zi , p 3 deﬁne ∆Xi = σ ti − ti−1Zi + b(ti − ti−1), i X 4 put Xi = ∆Xi . k=1 The simulation is exact on the grid in the sense that Xi has the same

law as Xti . Between Xi and Xi+1, one can used a linear interpolation.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 21 / 34 SIMULATION.

For the Brownian part Xt = σWt + γt : 1 split the interval [0, t] by a grid t0 = 0 < t1 < . . . < tn = t, 2 simulate n standard Gaussian r.v. Zi , p 3 deﬁne ∆Xi = σ ti − ti−1Zi + b(ti − ti−1), i X 4 put Xi = ∆Xi . k=1 To simulate the compound Poisson part on the interval [0, T ], the algorithm is : 1 simulate a Poisson r.v. N with parameter λT , 2 simulate N independent r.v. Ui with uniform law on [0, T ], 3 simulate the jumps : N independent r.v. Yi with distribution ν(dx)/λ, N X 4 put Xt = 1Ui

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 22 / 34 DEFINITION.

X Xt = AWt + γt + ∆Xs. 0

W : Brownian motion and A matrix. γ : vector. ν : ﬁnite measure on Rd . CHARACTERISTICEXPONENT : for any z ∈ Rd : Z 1 ∗ ihz,xi ψX (z) = − hz, AA zi + ihz, γi + (e − 1)ν(dx). 2 Rd

CHARACTERISTICTRIPLE : (A, ν, γ).

A. Popier (ENSTA) Jump-diffusion processes. January 2020 23 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 24 / 34 CONTINUITY OF THE LAW.

LEMMA −λt If X is a compound Poisson process, then P(Xt = 0) = e .

THEOREM Let X be a jump-diffusion process with triple (A, ν, γ). Equivalence :

1 P(Xt ) is continuous for every t > 0, 2 P(Xt ) is continuous for one t > 0, 3 A 6= 0.

COROLLARY Equivalence between :

1 P(Xt ) is discrete for every t > 0, 2 P(Xt ) is discrete for one t > 0, 3 A = 0 and ν discrete.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 25 / 34 EXISTENCEOFADENSITY.

PROPOSITION Let X be a d-dimensional jump-diffusion process with triple (A, ν, γ) with A of rank d. Then the law of Xt , t > 0 is absolutely continuous.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 26 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 27 / 34 SUBMULTIPLICATIVE FUNCTIONS.

DEFINITION A function on Rd is under-multiplicative if it is non-negative and if there exists a constant a > 0 s.t.

d 2 ∀(x, y) ∈ (R ) , g(x + y) ≤ ag(x)g(y).

LEMMA

1 The product of two submultiplicative functions is submultiplicative. α 2 If g is submultiplicative on Rd , then so is g(cx + γ) with c ∈ R, γ ∈ Rd and α > 0. 3 Let 0 < β ≤ 1. Then the following functions are submultiplicative.

|x| ∨ 1 = max(|x|, 1), exp(|x|β), ln(|x| ∨ e), ln ln(|x| ∨ ee).

| | 4 If g is submultiplicative and locally bounded, then g(x) ≤ bec x .

A. Popier (ENSTA) Jump-diffusion processes. January 2020 28 / 34 MOMENTSOFAJUMP-DIFFUSION PROCESS.

THEOREME Let g be a under-multiplicative function, locally bounded on Rd . Then equivalence between

there exists t > 0 s.t. E(g(Xt )) < +∞ for any t > 0, E(g(Xt )) < +∞. Moreover E(g(Xt )) < +∞ if and only if E(g(Y1)) < +∞.

n n Hence E(|Xt | ) < ∞ if and only if E(|Y1| ) < ∞. In particular

E(Xt ) = t (γ + λE(Y1)) ,

and ∗ Var Xt = t (A A + λVar (Y1)) .

A. Popier (ENSTA) Jump-diffusion processes. January 2020 29 / 34 EXPONENTIALMOMENTS.

THEOREM Let X be a jump-diffusion process with triple (A, ν, γ). Let Z d hc,xi hc,Y i C = c ∈ R , e ν(dx) < +∞ ⇔ Ee 1 < +∞ . Rd

1 C is convex and contains 0. h , i 2 c ∈ C if and only if E(e c Xt ) < +∞ for some t > 0 or equivalently for any t > 0. 3 If w ∈ Cd is s.t. Rew ∈ C, then 1 Z ψ(w) = hw, Awi + hγ, wi + (ehw,xi − 1)ν(dx) 2 Rd

h , i h , i ψ( ) has a sense, E(e w Xt ) < +∞ and E(e c Xt ) = et w .

A. Popier (ENSTA) Jump-diffusion processes. January 2020 30 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 31 / 34 PROCESSESWITHINDEPENDENTINCREMENTS.

PROPOSITION Let X be a process with independent increments. Then ! eihu,Xt i 1 for every u ∈ Rd , is a martingale. (eihu,Xt i) E t≥0 ! hu,Xt i h , i e 2 If E(e u Xt ) < ∞, ∀t ≥ 0, then is a martingale. (ehu,Xt i) E t≥0 3 If E(|Xt |) < ∞, ∀t ≥ 0, then Mt = Xt − E(Xt ) is a martingale (with independent increments). 2 2 4 In dimension 1, if Var (Xt ) < +∞, ∀t ≥ 0, then (Mt ) − E((Mt ) ) is a martingale.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 32 / 34 OUTLINE

1 JUMP-DIFFUSION PROCESSES Compound Poisson process Jump-diffusion processes

2 PROPERTIES OF JUMP-DIFFUSION PROCESSES Densities of the jump-diffusion processes. Moments Jump-diffusion processes and martingales. Poisson and Lévy measures

A. Popier (ENSTA) Jump-diffusion processes. January 2020 33 / 34 POISSONMEASURES.

DEFINITION Let (Ω, F, P) by a probability space, E ⊂ Rk , and ρ a measure on (E, E).A Poisson random measure on E with intensity ρ is a function with values in N : M :Ω × E → N s.t. (ω, A) 7→ M(ω, A),

1 ∀ω ∈ Ω,M(ω, .) Radon measure on E : ∀A ∈ E measurable and bounded, M(A) < +∞ r.v. with values in N ; 2 ∀A ∈ E,M(., A) = M(A) Poisson r.v. with parameter ρ(A) ;

3 for any A1,..., An disjointed sets, the r.v. M(A1),..., M(An) are independents.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 34 / 34 POISSONMEASURES.

For a compound Poisson process X with intensity λ and jump distribution π, (random) measure on [0, +∞[×Rd \{0} :

d ∀B ⊂ E = [0, +∞[×R \{0}, JX (ω, B) = #{(t, Xt (ω) − Xt− (ω)) ∈ B}.

PROPOSITION (POISSONMEASURE) d JX :Ω × E → N is a Poisson measure on E = [0, +∞[×R \{0} with intensity ρ(dt × dx) = dtν(dx) = λdtπ(dx). We obtain

ν(A) = E(JX ([0, 1] × A)) = E [#{t ∈ [0, 1], ∆Xt 6= 0, ∆Xt ∈ A}] .

Remember that ν is the Lévy measure of X.

A. Popier (ENSTA) Jump-diffusion processes. January 2020 34 / 34 POISSONMEASURES.

For a compound Poisson process X with intensity λ and jump distribution π, (random) measure on [0, +∞[×Rd \{0} :

d ∀B ⊂ E = [0, +∞[×R \{0}, JX (ω, B) = #{(t, Xt (ω) − Xt− (ω)) ∈ B}.

Moreover we have Z E(JX ([0, t] × A)) = tν(A) = tλ π(dx) A and X Z Xt = ∆Xs = xJX (ds × dx). d 0

A. Popier (ENSTA) Jump-diffusion processes. January 2020 34 / 34