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Theory of the Jump-Diffusion Processes

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Theory of the Jump-Diffusion Processes OUTLINE OF THE WHOLE LECTURES. PART 1 : JUMP-DIFFUSION PROCESSES. I Theory and simulation. I Stochastic calculus with jump-diffusion processes. I Applications in finance. PART 2 : GENERAL LÉVY PROCESSES. I Theory. I Simulation. I Stochastic calculus with semimartingales. I Applications in finance. A. Popier (ENSTA) Jump-diffusion processes. January 2020 1 / 34 INTRODUCTION. CONTINUOUS TIME MODELS WITH CONTINUOUS TRAJECTORIES. 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. CONTINUOUS TIME MODELS WITH DISCONTINUOUS TRAJECTORIES. 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. CONTINUOUS TIME MODELS WITH CONTINUOUS TRAJECTORIES. 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. CONTINUOUS TIME MODELS WITH DISCONTINUOUS TRAJECTORIES. 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 " > 0, lim P(jXt+h − Xt j ≥ ") = 0 h!0 5 there exists a subset Ω0 s.t. P(Ω0) = 1 and for every ! 2 Ω0, t 7! Xt (!) 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 (!) 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 fFt g is a larger filtration than (Ft ⊂ Ft ) and if Xt − Xs is independent of Fs, then fXt ; 0 ≤ t < +1g is a Lévy process under the large filtration. REMARKS ON THE HYPOTHESES : 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. REMARKS ON THE DEFINITIONS : X If Ft = Ft , the two definitions are equivalent. X If fFt g is a larger filtration than (Ft ⊂ Ft ) and if Xt − Xs is independent of Fs, then fXt ; 0 ≤ t < +1g is a Lévy process under the large filtration. REMARKS ON THE HYPOTHESES : 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 ENLARGED FILTRATION. For a process X = fXt ; t ≥ 0g, we define N1 = N the set of P-negligible events. X For any 0 ≤ t ≤ 1, augmented filtration : Ft = σ(Ft [N ). THEOREM Let X = fXt ; t ≥ 0g be a Lévy process. Then the augmented filtration fFt g is right-continuous. With respect to the enlarged filtration, fXt ; t ≥ 0g is still a Lévy process. A. Popier (ENSTA) Jump-diffusion processes. January 2020 6 / 34 CHARACTERISTIC FUNCTIONS. PROPOSITION d Let (Xt )t≥0 be a Lévy process in R . Then there exists a function : Rd ! R called characteristic exponent of X s.t. : d ihz;X i t (z) 8z 2 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 BROWNIAN MOTION. 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(−tjzj =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 POISSON PROCESS. 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)n2N 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 POISSON PROCESS. A. Popier (ENSTA) Jump-diffusion processes. January 2020 13 / 34 JUMP DISTRIBUTION. For any interval I, we denote by N(I) the number of jumps of Xt , t 2 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 POISSON PROCESSES. We consider a Poisson process (Pt )t≥0 with intensity λ and jump times d Tn, and a sequence (Yn)n2N∗ of R -valued r.v. s.t. 1 Yn are i.i.d. with distribution measure π ; 2 (Pt )t≥0 and (Yn)n2N∗ are independent. Define P +1 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 POISSON PROCESSES. PROPOSITION The process (Xt )t≥0 is a Lévy process, with piecewise constant trajectories and characteristic function : Z d ihz;X i ihz;xi 8z 2 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 2 B(Rd ): ν is called the Lévy measure of the compound Poisson process. Moreover ν(A) = E [#ft 2 [0; 1]; ∆Xt 6= 0; ∆Xt 2 Ag] : A. Popier (ENSTA) Jump-diffusion processes. January 2020 17 / 34 COMPOUND POISSON PROCESSES. DEFINITION The law µ of X1 is called compound Poisson distribution and has a Z characteristic function given by : µ^(z) = exp λ (eihz;xi − 1)π(dx) .
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