Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Calculus for L´evy Processes L´evy-Process Driven Financial Market Models Jump-Diffusion Models Merton-Model Kou-Model General L´evy 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. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Integral for L´evyProcesses Let (Xt ) be a L´evy process with L´evy-Khintchine triplet (α, σ, ν(dx)). By the L´evy-It´odecomposition 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! Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Canonical Decomposition 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<s≤t and Z t Z Z t Z Z X X E xµ (ds, dx) = xν (ds, dx) = t xν(dx). 0 R 0 R R Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Girsanov Theorem 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. Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Girsanov Theorem 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. Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Girsanov Theorem 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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; Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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; I in the general case X is a semi-martingale. Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options ItˆoFormula for L´evyProcesses 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<s≤t Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Exponential 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<s≤t 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<s≤t 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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<s≤t Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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<s≤t 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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 L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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 Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels 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)}.