Numerical Solution of Stochastic Differential Equations with Jumps - PowerPoint PPT Presentation

Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden, P.E. & Pl, E.: Numerical Solutions of Stochastic Differential Equations Springer, Applications of Mathematics 23 (1992,1995,1999). Pl, E. & Heath, D.: A Benchmark Approach to Quantitative Finance, Springer Finance (2006). Bruti-Liberati, N. & Pl, E.: Numerical Solutions of Stochastic Differential Equations with Jumps Springer, Applications of Mathematics (2008).

1 23 standing of the nature of stochastic volatility. important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models. The first part of the book describes the necessary tools from probability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part is devoted to financial modeling under the bench- mark approach. Various quantitative methods for the fair pricing and hedging of derivatives are explained. The general framework is used to provide an under- The book is intended for a wide audience that includes quantitative analysts, equivalent risk neutral pricing measure is not required. Instead, it leads to postgraduate students and practitioners in finance, economics and insurance. It aims to be a self-contained, accessible but mathematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quanti- tative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability. ������ ������ � pricing formulae with respect to the real world probability measure. This yields integrated risk management and insurance risk modeling. The existence of an Springer Finance Eckhard Platen A Benchmark Approach to Quantitative Finance 1 A Benchmark Approach to Quantitative Finance S F Platen · Heath David Heath It allows for a unified treatment of portfolio optimization, derivative pricing, Dieser Farbausdruck/pdf-file kann nur annähernd das endgültige Druckergebnis wiedergeben ! 63575 15.5.06 designandproduction GmbH – Bender Springer Finance E. Platen · D. Heath The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk neutral pricing theory. › springer.com ISBN 3-540-26212-1

Jump-Diffusion Multi-Factor Models Bj¨ ork, Kabanov & Runggaldier (1997) • continuous time • Markovian • explicit transition densities in special cases • benchmark framework • discrete time approximations • suitable for simulation • Markov chain approximations Eckhard Platen Bressanone07 1

Pathwise Approximations: • scenario simulation of entire markets • testing statistical techniques on simulated trajectories • filtering hidden state variables Pl. & Runggaldier (2005, 2007) • hedge simulation • dynamic financial analysis • extreme value simulation • stress testing ⇒ = higher order strong schemes predictor-corrector methods Eckhard Platen Bressanone07 2

Probability Approximations: • derivative prices • sensitivities • expected utilities • portfolio selection • risk measures • long term risk management ⇒ = Monte Carlo simulation, higher order weak schemes, predictor-corrector variance reduction, Quasi Monte Carlo, or Markov chain approximations, lattice methods Eckhard Platen Bressanone07 3

Essential Requirements: • parsimonious models • respect no-arbitrage in discrete time approximation • numerically stable methods • efficient methods for high-dimensional models • higher order schemes, predictor-corrector Eckhard Platen Bressanone07 4

Continuous and Event Driven Risk W k , k ∈ { 1 , 2 , . . . , m } • Wiener processes p k • counting processes intensity h k jump martingale q k � − 1 dW m + k = dq k dp k t − h k h k � � � t = 2 t dt t t k ∈ { 1 , 2 , . . . , d − m } t , . . . , q d − m W t = ( W 1 t , . . . , W m t , q 1 ) ⊤ t Eckhard Platen Bressanone07 5

Primary Security Accounts � � d dS j t = S j a j � b j,k dW k t dt + t t − t k =1 Assumption 1 � b j,k h k − m ≥ − t t k ∈ { m + 1 , . . . , d } . Assumption 2 Generalized volatility matrix b t = [ b j,k ] d j,k =1 invertible . t Eckhard Platen Bressanone07 6

• market price of risk t ) ⊤ = b − 1 θ t = ( θ 1 t , . . . , θ d [ a t − r t 1] t • primary security account � d � dS j t = S j � b j,k ( θ k t dt + dW k r t dt + t ) t − t k =1 • portfolio d � δ j t dS j dS δ t = t j =0 Eckhard Platen Bressanone07 7

• fraction S j π j δ,t = δ j t t S δ t • portfolio � � dS δ t = S δ r t dt + π ⊤ δ,t − b t ( θ t dt + dW t ) t − Eckhard Platen Bressanone07 8

Assumption 3 � h k − m > θ k t t • generalized GOP volatility  θ k k ∈ { 1 , 2 , . . . , m } for  t  c k t = θ k k ∈ { m + 1 , . . . , d } t for  ) − 1  t ( h k − m 1 − θ k 2 t • GOP fractions � ⊤ δ ∗ ,t ) ⊤ = � t b − 1 π δ ∗ ,t = ( π 1 δ ∗ ,t , . . . , π d c ⊤ t Eckhard Platen Bressanone07 9

• Growth Optimal Portfolio � � = S δ ∗ dS δ ∗ r t dt + c ⊤ t ( θ t dt + dW t ) t − t • optimal growth rate m = r t + 1 � g δ ∗ ( θ k t ) 2 t 2 k =1     d θ k θ k � h k − m t  + t −  ln  1 + t  � � h k − m h k − m − θ k k = m +1 t t t Eckhard Platen Bressanone07 10

• benchmarked portfolio t = S δ ˆ S δ t S δ ∗ t S δ is an Any nonnegative benchmarked portfolio ˆ Theorem 4 ( A , P ) -supermartingale. ⇒ = no strong arbitrage but there may exist: free lunch with vanishing risk (Delbaen & Schachermayer (2006)) free snacks or cheap thrills (Loewenstein & Willard (2000)) Eckhard Platen Bressanone07 11

Multi-Factor Model model mainly: • benchmarked primary security accounts t = S j t S j ˆ S δ ∗ t j ∈ { 0 , 1 , . . . , d } supermartingales, often SDE driftless, local martingales, sometimes martingales Eckhard Platen Bressanone07 12

savings account �� t � S 0 t = exp r s ds 0 ⇒ = GOP = S 0 t S δ ∗ t ˆ S 0 t ⇒ = stock S j t = ˆ S j t S δ ∗ t additionally dividend rates foreign interest rates Eckhard Platen Bressanone07 13

Example Black-Scholes Type Market d d ˆ S j t = − ˆ S j � σ j,k dW k t t − t k =1 h j t , σ j,k , r t t Eckhard Platen Bressanone07 14

Examples • Merton jump-diffusion model dX t = X t − ( µ dt + σ dW t + dp t ) , ⇓ N t X t = X 0 e ( µ − 1 2 σ 2 ) t + σW t � ξ i i =1 • Bates model � � � V t dW S dS t = S t − α dt + t + dp t � V t dW V dV t = ξ ( η − V t ) dt + θ t Eckhard Platen Bressanone07 15

3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 time Figure 1: Simulated benchmarked primary security accounts. Eckhard Platen Bressanone07 16

10 9 8 7 6 5 4 3 2 1 0 0 5 10 15 20 time Figure 2: Simulated primary security accounts. Eckhard Platen Bressanone07 17

4.5 GOP EWI 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 time Figure 3: Simulated GOP and EWI for d = 50 . Eckhard Platen Bressanone07 18

4.5 GOP index 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 time Figure 4: Simulated accumulation index and GOP. Eckhard Platen Bressanone07 19

Diversification • diversified portfolios K 2 � � � π j � ≤ � � δ,t 1 2 + K 1 d Eckhard Platen Bressanone07 20

Theorem 5 In a regular market any diversified portfolio is an approximate GOP. Pl. (2005) • robust characterization • similar to Central Limit Theorem • model independent Eckhard Platen Bressanone07 21

60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 5: Benchmarked primary security accounts. Eckhard Platen Bressanone07 22

450 400 350 300 250 200 150 100 50 0 0 5 9 14 18 23 27 32 Figure 6: Primary security accounts under the MMM. Eckhard Platen Bressanone07 23

100 EWI GOP 90 80 70 60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 7: GOP and EWI. Eckhard Platen Bressanone07 24

100 Market index 90 GOP 80 70 60 50 40 30 20 10 0 0 5 9 14 18 23 27 32 Figure 8: GOP and market index. Eckhard Platen Bressanone07 25

• fair security benchmarked security ( A , P ) -martingale ⇐ ⇒ fair • minimal replicating portfolio fair nonnegative portfolio S δ with S δ τ = H τ ⇒ = minimal nonnegative replicating portfolio • fair pricing formula � H τ � � V H τ ( t ) = S δ ∗ � � A t E � t S δ ∗ τ No need for equivalent risk neutral probability measure! Eckhard Platen Bressanone07 26

Fair Hedging S δ • fair portfolio t • benchmarked fair portfolio � H τ � � ˆ S δ � � A t t = E � S δ ∗ τ • martingale representation � H τ � τ d � H τ � � � x k H τ ( s ) dW k � A t = E + s + M H τ ( t ) � S δ ∗ S δ ∗ t τ τ k =1 M H τ - ( A , P ) -martingale (pooled) M H τ , W k � �� � E = 0 t F¨ ollmer & Schweizer (1991) No need for equivalent risk neutral probability measure! Eckhard Platen Bressanone07 27