《Risk Management - Simulation Valuation Techniques》.pdf

Risk Management Monte Carlo Simulation Techniques Copyright © 1997-2006 Investment Analytics Monte-Carlo Simulation Techniques Slide: 1 Monte-Carlo Simulation Techniques Pseudo-Random Number Generation Generating Pseudo-Random Variables Forecasting Volatilities and Correlations Monte Carlo Simulation of Diffusions Monte Carlo Options Pricing Variation Reduction Techniques Copyright © 1997-2006 Investment Analytics Monte-Carlo Simulation Techniques Slide: 2 Modeling Financial Processes Geometric Brownian Motion dSt ⎛ ⎡ 1 2 ⎤ ⎞ µdt + σdWt S S0 exp⎜ µ − σ t + σ W ⎟ S t ⎜ ⎢ 2 ⎥ t ⎟ t ⎝ ⎣ ⎦ ⎠ Hull White Stochastic Volatility Model dSt 1 dσ 2 2 µdt + σdWt t νdt +ξdWt St σ 2 t Copyright © 1997-2006 Investment Analytics Monte-Carlo Simulation Techniques Slide: 3 One Factor Interest Rate Models General Form: dr = m( r) dt + σ( r) dW Ito Process: • m: drift factor • σ: short rate volatility • dw: ε√t; ε ~ N(0,1) Model characteristics All rates move in same direction, but not by same amount Many different shapes possible (including inverted) Mean reversion can be built in Copyright © 1997-2006 Investment Analytics Monte-Carlo Simulation Techniques Slide: 4 Model Taxonomy Expected Mean Volatility Fits Term Str. change