Simulation &amp; Valuation Techniques

# Simulation &amp; Valuation Techniques - Risk Management...

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Copyright © 1997-2006 Investment Analytics Slide: 1 Monte-Carlo Simulation Techniques Risk Management Monte Carlo Simulation Techniques

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Copyright © 1997-2006 Investment Analytics Slide: 2 Monte-Carlo Simulation Techniques Monte-Carlo Simulation Techniques ¾ Pseudo-Random Number Generation ¾ Generating Pseudo-Random Variables ¾ Forecasting Volatilities and Correlations ¾ Monte Carlo Simulation of Diffusions ¾ Monte Carlo Options Pricing ¾ Variation Reduction Techniques
Copyright © 1997-2006 Investment Analytics Slide: 3 Monte-Carlo Simulation Techniques Modeling Financial Processes ¾ Geometric Brownian Motion ¾ Hull White Stochastic Volatility Model t t t dW dt S dS σ µ + = + = t t W t S S 2 0 2 1 exp 1 t t t dW dt S dS + = 2 2 2 t t t dW dt d ξ ν + =

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Copyright © 1997-2006 Investment Analytics Slide: 4 Monte-Carlo Simulation Techniques One Factor Interest Rate Models ¾ General Form: dr = m( r) dt + σ ( r) dW ± Ito Process: m: drift factor σ : short rate volatility d w: ε√ t; ε ~ N( 0,1 ) ¾ 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 Slide: 5 Monte-Carlo Simulation Techniques Model Taxonomy Expected Mean Volatility Fits Term Str. change in r Reversion of r Yield Vol m(r) s(r) Vasicek a[m - r] yes constant no no CIR a[m - r] yes σ r no no Brennan & a[b + L - r] yes f(r,L) no no Schwartz Ho & Lee g(t) no constant yes no BDT f(t,r, σ ) limited f(time) yes yes Hull & a(t)[m(t) - r] yes f(time) yes yes White

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Copyright © 1997-2006 Investment Analytics Slide: 6 Monte-Carlo Simulation Techniques Pseudo-Random Number Generation ¾ Simulation of price and return paths over time ± Requires the generation of sequences of random variables. ± This is known as Monte Carlo sampling ± Generation of “random” (i.e. almost independent) numbers ¾ Pseudo-Random Numbers ± Represent as decimal fractions ± Interpret as realizations U of the uniform distribution on the unit interval U(0,1)
Copyright © 1997-2006 Investment Analytics Slide: 7 Monte-Carlo Simulation Techniques ¾ Most Common Method of Sequential Generation ± m is a very large number -- the period of the generator ± a and c are parameters ± is the seed provided to start the recursive stream of numbers x 0 , x 1 , x 2 , . .. ¾ Construct uniform pseudo-random variates u i ~ U(0,1) Sequence is not independent due to the m-long cycle Okay when the sample number n is small relative to m xm m 0 01 1 { , ,. .., , } ... 2, 1, 0, / = = i m x u i i xa x c ii + =+ 1 ( ) modulo m i = 0, 1, 2, . .. Linear Congruential Generator

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Copyright © 1997-2006 Investment Analytics Slide: 8 Monte-Carlo Simulation Techniques Example For example, if x 0 := 35, a := 13, c := 65, and m := 100 the algorithm works as follows: Iteration 0 Set x 0 = 35, a = 13, c = 65, and m = 100.
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## This document was uploaded on 11/04/2011 for the course ECON 421 at CUNY York.

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Simulation &amp; Valuation Techniques - Risk Management...

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