MonteCarloSimulationLogNormalReturns

MonteCarloSimulationLogNormalReturns - MONTE CARLO...

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M ONTE C ARLO S IMULATION OF C ORRELATED , L OGNORMALLY D ISTRIBUTED A SSET P RICES Dynamic investment strategies based on continuous time stochastic models of asset prices frequently require that the portfolio manager simulate the portfolio process using the Monte Carlo simulation method. This appendix describes a simple approach to specifying and calibrating simulation models that have ex-post sample statistical properties that agree with the ex-ante properties of the assets. It is important to note that this approach does not work for all portfolio simulations and, if necessary, other model structures should be considered. Consider a collection of m assets that have asset prices, () , 1, , jt Sj m = . Their dynamic behavior is described by the following system of geometric Brownian motion processes. ,, , , 01 , , j j dS S dt S dZ t j m μσ =+ = (14A-1) The parameters j μ and j σ are the instantaneous growth rates and volatilities of the j st asset. The differential i dZ is a standard Wiener process (Neftci 2000) with the following properties {} , 2 , , 0 it ij Ed Z Z d t Z d Z d t i j ρ = = = ( 1 4 A - 2 ) The solution to the stochastic differential equation in (14A-1) is given by
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() 2 ,, , , 1 exp 2 0, j tj o j j j t jt SS t Z ZN t μσ ⎡⎤ ⎛⎞ = ⎢− + ⎝⎠ ⎢⎥ ⎣⎦ ( 1 4 A - 3 ) In order simulate sample paths of asset class prices, we need to estimate or specify the following parameters. 1. Instantaneous growth rates 1, , i im μ = 2. Instantaneous volatilities , i σ = 3. Correlations , , ; , ij jm ρ == …… These parameters are not directly observable in asset prices, interest rates and inflation, but can estimated from or specified as a function of the expected values, variances and covariances of the value relative 0 / ii t i YSS = . For convenience, we drop the explicit dependence on time. Subsequently we will refer to the 1 m × vector of these random variables 1 2 m Y Y Y = Y # The expected value of Y is the 1 m × vector Y μ . {} 1 2 m Y Y Y Y E Y # μ ( 1 4 A - 4 )
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The mm × matrix of variances and covariances for Y , denoted by Y C , is given by. () {} 11 2 1 21 2 2 12 2 2 2 m m m YY Y Y Y Y Y E σσ σ σ σ = −− = Y CY Y μμ " " ## % # " (14A-5) Prime denotes the transpose of a matrix or vector. We can then re-write equation (14A- 22) as follows ( ) exp ii YX = ( 1 6 ) The random variable i X is normally distributed with mean i X μ and standard deviation i X σ where 2 1 2 i Xii t ⎛⎞ =− ⎜⎟ ⎝⎠ ( 1 7 ) 22 i X i t = ( 1 8 ) We will also make reference to the 1 m × vector of these normally distributed random variables.
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MonteCarloSimulationLogNormalReturns - MONTE CARLO...

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