FRM: Monte carlo simulation: Brownian motion

[Music] hello this is David Harper Evonik turtle in the last few days I Illustrated a parametric and a historical simulation approach to estimating value at risk or var today I'd like to introduce a basic approach in the third way to estimating value at risk and that is the Monte Carlo simulation so I'll start here with a basic Monte Carlo simulation and that is in regard to the asset class that is equities or stocks the fr-s frm candidates were following John hull and he employs a popular in common process to model stock returns or stock prices and that is Brownian motion so that's what this Montecarlo simulation is a simple Brownian motion application and so just to show you how this works here's the formula from john hall that describes Brownian motion and it says on the left here that the natural log of today's price divided by yesterday's price that's if the periods are daily price T divided by price t1 take the natural log that's the continuously compounded periodic return for the stock and so we're saying the periodic return on a continuously compounded frequency is approximately normally distributed with mean here of drift minus 1/2 of the variance over time and volatility here of the volatility multiplied by the square root of time that's our square root roll where we say volatility scales with the square root of time so we've got a periodic return on a continuous basis that is approximately normally distributed and this is why we say that price levels are log normal that's because the natural log of the returns is normally distributed if the natural log is normally distributed we say here the price levels are ratio of prices is log normally distributed so this Brownian motion ends up being a what we could call a log normal diffusion process so I won't go into more detail on that formula here but show you the simpler at least equivalent implementation here in the model of the Monte Carlo simulation so here really is the same thing and again we've got the periodic return natural log of the price over yesterday's price is equal to two components here a deterministic component and a stochastic de Punk component the deterministic component is the drift we're expecting the stock price to drift upward it has some positive expected return over time so that's the fixed piece but then there is a random shock it's going to be a function of the volatility x here Z which is a random variable so that random variable is scaled by volatility and that allows us to model this as a stochastic process here I'm hitting recalc so I get a different series each time and that random Z's allow me to do that so let me just show you how this is implemented in the excel here I only need three assumptions I hat those are in yellow an annual drift or expected return for the stock I'm going to assume 10% an annual volatility the stock I'll assume 40% and then I'm going to assume the stock starts at $100 so I need to do a few conversions here first i compute excuse me I convert the annual drift to a daily drift and to do that I simply divide it by the annual drift by 252 I'm assuming that there's 252 trading days in a year so you can see I just divided that into it converted the annual drift into a daily drift and then I also did the same thing for the volatility converted the annual volatility 40% into a daily volatility but remember I don't divide by 252 there I'm dividing by the square root of 252 the square root rule volatility scales with the square root of time because variance scales with time directly so that gives me the daily volatility and I'm almost ready I just need to take this daily drift and convert it into an expected daily drift by subtracting 1/2 the variance so what I like to say about that in words is that at least under this geometric averaging the volatility is eroding returns this goes to the ambiguous definition of return but for now without going to detail that let's just note that the daily drift is experiencing some drag as a function of the variance and we end up with the 2 things we really need to model the Brownian motion which is a daily volatility and a daily drift so now I'm going to model these down and what I have is one column for each day so this is tomorrow and the next day and the next day and the next day and they're stitched together in sequence and here's the plot each time I hit f9 I get a new plot if I take a look at the first day the first thing I want to do is compute this random Z here this is a function and we've seen this a lot and quantitative finance norm SN of random random gives me the probability between 0 and 1 and norm s and then translates that into the inverse standard normal cumulative distribution so it's going to give me a value generally between negative 3 and 3 and the idea of that is I get to randomize my a volatility and then I'm having done that I can now apply implement this formula for Brownian motion here if we take a look at this on my first day you'll see how simple this is I'm saying tomorrow it's the price is going to be a function of drift plus my volatility multiplied by that random Z that I just calculated so we can see my drift is here that's my expected daily return so if I just stop there what I would get here is a straight lot straight slightly upward sloping line my drift will be constant but then I enter then I add to that the random shock so it's going to be my volatility my daily volatility that's right here two point five two percent multiplied by this random Z right up here in this case happens to be negative one point not 0.19 remember that's going to tend to be between negative three to three so that's going to random that effectively randomizes my volatility and adds a shock to the constant drift so that's the essence here constant drift plus plus random shock as a function of the volatility and that gives me my log return and finally here I simply multiply that today's price of a hundred by E raised to that return or the exponential function of and that gives me tomorrow's price and then on the next day I'm going to do the same thing I'm just going to take that calculate that that log return which is drift plus randomized shock and I'm going to take the previous price and multiply it by the exponential function of this log return and so each day I get a new price the that's a function of the previous price but it's randomized because I've got these random variables which will change here each time and as I hit f9 each time to recalc I get a new series based on my Brownian motion for a model of the stock price so that's an introduction to a real basic Monte Carlo simulation this is David Harper the Bionic turtle thanks for your time [Music] you

Channel: Bionic Turtle

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Keywords: Finance, monte, carlo, simulation, value, at, risk, Monte Carlo Method (Ranked Item), Brownian Motion, Market, frm, financial risk management, bionic turtle, garp, Financial Risk Manager

Id: e79OtCamxD0

Length: 9min 27sec (567 seconds)

Published: Fri Jul 25 2008