What is the meaning of time variable in GBM sample paths

[nick.snels]

I use the code (taken from https://mhittesdorf.wordpress.com/2013/12/29/introducing-quantlib-modeling-asset-prices-with-geometric-brownian-motion/) below to generate GBM sample paths. The code is for a sample path of one year. I would however like to make sample paths for 100 trading days. So I adjusted time_steps to 100, but I don’t know what to do with the length variable. When I set it to 100/255, the extreme values of the sample paths are much bigger and more frequent then when I just leave it at 1 (which to me seems counterintuative). So what is the meaning of the length variable and to which value should I set it? Thank you.

 

 

Real starting_price = 20.16; //closing price for INTC on 12/7/2012

Real mu = .2312; //INTC one year historical annual return

Volatility hv = 0.2116; //INTC one year historical volatility

Size time_steps = 255; //trading days in a year (U.S.)

Time length = 1; //one year

 

//instantiate Geometric Brownian Motion (GBM) stochastic process

const boost::shared_ptr<StochasticProcess>& gbm = boost::shared_ptr<StochasticProcess> (new GeometricBrownianMotionProcess(starting_price, mu, hv));

 

//generate a sequence of normally distributed random numbers from a

//uniform distribution using Box-Muller transformation

BigInteger seed = SeedGenerator::instance().get();

MersenneTwisterUniformRng mersenneRng(seed);

BoxMullerGaussianRng<MersenneTwisterUniformRng> boxMullerRng(mersenneRng);

RandomSequenceGenerator<BoxMullerGaussianRng<MersenneTwisterUniformRng>> gsg(time_steps, boxMullerRng);

 

//generate simulated path of stock price using GBM stochastic process

PathGenerator<RandomSequenceGenerator<BoxMullerGaussianRng<MersenneTwisterUniformRng>> > gbmPathGenerator(gbm, length, time_steps, gsg, false);

const Path& samplePath = gbmPathGenerator.next().value;

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[Luigi Ballabio]

It's the length of the paths as a fraction of year, so 100/255.0 should work.  I haven't checked for extreme values, but when I try your code, extract 10000 sample paths and collect the end points, I get a standard deviation between 5.4 and 5.5 for 255 time steps and length = 1, and a std dev around 2.9 for 100 time steps and length = 100/255.0.

Luigi

On Mon, Jun 27, 2016 at 2:31 PM <[hidden email]> wrote:

I use the code (taken from https://mhittesdorf.wordpress.com/2013/12/29/introducing-quantlib-modeling-asset-prices-with-geometric-brownian-motion/) below to generate GBM sample paths. The code is for a sample path of one year. I would however like to make sample paths for 100 trading days. So I adjusted time_steps to 100, but I don’t know what to do with the length variable. When I set it to 100/255, the extreme values of the sample paths are much bigger and more frequent then when I just leave it at 1 (which to me seems counterintuative). So what is the meaning of the length variable and to which value should I set it? Thank you.

 

 

Real starting_price = 20.16; //closing price for INTC on 12/7/2012

Real mu = .2312; //INTC one year historical annual return

Volatility hv = 0.2116; //INTC one year historical volatility

Size time_steps = 255; //trading days in a year (U.S.)

Time length = 1; //one year

 

//instantiate Geometric Brownian Motion (GBM) stochastic process

const boost::shared_ptr<StochasticProcess>& gbm = boost::shared_ptr<StochasticProcess> (new GeometricBrownianMotionProcess(starting_price, mu, hv));

 

//generate a sequence of normally distributed random numbers from a

//uniform distribution using Box-Muller transformation

BigInteger seed = SeedGenerator::instance().get();

MersenneTwisterUniformRng mersenneRng(seed);

BoxMullerGaussianRng<MersenneTwisterUniformRng> boxMullerRng(mersenneRng);

RandomSequenceGenerator<BoxMullerGaussianRng<MersenneTwisterUniformRng>> gsg(time_steps, boxMullerRng);

 

//generate simulated path of stock price using GBM stochastic process

PathGenerator<RandomSequenceGenerator<BoxMullerGaussianRng<MersenneTwisterUniformRng>> > gbmPathGenerator(gbm, length, time_steps, gsg, false);

const Path& samplePath = gbmPathGenerator.next().value;

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