Random Walk

[andreww]

Hi,

I've just been looking at random walk for discrete time and have a question. Using the formula to model the random walk:

S[i+1] = S * (1+μ*t + σ*φ*t^1/2)

where t = small increment of time

When I simulate this on a spreadsheet, with high volatility(σ), and relatively large timestep e.g. t=0.5, and drift (μ) = 0.15, I get negative asset price.

Looking at the formula, it is certainly possible to get negative asset price. e.g. timestep = 1, the drift is low, volatility is high enough to make the asset terms in the brackets a negative value.

Why is this so? I thought the model should not allow the modelled stock price to be negative?

regards,

 

[koupparis]

The actual model doesn't go negative, but the simulation may. You're approximating the evolution of the stock price. If you're stressing your approximation, ie large time steps and large vol, you're going to get negative S.

 

[quotes]

Because your steps are not dense enough.
Besides, this multiplicative model has error simulating a Brownian motion, try additive model instead, which simulates log(S(t)) and you will never get a negative S(t).

 

[Daniel Duffy]

The value can become negative because you are using explicit Euler method.

http://www.maths.lth.se/na/courses/FMN081/FMN081-05/lecture22.pdf

 

[finengtoo]

Quotes is right. If you do it properly with logs, asset prices will always stay positive. This could be done in a spreadsheet too the way you did it in your model.

Your model is not very different from the one with the logs. It's the parametrization rather than the model if your S(t)s are deep in the negative. If it's just sometimes slightly negative, it's not bad.

 

[finengtoo]

Too lazy to type, so here is copied from Duke's site.

Geometric random walk model: Application of the random walk model to the logged series implies that the forecast for the next month's value of the original series will equal the previous month's value plus a constant percentage increase. To see this, note that the random walk model for LOG(Y) is given by the equation:

where the constant term (alpha) is the average monthly change in LOG(Y), which is approximately the average monthly percentage change in Y. For the S&P 500 series, alpha is equal to 0.0056, representing an average monthly increase of 0.56%. Exponentiating both sides of the preceding equation, and using the fact that EXP(x) is approximately equal to 1+x for small x, we obtain:

This forecasting model is known as a geometric random walk model, and it is the default model commonly used for stock market data.

This is a random-walk-with-growth model (i.e., an ARIMA model with one nonseasonal difference and a constant term) in conjunction with a log transformation.

The random walk model remains appealing because it leads to smaller forecasting errors than most other models. It's simple. Big positive.

 

[peter pan]

i spent a considerable amount of time a few years back reading about random walk stuff...

here is some thing that i think may or may not stimulate your senses while you are studying the random walk papers...

http://en.wikipedia.org/wiki/Monty_Hall_problem

enjoy.