Geometric Brownian Motion - increasing simulations or smaller step size

Question

I am running Monte Carlo simulations to estimate future share prices of some stocks.

For stock A, I need 1 share price exactly one year from now.

For stock B, I need daily prices for each trading day for the coming year.

Both models are simulated, lets say, 1000 times.

As dt is smaller for B, this increases the accurateness on the share price on the date one year from now. But how to prove this? And, what is the relation between the number of simulations and the time step size?

Edit: I am using 3 year lognormal daily returns to estimate volatility; drift is based on a zero-coupon bond with the term equal to the term of the option/share (in this case 1 year). Both remain constant during the simulation year. Random numbers are generated using Mersenne Twister algorithm.

Stock price at time t is being calculated by:

For A, dt = 1 $$ S_{t}= S_0 \cdot exp((r-\frac{1}{2}\sigma^2)dt+\sigma\sqrt{dt}Z) $$

For B, I am using Euler's discretization, dt = 1/255 $$ S_{t+dt}= S_t \cdot exp((r-\frac{1}{2}\sigma^2)dt+\sigma\sqrt{dt}Z) $$

Answer 1

Since you are using geometric brownian motion (GBM) as your model, there is a strong (and therefore weak) solution to the SDE. That is to say, your simulation that presumably looks like

$$ S^A_T \sim S^A_0 \exp\left( \left(r-q-\frac12 \sigma^2\right) T + z \sigma \sqrt{T} \right) $$

for standard gaussian $z$ has precisely the correct distribution.

Because sums of gaussian variables are themselves gaussian, your chained simulation for stock B follows the formula

$$ S^B_{t_i} \sim S^B_{t_{i-1}} \exp\left( \left(r-q-\frac12 \sigma^2\right) (t_{i}-t_{i-1}) + z \sigma \sqrt{t_{i}-t_{i-1}} \right) $$

which telescopes into

$$ S^B_{t_N} \sim S^B_0 \prod_{i=1}^N \exp\left( \left(r-q-\frac12 \sigma^2\right) (t_{i}-t_{i-1}) + z \sigma \sqrt{t_{i}-t_{i-1}} \right) $$

or

$$ S^B_{t_N} \sim S^B_0 \exp\left( \sum_{i=1}^N \left(r-q-\frac12 \sigma^2\right) (t_{i}-t_{i-1}) + z \sigma \sqrt{t_{i}-t_{i-1}} \right) $$

and also is precisely correct in distributional terms.

Therefore the simulation for stock A with a 1 year time interval is no less "accurate" than the simulation for stock B.

Now, if you were using an Euler or Milstein discretization of the GBM of the stock B SDE, then you would have cause to worry about the relative accuracy.

Answer 2

a) there is no point to make any simulations between NOW and 1 YEAR. simulate 1 year stock price directly.

b) here you concern about PATH of stoch process, so simulate each day, but do not simulate "between" or "inside" days.

effectively procedure b uses algo from procedure a with appropriate adjustments for drift and vol

to measure "accuracy" calculate confidence intervals. your estimates have normal distribution, sample variance is proportional to SQRT (N), where N - number of simulations. so if you make 4 times more simulation you get twice more accurate estimate SQRT(4)=2. increase of N is very costly, better use Variance reduction techniques (see wiki).