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Choosing Monte Carlo Time Steps Independently of Day-Count Conventions

Article Quant Q&A · Author: dmitry
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Summary

The document clarifies how to choose the time increment in a Monte Carlo simulation when the stock follows geometric Brownian motion and the risk-free rate follows a CIR process. It distinguishes model time from bond day-count conventions such as Actual/360 or Actual/Actual. Those conventions govern calculations such as accrued interest; they do not determine the simulation’s time increment.

For a simulation horizon T divided into N equally spaced intervals, set the step size to T/N. If the model must include particular dates, such as option exercise dates, add those dates to the time grid and use each interval’s actual length, t(k+1) minus t(k), when advancing the process. The guidance is about constructing the simulation timeline. It does not specify calibration or discretization schemes for either stochastic process, nor how to map calendar dates into a chosen model-time unit.

Key ideas

  • Bond day-count conventions determine interest accrual calculations, not the Monte Carlo model time step.
  • For a uniform grid over horizon T with N intervals, use a step size of T/N.
  • Add required dates, including exercise dates, to the simulation timeline.
  • Use the actual spacing between adjacent grid points when simulating each step.
  • The guidance does not prescribe process calibration or a stochastic discretization scheme.

Tags

OptionsStatisticsDerivatives pricing
Full text
# Day counts and time increment in Monte Carlo


# Day counts and time increment in Monte Carlo












Suppose the evolution of the stock price is given by Geometric Brownian Motion. Futher I assume that the risk free rate process is given by CIR model. In both models there is a time increment dt. To my understanding dt is dependent on day count convention. There are 252 bussiness days in one year. For rates one should take into account their day count conventions e.g. Act/Act, Actual/360 and so on.

How do people deal with this issues in real world applications?

Thanks in advance.

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/34595

Day count conventions such as Act/Act or Act/360 are used for bond math, e.g. interest accrual calculation. The $dt$ in your Monte Carlo simulation is just model time increment and is unrelated to day count conventions. If $T$ is your time horizon for the simulation and you want a uniformly spaced time line $\{t_k\}$ with $N$ time points simply set $dt = T/N$. If you need to add specific dates (e.g. option exercice dates) to your time line just do so and make sure you use the correct time increment $t_{k+1}-t_k$ when simulating from time $t_k$ to $t_{k+1}$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.