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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.

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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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  • $\begingroup$ Thanks. Assume that the Future price at time t (some day) is given by F=Sexp(rdt) where dt is calculated as the time between dates in accordance with some day count convention. Then the mean of simulated prices at time t should converge with F. If one uses dt=T/N the price will not converge, won't it? $\endgroup$
    – dmitry
    Jun 8 '17 at 9:46
  • $\begingroup$ When you compute $\exp(r dt)$ the rate $r$ is a continuous rate so it does not follow any market day count convention, it simply has to be consistent with how you measure time in your model. Most applications measure model time in years and compute the model time between 2 dates as $(d_2 - d_1)/365$ or $(d_2 - d_1)/365.25$, but you can also design an application where model time is in days. $\endgroup$
    – Antoine Conze
    Jun 8 '17 at 9:56
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    $\begingroup$ @AntoineConze in your comment you mention that it is important to be consistent - i think this one of the more important aspects. If you're takign your option implied vol surface using $t=(d_2 - d_1)/365.25$ then you just need to make sure that your treatment of time is the same everywhere - then any issues from it will go away. $\endgroup$
    – will
    Jun 8 '17 at 10:15
  • $\begingroup$ @will Yes you're absolutely right you need model time to be the same everywhere. Then model parameters (vols, mean rev, etc.) are homogeneous to the chosen time unit, and it has no impact on pricing since in the end all models are calibrated to market prices. $\endgroup$
    – Antoine Conze
    Jun 8 '17 at 11:41

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