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Assume a financial instrument which has a (roughly) log-normal price distribution and behaves like a random walk. I would like to generate some possible scenarios for where the price might be tomorrow.

Knowing that the prices are log-normally distributed (and hence skewed, i.e. an increase in price is usually greater in magnitude than a decrease in price), would it be more correct to generate asymmetric scenarios (with equal probability) around today's market price, for ex. by estimating a log-normal distribution based on the historical data? Or would it be more correct to generate symmetric scenarios around today's market price (in a way assuming that the log-returns are normally distributed and hence symmetric)?

Does generating asymmetric scenarios in any way bias my future view of the market, by making me believe that one direction is "more likely" than the other?

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Numerical simulation is used in Monte Carlo simulation with applications to option pricing but in option pricing, we simulate prices (which is assumed to follow a log normal distribution) have what is known as a drift equal to the risk free rate.

In your case, since it doesn't seem like you are doing this for the sake of pricing options, if prices are assumed to follow a log normal distribution, then its dynamic follows the following stochastic differential equation:

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

where $W_t$ is Brownian motion. This fortunately has an analytical solution which can be solved using Ito's lemma:

$$S_t = S_o \exp\bigg(\big(\mu - \frac{\sigma^2}{2}\big)t+\sigma W_t\bigg)$$

You can simulate $W_t$ using random normal variates. If you have positive drift $\mu$ then future price will drift upwards.

Example, today's spot price, $S_0 = 100$, drift $\mu = 0.2$, volatility $\sigma = 0.01$, time (in years) $t=1/252$ and $W_t=0.28499$ (generated using normal random number generator), plug all that into the equation to generate one possible scenario. Generating thousands of scenarios will give you a distribution of where tomorrow's price may lie.

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  • $\begingroup$ How do you determine the distribution for $W_t$, mean and variance? $\endgroup$ Feb 3, 2022 at 10:49
  • $\begingroup$ $W_t$ is normally distributed with mean 0 and standard deviation $\sqrt{1/252}$ $\endgroup$ Feb 3, 2022 at 10:53

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