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Suppose we know stock price volatility is normally distributed with mean = 0 and annual volatility say 20%. Let's assume markets never close and we can trade at 1 second intervals. Let's assume stock at $t_0$ is \$100.

How to estimate the number of round trips in a given time period (day, month,...) with a basic strategy of opening a trade $x$ standard deviations away from \$100 and closing it at \$100? We can have a max open position of 1 unit, so basically this trades around stock at $t_0$.

Then how to estimate with $x\%$ confidence that we will not be underwater more than $y$ dollars with this basic strategy?

I've never seen this type of problem expressed before but it has practical implications for what I am trying to do. I'm not sure what I'm getting myself into, if this is a hard problem or not, if it is some recommended stuff to look into will be helpful.

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  • $\begingroup$ After thinking more about this, it's actually succinctly a problem called the random walk. Mean = 0, but variance continues to grow linearly with steps. $\endgroup$ – user38066 Dec 23 '18 at 3:57
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    $\begingroup$ I can’t understand the question. How can volatility have a mean of zero ? What exactly do you mean by “round trip” ? Very unclear $\endgroup$ – dm63 Dec 23 '18 at 14:58
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To approximate the number of transactions based on volatility you will need to make many simulations and run your trading strategy in all the simulations and then take the average or the median as an approximation. To make the simulations you can use the Random Walk approach.

You can use GBM equation to create a Random walk simulation as shown below:

A Geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift.

To simulate the stock prices we can use the SDE or Stochastic Differential Equation of St (a stochastic process).

SDE===> dSt = 𝛍St dt + 𝞼St dWt

Where,
St is a stochastic process
𝛍 is the percentage drift
𝞼 is the percentage of volatility
Wt is a Weiner’s process or Brownian motion

If you want to link this equation to a stock data then you can think of St as the stock price at time step t, 𝛍 as the average daily return and 𝞼 as the average daily volatility of the stock. Let us try to simulate the stock prices from the above equation by expanding it further using the Ito’s interpretation.

St = St-1* exp((𝛍-(𝞼^2/2))*t + 𝞼*Wt)

Where,
St is stock price at time t
St-1 is stock price at time t-1
𝛍 is the mean daily returns
𝞼 is the mean daily volatility
t is the time interval of the step
Wt is random normal noise

These simulations are very useful when one is interested in finding the VaR or the expected shortfall for a particular stock with a certain degree of confidence. This exactly what your question is. To calculate the VaR you just need to take the lower x% percentile price(p) of all the simulated prices. This should give you the confidence-x and price p associated with it. For a given confidence value, you can get the price associated with it.

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