Calculating Probabilities of Exchange Rate Fluctuations with Markov Processes

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Discussion Overview

The discussion revolves around calculating the probabilities of currency exchange rate fluctuations using Markov processes. Participants explore the statistical properties of daily changes in exchange rates, specifically focusing on the mean and variance of these changes over time. The conversation includes considerations of how to set up the problem and compute probabilities based on normal distributions.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant states that daily changes in currency exchange rates are modeled as a random variable with mean 0 and variance v, suggesting a recursive relationship for the exchange rate over days.
  • Another participant reiterates the model and discusses the implications for calculating probabilities over multiple days, indicating that the mean change remains 0 while the variance scales with the number of days.
  • A participant seeks clarification on how to compute the probability of the exchange rate falling below a certain value after a specified number of days, questioning whether to use a differential equation or probability tables.
  • There is a suggestion that the change in exchange rate over n days can be represented as a normal distribution, specifically N(0, nv), and that probability tables may be used for calculations.

Areas of Agreement / Disagreement

Participants generally agree on the statistical model for daily changes in exchange rates, but there is uncertainty regarding the method for calculating probabilities, with some suggesting different approaches.

Contextual Notes

The discussion includes assumptions about the independence and distribution of daily changes, and there are unresolved questions about the appropriate computational methods for probability calculations.

Poirot1
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It is widely believed that the daily change in currency exchange rates is a random variable with mean 0 and variance vThat is, if Yn represents the exchange rate on the nth day, Yn = Yn1 + Xn, n = 1, 2, . . . where X1,X2, . . . are independent and identically
distributed normal random variables with mean 0 and variance v. Suppose that today’s exchange rate is 1.55 and v = 0.0025, then what is the probability that the exchange rate will be below 1.4 in 9 days and in 25 days?

Am I meant to setup a diffferential equation, I'm not sure?
 
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Poirot said:
It is widely believed that the daily change in currency exchange rates is a random variable with mean 0 and variance v​
That is, if Yn represents the exchange rate on the nth day, Yn = Yn1 + Xn, n = 1, 2, . . . where X1,X2, . . . are independent and identically

distributed normal random variables with mean 0 and variance v. Suppose that today’s exchange rate is 1.55 and v= 0.0025, then what is the probability that the exchange rate will be below 1.4 in 9 days and in 25 days?

Am I meant to setup a diffferential equation, I'm not sure?


The change over \(n\) days is the sum of \(n\) iid RV's and so the mean change is \(n\) times the mean change, in this case \(0\) and the variance of the change is \(n\) times the single day variance, so in this case is \(n.v\)

CB
 
Thanks CaptainBlack, but I am looking for a probability. If we let Z denote the change in exchange rate over n days, so Z-normal(0,nv), do I compute probability by normal table?
 
Poirot said:
Thanks CaptainBlack, but I am looking for a probability. If we let Z denote the change in exchange rate over n days, so Z-normal(0,nv), do I compute probability by normal table?

The distribution is N(0,nv) and you use probability tables for the computations.

CB
 

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