Calculating Probabilities of Exchange Rate Fluctuations with Markov Processes

[Poirot1]

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 = Yn−1 + 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?

 

[CaptainBlack]

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 = Yn−1 + 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

 

[Poirot1]

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?

 

[CaptainBlack]

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

 

[blue_raver22]

I would approach this problem by using Markov processes to calculate the probabilities of exchange rate fluctuations. Markov processes are a mathematical tool used to model random processes, such as the daily changes in currency exchange rates. In this case, we can use the given information about the mean, variance, and starting exchange rate to calculate the probabilities of the exchange rate being below 1.4 in 9 days and 25 days.

To begin, we can set up a Markov chain with two states: above 1.4 and below 1.4. We can then use the given information about the exchange rate being a random variable with mean 0 and variance v to calculate the transition probabilities between the two states. For example, the probability of transitioning from above 1.4 to below 1.4 in one day can be calculated using the cumulative distribution function of a normal distribution with mean 0 and variance v.

Once we have these transition probabilities, we can use them to calculate the probability of the exchange rate being below 1.4 in 9 days and 25 days. This can be done by multiplying the transition probabilities over the desired time period. For example, to calculate the probability of the exchange rate being below 1.4 in 9 days, we would multiply the transition probabilities for 9 days. Similarly, to calculate the probability for 25 days, we would multiply the transition probabilities for 25 days.

Therefore, the probability of the exchange rate being below 1.4 in 9 days and 25 days can be calculated using Markov processes. This approach allows us to account for the randomness and volatility of exchange rates, making it a useful tool for predicting future exchange rate fluctuations.