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Recently, I asked myself the question: "Why is the expected value of e^sX (X ~ N(0,1)) equal to e^(0.5*s^2)?". You know that compounding a sum of money (A) at a continously compounded rate X for a period involves multiplying it by e^X. In case X is non-random, the expected value is A*e^X. In case X is random, calculating the expected value becomes quite difficult. The solution for a standard normally distributed X can be derived as follows:

1. Write down the equation (μ = E[e^sX] where X ~ N(0,1)):
gau012. Substitute y = x - s and you get:gau2More general, if X ~ N(μ*, σ2), then X = μ* + σZ with Z ~N(0,1). It follows thatgau03Of course, practitioners often know the parameters they are interested in (e.g. s = 0.3), so they can take a shortcut and run a monte carlo analysis:

s <- 0.3
n <- 10000
x <- rnorm(n)
answer <- mean(exp(s*x))

Or a bit more sophisticated and less arbitrary:

s <- 0.3
n <- 10000
i <- (1:n)/(n+1)
x <- qnorm(i)
answer <- mean(exp(s*x))

NB: The expected value can be very misleading.
Source: Stochastische Grundlagen der Finanzmathematik, Klaus Pötzelberger
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E[e^sX] when X Well Explained
When writing such forms of equation in mathematics, care has to be taken to ensure the research being undertaken is precise and conscise. This equaltion are well explained from the top to the bottom as a way of guiding any person who needs to undertake research paper help or wants to understand the expected value in such calculations. Listing the source adds more guidance to anyone doing more research on such particular research