Normal Sums and the Limit of a Discrete Price Model

Summary

The question considers a discrete price process with normally distributed independent shocks and asks whether the affine closure property of normal random variables can replace the Central Limit Theorem in deriving a continuous-time limit. It specifies an additive update in which drift and a scaled random shock are proportional to the current price, and states that the process converges in distribution to a log-normal variable as the time step shrinks.

The document raises a mathematical distinction: sums of independent normal variables remain normal, but the price process is multiplicative across time, so taking logarithms and analyzing the accumulated increments is central to the limit argument. No answer, derivation, or conditions are provided. As a result, it identifies the issue but does not establish whether the proposed reformulation works or explain how to prove convergence.

Key ideas

• The discrete update scales both drift and random shocks by the current price.
• The question states that the shrinking-step process converges to a log-normal distribution.
• Affine combinations of independent normal variables remain normal, but the price recursion is multiplicative.
• The document supplies no derivation or answer resolving the proposed alternative proof.

Tags

# 48966


# is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?












I have proven the distribution of a discrete time model, evolving over a uniform mesh with $\delta t = T/L$ is given by

$$S(t_{i+1}) = S(t_i) + \mu \delta t S(t_i) + \sigma\sqrt{\delta t}S(t_i)Y_i,$$

for $i = 0, . . . , M − 1$, where $Y_i$ is an i.i.d. N(0, 1) sequence, converges to that of a log-normal random variable as $\delta\to0$(and hence as $L\to \infty$). This required an application of the Central Limit Theorem.

So now if I let $X_1, X_2, . . . , X_n$ be a set of independent Normal random variables, with means $\mu_i$ and variances $θ^2_i$ for $i = 1, . . . n$ respectively is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.