The course Financial Math 3 includes 70 exercises that are performed individually or in small groups. Sample exercises are indicated here. In the course, you will learn how to solve problems like these—and many more!

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Mean Squared Error = (Standard Error)² + Bias²

(Hint: use the fact that var(X) = E(X²) – E(X)².)

Sample 2. Consider a modification of sample variance as an estimator for the variance of a random variable X. Suppose we happen to know the mean of X. Replacing the sample mean  with that known mean in the formula for S², we obtain a new estimator for variance:

What is the bias of this estimator?

Sample 3: Consider the data:

Use general regression to fit a curve of the form:

to the data.

Sample 4: Solve for y as a function of x:

Sample 5: Consider the exponential Brownian motion:

X = exp(W + t)

Apply Ito's lemma to calculate dX.

Sample 6: If

find the stochastic differential equations satisfied by

a. f(X) = AX
b. f(X) = AXⁿ.