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CFRM505: Monte Carlo Methods in Finance

创建日期：2024-07-06 17:34:27

所属分类：金融Finance

标签： CFRM505

Monte Carlo Methods in Finance

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CFRM505: Monte Carlo Methods in Finance

Midterm Exam

Practice Questions

1. Consider estimating the integral:

First, show how to use inverse transform. to simulate N IID exp(2) exponential random variables.

Now given that you have already obtained a large sample of IID exp(2) exponen-tials: (v1, . . . , vN ). Now how to use only (v1, . . . , vN ) as your sample to estimate the above integral.

2. Consider a random variable X that has probability density function

(a) Write down the CDF of X and determine the value of α analytically.

(b) Apply the inverse transform. method to simulate X. Provide the psuedo-algorithm.

(c) Instead of using inverse transform. method, let’s generate samples of X using the acceptance-rejection algorithm. To do so, consider two approaches:

(i) Use the uniform. density function for g. What’s the minimum value of a you can use? Determine it analytically.

(ii) Alternatively, for g use the pdf of the exponential random variable exp(λ), for some rate parameter λ > 0. What value of λ would make your algorithm most efficient? Derive the corresponding value of a.

Lastly, which method, (i) or (ii), is more efficient? Explain why.

3. Consider the acceptance-rejection method to simulate a standard normal N(0, 1) r.v. (the target r.v.) whose PDF is denoted by f(x). Do so with the function g(x) being the Cauchy PDF, that is

Determine the minimum value of the constant a such that f(x) ≤ ag(x) ∀x. Give a sketch of f(x) and ag(x) over x on the same graph (label clearly).

Now, explain whether we can do the reverse. That is, simulate a Cauchy r.v. (the target r.v.) by acceptance-rejection using the standard normal distribution.

4. Consider the following simulation algorithm:

Show that this simulation algorithm will generate samples of X1 and X2 that are independent standard normals.

5. Consider a non-homogeneous Poisson process with intensity function

λ(t) = a + bt, t ≥ 0,

with constants a, b > 0. Denote by (Tk)k=1,2,... the arrival times for this process. Recall from the definition of NHPP that

Using this fact, describe an algorithm that generates the times of the first m arrivals of this NHPP using exactly m uniforms. (Hint: Use the CDF of inter-arrival time(s).)

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