代写HOMEWORK #4 Spring 2025代写R编程

HOMEWORK #4

Spring 2025

Total Points: 70

Assignment Date: Monday, April 7th.

Due Date: Tuesday, April 22nd at 11:59pm (EST).

Instructions

Read each question carefully and complete all the requirements. While only a subset of the questions may be graded, it is recommended that you attempt all of them. You may discuss homework problems with your peers, but the submitted work must be your own. Identical submissions will receive a zero.

Submission Instructions: Submit a PDF file of your solutions to Canvas by the stated deadline. Scanned handwritten solutions in PDF format will be accepted but must be written neatly. It is your responsibility to ensure that the solutions are clear and legible. Late submissions will NOT be accepted!

Problem Set

Problem 1 (10 points). Let W = (Wt)t≥0 be a Brownian motion and consider the SDEs defining the Geometric Brownian Motion (GBM). S = (St)t≥0, and OU process, X = (Xt)t≥0:

dSt = µStdt + σStdWt , S0 = s0,

dXt = κ(θ − Xt)dt + σdWt , X0 = x0.

Fix ∆t > 0, and let µ = 0.25, κ = 0.2, θ = 1, σ = 0.5 and s0 = x0 = 1.2. From the SDEs we have that:

Using this observation, simulate (and plot) 10 paths of the GBM and OU process. You may use any programming language that you are familiar with for this task. Include your code as a separate file in your submission.

Problem 2 (10 Points). Let us use the OU process:

dXt = κ(θ − rt)dt + σdWt , X0 = x0 ∈ R.

where W = (Wt)t≥0 is a P-Brownian motion as a model for the USD/EUR exchange rate. For this purpose download 5 years worth of data from the St. Louis Fed: https://fred.stlouisfed.org/series/DEXUSEU. The coefficients of the OU process can be inferred from a Regression (AR) Model. To be explicit, by letting ∆ = ti+1 − ti a discretization of the OU solution reveals that:

where Fit this regression model to the data using a programming language of your choice and report the implied values of κ, θ and σ that you obtain by rearranging the equations for α, β and γ. Then, simulate a few trajectories from your calibrated model and compare them to the original price series.

Problem 3 (10 Points). Let W = (Wt)t≥0 be a Brownian motion and let X = (Xt)t≥0 be an Ornstein Uhlenbeck (OU) process; i.e., X satisfies:

dXt = κ(θ − Xt)dt + σdWt , X0 = x0.

Show that the covariance function of X, c(s, t), is:

Problem 4 (15 Points). Compute the mean and variance of the following stochastic integrals. Then, comment on whether or not the integral in ques-tion is normally distributed.

Problem 5 (5 Points). Let W = (Wt)t≥0 be a Brownian motion. Argue for the following integration by parts results:

Problem 6 (10 points). Let W = (Wt)t≥0 and B = (Bt)t≥0 be Brownian motions with instantaneous correlation ρ. Apply Itˆo’s formula to get the dynamics of:

1. Xt = cos(Wt).

2. Xt = sin(tWt).

3. Xt = (St) p for p > 0 where S = (St)t≥0 is a Geometric Brownian

Motion:

dSt = µStdt + σStdWt .

4. Xt = exp{Yt} where Y is an OU process:

dYt = κ(θ − Yt)dt + σdWt .

5. Xt = StPt where S = (St)t≥0 and P = (Pt)t≥0 are Geometric Brownian Motions:

dSt = µStdt + σStdWt ,

dPt = νPtdt + ηPtdBt .

Problem 7 (10 points). Solve the following Stochastic Differential Equations (SDEs):

1. dSt = µ(t)Stdt + σ(t)StdWt where S0 = s0 > 0 and µ(·), σ(·) are deterministic functions of time.

2. dXt = κ(θ(t) − Xt)dt + σdWt where X0 = x0 ∈ R and θ(·) is a deterministic function of time.

Hint: Use the same approach as in class for the standard GBM and OU processes.