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MMF1941H: Stochastic Analysis

创建日期：2024-04-19 15:58:30

所属分类：数学mathematics

标签： MMF1941H

Stochastic Analysis

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MMF1941H: Stochastic Analysis - Assignment # 1

25 Oct 2021
1 Instructions
1. Assignments are due on Monday Nov 8th, 7:30pm EST. Submissions are accepted late for 1
day with a 10% penalty, submissions after Nov 2nd will not be accepted.
2. Please have your final report typeset using LATEX and submit your report individually. Pro-
vide code separate in a Python script file that you attach in your submission.
3. You may discuss these questions with your fellow students, however the write-up must be
yours and yours alone, sharing of the write-up before the deadline is not allowed.
2 Problem: Bachelier Call Option Pricing
Let X be a standard normal random variable and let µX ∈ R and variance σ2X > 0 parameters.
We are looking at the value of a call option in the Bachelier Model ie
v = E
(
(µX + σXX− K)+
)
for a given strike K.
1. (10 pts) Show that for K ∈ R
E((X− K)+) = ϕ(K)− K(1−Φ(K))
holds where ϕ is the pdf of the standard normal distribution and Φ the respective cdf.
(Hint: you can exploit that ϕ′(x) = −xϕ(x) holds).
2. (5 pts) Use the previous result to show that analytically
v = σXϕ(
K− µX
σX
)− (K− µX)
(
1−Φ(K− µX
σX
)
)
holds.
1
MMF1941H: Stochastic Analysis
3. (10 pts) For a parameter a ∈ R we can define a measure Qa via the definition
Qa(A) = E
(
1A
exp(aX)
E(exp(aX))
)
.
For x ∈ R calculate the Qa (X ≤ x) and conclude that under Qa X again follows a normal
distribution and determine its parameters.
4. (5 pts) Write Python code to simulate the option value 1000 times under the measure P
with a sample size of 5000 simulations each for µX = 4, σ2X = 4 and K = 8. Share the code
and provide a histogram of the results. Also calculate the exact value analytically per the
above formula.
5. (5 pts) Denoting vj a single MC estimate (based on 5000 simulations) for j = 1, . . . , M with
M = 1000 we can define the sample variance as
s =
1
M− 1
M
∑
j=1
(
vj − 1M
M
∑
k=1
vk
)2
.
Calculate the sample variance for your previous experiment.
6. (10 pts) Note that for any a the option value can be re-written as
v = EQa
(
dP
dQa
(µX + σXX− K)+
)
which mathematically will yield the same answer for any a. Define
g(a) = EQa
((
dP
dQa
(µX + σXX− K)+
)2)
which can be evaluated through MC simulation given the fact the distribution of X under
the measure Qa is known. Write Python code plot the function g(a) for the above selection
of parameters and a ∈ [0, 5]. Note that equivalently,
g(a) = E
(
dP
dQa
(
(µX + σXX− K)+
)2)
holds which you could use alternatively for implementation purposes.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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