This document contains a group project report for Group 28. It includes 5 parts: (1) calculating option values using binomial and Black-Scholes models, (2) confirming the Black-Scholes option value and calculating Greeks, (3) generating a random stock price path, (4) calculating option values and Greeks along the path over time, and (5) plotting the changing option value as it approaches expiration. The group members and their student numbers are listed at the top.
The document summarizes key policy options for transforming Kenya's food systems to support economic growth and prosperity under the new Bottom-Up Economic Plan. The priorities include broadening from a food security to a food systems approach, accelerating industrialization and commercialization of the food system through mechanization, productivity and value addition, expanding access for smallholders through improved fertilizer access and credit, leveraging digital technologies, improving nutrition, and empowering women's participation. The transformations aim to reduce poverty, create jobs, improve diets and harness the food system's potential to address challenges from population growth and shocks like drought and the pandemic.
This document provides an overview of the 2022/2023 cereal production figures and regional market situation in West Africa. Cereal production reached 77 million tonnes, a 7.7% increase over 2021 but below the five-year average. Key points include maize and rice production increases but deficits for countries like Chad, Gambia, and Cabo Verde. Regional markets face high demand, inflation, and insecurity impacts. Recommendations are needed to address food security challenges in the region.
PremierSolutions team-Stanley case-final memoRaktim Ray
PremierSolutions proposes two exotic options to Stanley Investments to trade on their views of future interest rates: 1) A bull call spread on the 3-month interest rate to profit if rates increase within a band. 2) A yield curve spread option to profit if the 1-year to 3-month rate spread decreases. PremierSolutions models the rates using HJM and calibrates parameters to price the options. They estimate total profit of $68,590 for Stanley by combining the options. Risks include interest rate and yield spread risk, which PremierSolutions plans to hedge by entering offsetting positions.
Home Work; Chapter 8; Forecasting Supply Chain RequirementsShaheen Sardar
Home Work; Chapter 8; Forecasting Supply Chain Requirements
Book reference: Ballou, Ronald H. (2004). “Business Logistics/ Supply Chain Management: Planning, Organizing, and Controlling the Supply Chain.” (5th Edition).
Original reference of this document: http://wweb.uta.edu/insyopma/prater/ballou08_im.pdf
PA 1c. Decision VariablesabcdCalculated values0.21110.531110.09760.docxgerardkortney
PA 1c. Decision VariablesabcdCalculated values0.21110.531110.09760.16019TotalObjective Function0.860.940.930.850.90772Constraints1111110.774-0.094-0.093-0.0850.09077>=0-0.0860.846-0.093-0.0850.40847>=0-0.086-0.0940.837-0.0851.90E-17>=0-0.086-0.094-0.0930.7650.04539>=00.94-2.790.22693>=00.86-1.86-2.00E-16>=0-0.129-0.141-0.13950.72256.90E-17>=0
a.
Let the weights be a, b, c and d to midterm, final, individual assignment and Participation respectively.
Korey would like to maximize the course grade. Therefore the course grade (Maximization):
=0.86a + 0.94b + 0.93c + 0.85d
Restrictions to course grade working: a+b+c+d=1
The weights must be non-negative, Non negativity constraints: a, b, c, d ≥ 0
The four components for each should determine 10% of the sum of the grade at least.
0.86a ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.86a ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.774a – 0.094b – 0.093c -0.085d ≥ 0
0.94b ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0846b ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.846b – 0.086a – 0.093c – 0.085d ≥ 0
0.93c ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.93c ≥ 0.086a +0.094b +0.093c + 0.085d
0.837c – 0.086a – 0.094b – 0.085d ≥ 0
0.85d ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.85d ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.765d – 0.086a – 0.094b – 0.093c ≥ 0
Here it is three times the particular assignment grade.
0.94b ≥ 3(0.93c)
0.94b ≥ 2.79c
0.94b – 2.79c ≥ 0
Midterm grade must count at least twice as much as the individual assignment score.
0.86a ≥ 2(0.93c)
0.86a ≥ 1.86c
0.86a – 1.86c ≥ 0
The presence of the grade should be less than the 15% of the whole grade.
0.85d ≤ 0.15(0.86a + 0.94b +0.93c +0.85d)
0.85d ≤ 0.129a + 0.141b +0.1395c + 0.1275d
0.7225d – 00.129a – 0.141b – 0.1395c ≥ 0
b.
The complete optimization model is Course grade (Maximization):
= 0.86a + 0.94b + 0.93c + 0.85d
a+b+c+d=1
0.774a – 0.094b - 0.093c – 0.085d ≥ 0
0.846b – 0.086a – 0.093c – 0.085d ≥ 0
0.837c – 0.086a – 0.094b – 0.085d ≥ 0
0.765d – 0.086a – 0.094b – 0.093c ≥ 0
0.94b – 2.79c ≥ 0
0.86a – 1.86c ≥ 0
0.7225d – 0.129a – 0.141b – 0.1395c ≥ 0
c.
Therefore midterm weights should be 21%, final weights 53%, individual assignment 10%, Participation should be 16%.
The maximum course grade is 90%.
PA 5b.Rosenberg Land DevelopmentDataOneTwoThreeBedroomBedroomBedroomUnitUnitUnit1BR2BR3BRAvailableConstruction cost$450,000$600,000$750,000$180,000,000Total units325Profit/ unit$45,000$60,000$75,000Minimum15%25%25%ModelTotalUnits Build4067162270Minimum406767Construction cost$18,202,247$40,449,438$121,348,315$180,000,000Contribution in profit$1,820,225$4,044,944$12,134,831$18,000,000c.ModelTotalUnits Build4981195325Minimum498181Construction cost$21,937,500$48,750,000$146,250,000$216,937,500Contribution in profit$2,193,750$4,875,000$14,625,000$21,693,750
a.
1BR = number of one bedroom units produced
2BR = number of two bedroom units produced
3BR = number of three bedroom units produced
Maximize Total Profit = $45,000 (1BR) + $60,000 (2BR) + $75,000 (3BR)
(1BR) + (2BR) + (.
A Study on the Short Run Relationship b/w Major Economic Indicators of US Eco...aurkoiitk
The objective of this study
was to develop an economic indicator system for the US
economy that will help to forecast the turning points in the
aggregate level of economic activity. Our primary concern
is to study the short run relationship between the major
economic indicators of US economy (eg: GDP, Money
Supply, Unemployment Rate, Inflation rate, Federal Fund
Rate, Exchange Rate, Government Expenditure &
Receipt, Crude Oil Price, Net Import & Export).
This document provides an outline and learning objectives for Chapter 5 of a statistics textbook on discrete distributions. The chapter will:
1. Distinguish between discrete and continuous random variables and distributions.
2. Explain how to calculate the mean and variance of discrete distributions.
3. Cover the binomial distribution and how to solve problems using it.
4. Cover the Poisson distribution and how to solve problems using it.
5. Explain how to approximate binomial problems with the Poisson distribution.
6. Cover the hypergeometric distribution and how to solve problems using it.
The document summarizes key policy options for transforming Kenya's food systems to support economic growth and prosperity under the new Bottom-Up Economic Plan. The priorities include broadening from a food security to a food systems approach, accelerating industrialization and commercialization of the food system through mechanization, productivity and value addition, expanding access for smallholders through improved fertilizer access and credit, leveraging digital technologies, improving nutrition, and empowering women's participation. The transformations aim to reduce poverty, create jobs, improve diets and harness the food system's potential to address challenges from population growth and shocks like drought and the pandemic.
This document provides an overview of the 2022/2023 cereal production figures and regional market situation in West Africa. Cereal production reached 77 million tonnes, a 7.7% increase over 2021 but below the five-year average. Key points include maize and rice production increases but deficits for countries like Chad, Gambia, and Cabo Verde. Regional markets face high demand, inflation, and insecurity impacts. Recommendations are needed to address food security challenges in the region.
PremierSolutions team-Stanley case-final memoRaktim Ray
PremierSolutions proposes two exotic options to Stanley Investments to trade on their views of future interest rates: 1) A bull call spread on the 3-month interest rate to profit if rates increase within a band. 2) A yield curve spread option to profit if the 1-year to 3-month rate spread decreases. PremierSolutions models the rates using HJM and calibrates parameters to price the options. They estimate total profit of $68,590 for Stanley by combining the options. Risks include interest rate and yield spread risk, which PremierSolutions plans to hedge by entering offsetting positions.
Home Work; Chapter 8; Forecasting Supply Chain RequirementsShaheen Sardar
Home Work; Chapter 8; Forecasting Supply Chain Requirements
Book reference: Ballou, Ronald H. (2004). “Business Logistics/ Supply Chain Management: Planning, Organizing, and Controlling the Supply Chain.” (5th Edition).
Original reference of this document: http://wweb.uta.edu/insyopma/prater/ballou08_im.pdf
PA 1c. Decision VariablesabcdCalculated values0.21110.531110.09760.docxgerardkortney
PA 1c. Decision VariablesabcdCalculated values0.21110.531110.09760.16019TotalObjective Function0.860.940.930.850.90772Constraints1111110.774-0.094-0.093-0.0850.09077>=0-0.0860.846-0.093-0.0850.40847>=0-0.086-0.0940.837-0.0851.90E-17>=0-0.086-0.094-0.0930.7650.04539>=00.94-2.790.22693>=00.86-1.86-2.00E-16>=0-0.129-0.141-0.13950.72256.90E-17>=0
a.
Let the weights be a, b, c and d to midterm, final, individual assignment and Participation respectively.
Korey would like to maximize the course grade. Therefore the course grade (Maximization):
=0.86a + 0.94b + 0.93c + 0.85d
Restrictions to course grade working: a+b+c+d=1
The weights must be non-negative, Non negativity constraints: a, b, c, d ≥ 0
The four components for each should determine 10% of the sum of the grade at least.
0.86a ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.86a ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.774a – 0.094b – 0.093c -0.085d ≥ 0
0.94b ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0846b ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.846b – 0.086a – 0.093c – 0.085d ≥ 0
0.93c ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.93c ≥ 0.086a +0.094b +0.093c + 0.085d
0.837c – 0.086a – 0.094b – 0.085d ≥ 0
0.85d ≥ 0.1 (0.86a + 0.94b + 0.93c + 0.85d)
0.85d ≥ 0.086a + 0.094b + 0.093c + 0.085d
0.765d – 0.086a – 0.094b – 0.093c ≥ 0
Here it is three times the particular assignment grade.
0.94b ≥ 3(0.93c)
0.94b ≥ 2.79c
0.94b – 2.79c ≥ 0
Midterm grade must count at least twice as much as the individual assignment score.
0.86a ≥ 2(0.93c)
0.86a ≥ 1.86c
0.86a – 1.86c ≥ 0
The presence of the grade should be less than the 15% of the whole grade.
0.85d ≤ 0.15(0.86a + 0.94b +0.93c +0.85d)
0.85d ≤ 0.129a + 0.141b +0.1395c + 0.1275d
0.7225d – 00.129a – 0.141b – 0.1395c ≥ 0
b.
The complete optimization model is Course grade (Maximization):
= 0.86a + 0.94b + 0.93c + 0.85d
a+b+c+d=1
0.774a – 0.094b - 0.093c – 0.085d ≥ 0
0.846b – 0.086a – 0.093c – 0.085d ≥ 0
0.837c – 0.086a – 0.094b – 0.085d ≥ 0
0.765d – 0.086a – 0.094b – 0.093c ≥ 0
0.94b – 2.79c ≥ 0
0.86a – 1.86c ≥ 0
0.7225d – 0.129a – 0.141b – 0.1395c ≥ 0
c.
Therefore midterm weights should be 21%, final weights 53%, individual assignment 10%, Participation should be 16%.
The maximum course grade is 90%.
PA 5b.Rosenberg Land DevelopmentDataOneTwoThreeBedroomBedroomBedroomUnitUnitUnit1BR2BR3BRAvailableConstruction cost$450,000$600,000$750,000$180,000,000Total units325Profit/ unit$45,000$60,000$75,000Minimum15%25%25%ModelTotalUnits Build4067162270Minimum406767Construction cost$18,202,247$40,449,438$121,348,315$180,000,000Contribution in profit$1,820,225$4,044,944$12,134,831$18,000,000c.ModelTotalUnits Build4981195325Minimum498181Construction cost$21,937,500$48,750,000$146,250,000$216,937,500Contribution in profit$2,193,750$4,875,000$14,625,000$21,693,750
a.
1BR = number of one bedroom units produced
2BR = number of two bedroom units produced
3BR = number of three bedroom units produced
Maximize Total Profit = $45,000 (1BR) + $60,000 (2BR) + $75,000 (3BR)
(1BR) + (2BR) + (.
A Study on the Short Run Relationship b/w Major Economic Indicators of US Eco...aurkoiitk
The objective of this study
was to develop an economic indicator system for the US
economy that will help to forecast the turning points in the
aggregate level of economic activity. Our primary concern
is to study the short run relationship between the major
economic indicators of US economy (eg: GDP, Money
Supply, Unemployment Rate, Inflation rate, Federal Fund
Rate, Exchange Rate, Government Expenditure &
Receipt, Crude Oil Price, Net Import & Export).
This document provides an outline and learning objectives for Chapter 5 of a statistics textbook on discrete distributions. The chapter will:
1. Distinguish between discrete and continuous random variables and distributions.
2. Explain how to calculate the mean and variance of discrete distributions.
3. Cover the binomial distribution and how to solve problems using it.
4. Cover the Poisson distribution and how to solve problems using it.
5. Explain how to approximate binomial problems with the Poisson distribution.
6. Cover the hypergeometric distribution and how to solve problems using it.
Model Presolve, Warmstart and Conflict Refining in CP OptimizerPhilippe Laborie
The IBM constraint programming optimization system CP Optimizer was designed to provide automatic search and a simple modeling of scheduling problems. It is used in industry for solving operational planning and scheduling problems. We present three features that we recently added to CP Optimizer to accelerate problem solving and make the solver more interactive. These are model presolve, warm-start and conflict refinement. The aim of model presolve is to reformulate and group constraints to obtain a stronger model that will be solved more rapidly. We give examples of some interesting model reformulations. Search warm-start starts search from a known - possibly incomplete - solution given by the user in order to further improve it or to help to guide the engine towards a first solution. Finally the conflict refiner helps to identify a reason for an inconsistency by providing a minimal subset of an infeasible model. All these features are illustrated on concrete examples.
This Presentation course will help you in understanding the Machine Learning model i.e. Generalized Linear Models for classification and regression with an intuitive approach of presenting the core concepts
1) The document contains calculations related to interest rates, compounding periods, and time value of money. Various interest rates, discount rates, and yields are computed.
2) Quadratic and logarithmic equations are set up and solved to relate future and present values under different interest rates and time periods.
3) Formulas are provided and derived for simple and compound interest, continuous compounding, and calculating time periods between dates.
This document describes a regression analysis conducted on data containing 97 observations of PSA levels and 7 predictor variables. Initially, a full regression model was fit using the first 65 observations. Diagnostic plots of the residuals showed some lack of randomness, indicating a need for transformation. A Box-Cox transformation with lambda=0.5 was applied to the response variable before refitting the model. The transformed model will be validated using the remaining 32 observations to select the best regression model for predicting PSA levels from this data.
This document contains output from statistical analyses performed on panel data using Stata. The analyses include:
1. Correlation analysis, pooled OLS regression, and tests for multicollinearity to examine the relationship between variables.
2. Specification error tests to check if the model is correctly specified.
3. Tests for normality of residuals to check model assumptions.
4. Panel regression using fixed effects and random effects models.
5. Tests to compare the fixed and random effects models and check for heteroskedasticity and autocorrelation.
In summary, the document analyzes relationships between variables in panel data and tests assumptions and specifications of regression models fit to the data.
I presented these slides at a meeting of ACM data mining group. I discuss using data mining to improve performance of an existing trading system. The presentation was video taped. You can see the video at:
http://fora.tv/2009/05/13/Michael_Bowles_Neural_Nets_and_Rule-Based_Trading_Systems
if you have any questions or comments contact me: mike@mbowles.com or
http://www.linkedin.com/in/mikebowles
This document contains sample data and calculations for determining statistical properties of distributions. It includes:
1) A sample data set with calculations to determine the mean, standard deviation, and normal distribution parameters.
2) A second sample data set presented as a histogram with calculations to fit both a normal and lognormal distribution.
3) Examples of using common statistical equations like the CDF and PDF for uniform and normal distributions to analyze sample data sets.
Precautionary Savings and the Stock-Bond CovarianceEesti Pank
The document discusses the relationship between stock and bond returns and how their covariance varies over time. It presents several empirical findings, including that negative stock-bond covariance is associated with lower Treasury yields, wider corporate spreads, and negative subsequent returns on stocks and bonds. It then introduces a two-factor term structure model where the price of risk varies over time, which can generate time-varying stock-bond covariance as in the data. Simulation results from the model match several key empirical moments.
This document describes a finite element analysis of a rigid jointed frame structure. It provides the geometry, material properties, boundary conditions and loading of the frame. It then outlines the steps to model the frame in finite element software, apply the loading, perform the analysis, and view the results for support reactions, axial forces, shear forces, and bending moments. Hand calculations are also shown to verify the analysis results.
This document provides an overview of demand forecasting methods. It discusses qualitative and quantitative forecasting models, including time series analysis techniques like moving averages, exponential smoothing, and adjusting for trends and seasonality. It also covers causal models using linear regression. Key steps in forecasting like selecting a model, measuring accuracy, and choosing software are outlined. The homework assigns practicing examples on least squares, moving averages, and exponential smoothing from a textbook.
Feature scaling is a technique used in machine learning to standardize the range of independent variables or features of data. There are several common feature scaling methods including standardization, min-max scaling, and mean normalization. Standardization transforms the data to have a mean of 0 and standard deviation of 1. Min-max scaling scales features between 0 and 1. Mean normalization scales the mean value to zero. The document then provides the formulas and R code examples for implementing each of these scaling methods.
This document discusses probability distributions and related concepts. It begins by defining key terms like probability distribution, random variable, discrete and continuous distributions. It then focuses on several specific discrete probability distributions - binomial, hypergeometric, and Poisson. For each, it provides the characteristics and formulas for calculating probabilities. Several examples are worked through to demonstrate calculating probabilities, means, variances and more for problems that fit each distribution.
Home Work; Chapter 9; Inventory Policy DecisionsShaheen Sardar
Book reference: Ballou, Ronald H. (2004). “Business Logistics/ Supply Chain Management: Planning, Organizing, and Controlling the Supply Chain.” (5th Edition).
Original reference of this document: http://wweb.uta.edu/insyopma/prater/ballou09_im.pdf
Week 2 Individual Assignment 2: Quantitative Analysis of Credit -
Solution
s
This assignment is based on the data we used during our two live sessions, but it has been updated to include a splitting variable (credit2.xlsx). In the spreadsheet under the tab “Data," you will find data
pertaining to 1,000 personal loan accounts. The tab “Data Dictionary” contains a description of what the various variables mean.
As a part of a new credit application, the company collects information about the applicant. The company then decides an amount of the credit extended (the variable CREDIT_EXTENDED). For these 1,000 accounts, we also have information on how profitable each account turned out to be (the variable NPV). A negative value indicates a net loss, and this typically happens when the debtor defaults on his/her payments.
The goal in this assignment is to investigate how one can use this data to better manage the bank's credit extension program. Specifically, our goal is to develop a classification model to classify a new credit account as “profitable” or “not profitable." Secondly we want to compare its performance in the context of decision support to a linear regression model that predicts NPV directly.
Please answer all the questions. Supply supporting documentation and show calculations as
needed. Please submit a single, well-formatted PDF or Word file. The instructor should not need to go searching for your answers! In addition, please upload an Excel file with your model outputs – the file will not be graded, but will help the instructor give you feedback, if your model differs substantially from the solutions.
For extra assistance, you may want to access the tutorials located on the course resource center page.Data Preparation
The data preparation repeats the steps from the live session:
a) The goal is to predict whether or not a new credit will result in a profitable account. Create a new variable to use as the dependent variable.
b) Create dummy variables for all categorical variables with more than 2 values (or if you prefer, you can sort your variables into numerical and categorical when you run the model).
c) Split the data into 2 parts using the splitting variable that has been added to the data set. This is to ensure a more balanced split between the validation and training samples. Note that Analytic Solver Data Mining only allows 50 columns in the analysis, so leave out your base dummies (if you created them) when partitioning. After the data partition, you should have 666 rows in your training data and 334 in your validation data.
The Assignment
1. Applying Logistic Regression
If one fits a Logistic Regression Model using all the independent variables, one observes a) a gap in the classification performance between the training data and the validation data, and b) very
high p-values for some of the variables. The performance gap between the training and validation may be a sign of overfitting, and the high p-values may b ...
The document demonstrates various operations that can be performed on vectors and data frames in R. It shows how to create, subset, reorder, and modify vectors and data frames. Key operations include subsetting vectors and data frames using indices or logical vectors, applying functions to entire vectors or selected elements, and reordering a data frame based on the values in one of its columns.
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 The Binary Logistic...J. García - Verdugo
The document discusses binary logistic regression and provides an example. It analyzes data from a study of 100 men investigating the relationship between age and risk of coronary heart disease. Logistic regression is used to estimate the effect of age on the probability of disease. The analysis finds that for each one year increase in age, the odds of disease increase by 13% (odds ratio of 1.13).
The document provides important facts and formulae related to decimal fractions. Some key points:
- Decimal fractions have denominators that are powers of 10 (e.g. 1/10 = 0.1)
- To convert a decimal to a vulgar fraction, write the denominator as successive powers of 10 and remove the decimal point
- Operations like addition, subtraction, multiplication and division can be performed on decimal fractions by considering the number of decimal places
- A recurring decimal has a digit or set of digits that repeat continuously, and is expressed using a bar or dot above the recurring digits
- Pure and mixed recurring decimals can be converted to vulgar fractions using different methods outlined in the document
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
The document discusses using a normal distribution to analyze inventory data from multiple years to better understand inventory levels. It includes calculations of the mean, median, standard deviation, and other measures to compare the data sets and determine which metrics most accurately depict central tendency and dispersion. The analysis would help improve the company's inventory system by better modeling demand variability based on historical data.
Problem 1(a) Complete the following ANOVA table based on 20 obs.docxChantellPantoja184
Problem 1:
(a) Complete the following ANOVA table based on 20 observations for the regression equation
(a) Is the overall regression significant? Fill in the missing values in the table.
Source DF SS MS F
Regression ___ 350 ____ ____
Error ___ _____
Total 500
(b) Suppose that you have computed the following sequential sums of squares due to regression:
Regressor Variables in Model SS Regression
………………………………………. 300
……………………………………… 250
…………………………………….. 340
……………………………………. 325
Fill in the missing values in the following “computer output”:
Source DF Partial SS F-value Pr>F
……………………………………………………………………………………….. 0.1245
………………………………………………………………………………………. 0.3841
………………………………………………………………………………………. 0.0042
………………………………………………………………………………………. 0.0401
Problem 2:
The time required for a merchandise to stock a grocery store shelf with a soft drink product as well as the number of cases of product stocked are given below. Consider a linear regression of delivery time against number of cases.
X=number of cases
Y=delivery time
Delivery time number of cases Hat diagonals
1.41 4 0.5077
2.96 6 0.3907
6.04 14 0.2013
7.57 19 0.3092
9.38 24 0.5912
Observations used L.S. Model
4,6,14,19,24
6,14,19,24
4,14,19,24
4,14,19,24
4,6,14,24
4,6,14,19
(a)
Calculate the PRESS statistic for the model .
(b) Calculate the regular residual for the model above. Then, compare these residuals with the PRESS residuals for this model.
Exercises from the Text
Use SAS whenever possible to do these exercises:
# 3.4 on p 122
# 3.5
# 3.8
# 3.15
# 3.21
# 3.27
# 3.28
# 3.31
# 3.38
# 3.39
Example with SAS on Sequential and Partial Sum of Squares
Data Weather;
Title 'Lows and Highs from N&O Jan 28,29,30 1992';
Title2 'using actual numbers (yesterday values)';
input city $ hi2 lo2 yhi ylo thi tlo;
* Mon Tues Wed ;
cards;
seattle 51 44 52 44 59 47
.
.
.
;
proc reg; model thi = yhi hi2 tlo ylo lo2/ss1 ss2;
test tlo=0, ylo=0, lo2=0;
/*-----------------------------------------------
| Showing sequential and partial sums of squares|
| Note t**2 = F relationship for partial F. By |
| hand, construct F to leave out .
Model Presolve, Warmstart and Conflict Refining in CP OptimizerPhilippe Laborie
The IBM constraint programming optimization system CP Optimizer was designed to provide automatic search and a simple modeling of scheduling problems. It is used in industry for solving operational planning and scheduling problems. We present three features that we recently added to CP Optimizer to accelerate problem solving and make the solver more interactive. These are model presolve, warm-start and conflict refinement. The aim of model presolve is to reformulate and group constraints to obtain a stronger model that will be solved more rapidly. We give examples of some interesting model reformulations. Search warm-start starts search from a known - possibly incomplete - solution given by the user in order to further improve it or to help to guide the engine towards a first solution. Finally the conflict refiner helps to identify a reason for an inconsistency by providing a minimal subset of an infeasible model. All these features are illustrated on concrete examples.
This Presentation course will help you in understanding the Machine Learning model i.e. Generalized Linear Models for classification and regression with an intuitive approach of presenting the core concepts
1) The document contains calculations related to interest rates, compounding periods, and time value of money. Various interest rates, discount rates, and yields are computed.
2) Quadratic and logarithmic equations are set up and solved to relate future and present values under different interest rates and time periods.
3) Formulas are provided and derived for simple and compound interest, continuous compounding, and calculating time periods between dates.
This document describes a regression analysis conducted on data containing 97 observations of PSA levels and 7 predictor variables. Initially, a full regression model was fit using the first 65 observations. Diagnostic plots of the residuals showed some lack of randomness, indicating a need for transformation. A Box-Cox transformation with lambda=0.5 was applied to the response variable before refitting the model. The transformed model will be validated using the remaining 32 observations to select the best regression model for predicting PSA levels from this data.
This document contains output from statistical analyses performed on panel data using Stata. The analyses include:
1. Correlation analysis, pooled OLS regression, and tests for multicollinearity to examine the relationship between variables.
2. Specification error tests to check if the model is correctly specified.
3. Tests for normality of residuals to check model assumptions.
4. Panel regression using fixed effects and random effects models.
5. Tests to compare the fixed and random effects models and check for heteroskedasticity and autocorrelation.
In summary, the document analyzes relationships between variables in panel data and tests assumptions and specifications of regression models fit to the data.
I presented these slides at a meeting of ACM data mining group. I discuss using data mining to improve performance of an existing trading system. The presentation was video taped. You can see the video at:
http://fora.tv/2009/05/13/Michael_Bowles_Neural_Nets_and_Rule-Based_Trading_Systems
if you have any questions or comments contact me: mike@mbowles.com or
http://www.linkedin.com/in/mikebowles
This document contains sample data and calculations for determining statistical properties of distributions. It includes:
1) A sample data set with calculations to determine the mean, standard deviation, and normal distribution parameters.
2) A second sample data set presented as a histogram with calculations to fit both a normal and lognormal distribution.
3) Examples of using common statistical equations like the CDF and PDF for uniform and normal distributions to analyze sample data sets.
Precautionary Savings and the Stock-Bond CovarianceEesti Pank
The document discusses the relationship between stock and bond returns and how their covariance varies over time. It presents several empirical findings, including that negative stock-bond covariance is associated with lower Treasury yields, wider corporate spreads, and negative subsequent returns on stocks and bonds. It then introduces a two-factor term structure model where the price of risk varies over time, which can generate time-varying stock-bond covariance as in the data. Simulation results from the model match several key empirical moments.
This document describes a finite element analysis of a rigid jointed frame structure. It provides the geometry, material properties, boundary conditions and loading of the frame. It then outlines the steps to model the frame in finite element software, apply the loading, perform the analysis, and view the results for support reactions, axial forces, shear forces, and bending moments. Hand calculations are also shown to verify the analysis results.
This document provides an overview of demand forecasting methods. It discusses qualitative and quantitative forecasting models, including time series analysis techniques like moving averages, exponential smoothing, and adjusting for trends and seasonality. It also covers causal models using linear regression. Key steps in forecasting like selecting a model, measuring accuracy, and choosing software are outlined. The homework assigns practicing examples on least squares, moving averages, and exponential smoothing from a textbook.
Feature scaling is a technique used in machine learning to standardize the range of independent variables or features of data. There are several common feature scaling methods including standardization, min-max scaling, and mean normalization. Standardization transforms the data to have a mean of 0 and standard deviation of 1. Min-max scaling scales features between 0 and 1. Mean normalization scales the mean value to zero. The document then provides the formulas and R code examples for implementing each of these scaling methods.
This document discusses probability distributions and related concepts. It begins by defining key terms like probability distribution, random variable, discrete and continuous distributions. It then focuses on several specific discrete probability distributions - binomial, hypergeometric, and Poisson. For each, it provides the characteristics and formulas for calculating probabilities. Several examples are worked through to demonstrate calculating probabilities, means, variances and more for problems that fit each distribution.
Home Work; Chapter 9; Inventory Policy DecisionsShaheen Sardar
Book reference: Ballou, Ronald H. (2004). “Business Logistics/ Supply Chain Management: Planning, Organizing, and Controlling the Supply Chain.” (5th Edition).
Original reference of this document: http://wweb.uta.edu/insyopma/prater/ballou09_im.pdf
Week 2 Individual Assignment 2: Quantitative Analysis of Credit -
Solution
s
This assignment is based on the data we used during our two live sessions, but it has been updated to include a splitting variable (credit2.xlsx). In the spreadsheet under the tab “Data," you will find data
pertaining to 1,000 personal loan accounts. The tab “Data Dictionary” contains a description of what the various variables mean.
As a part of a new credit application, the company collects information about the applicant. The company then decides an amount of the credit extended (the variable CREDIT_EXTENDED). For these 1,000 accounts, we also have information on how profitable each account turned out to be (the variable NPV). A negative value indicates a net loss, and this typically happens when the debtor defaults on his/her payments.
The goal in this assignment is to investigate how one can use this data to better manage the bank's credit extension program. Specifically, our goal is to develop a classification model to classify a new credit account as “profitable” or “not profitable." Secondly we want to compare its performance in the context of decision support to a linear regression model that predicts NPV directly.
Please answer all the questions. Supply supporting documentation and show calculations as
needed. Please submit a single, well-formatted PDF or Word file. The instructor should not need to go searching for your answers! In addition, please upload an Excel file with your model outputs – the file will not be graded, but will help the instructor give you feedback, if your model differs substantially from the solutions.
For extra assistance, you may want to access the tutorials located on the course resource center page.Data Preparation
The data preparation repeats the steps from the live session:
a) The goal is to predict whether or not a new credit will result in a profitable account. Create a new variable to use as the dependent variable.
b) Create dummy variables for all categorical variables with more than 2 values (or if you prefer, you can sort your variables into numerical and categorical when you run the model).
c) Split the data into 2 parts using the splitting variable that has been added to the data set. This is to ensure a more balanced split between the validation and training samples. Note that Analytic Solver Data Mining only allows 50 columns in the analysis, so leave out your base dummies (if you created them) when partitioning. After the data partition, you should have 666 rows in your training data and 334 in your validation data.
The Assignment
1. Applying Logistic Regression
If one fits a Logistic Regression Model using all the independent variables, one observes a) a gap in the classification performance between the training data and the validation data, and b) very
high p-values for some of the variables. The performance gap between the training and validation may be a sign of overfitting, and the high p-values may b ...
The document demonstrates various operations that can be performed on vectors and data frames in R. It shows how to create, subset, reorder, and modify vectors and data frames. Key operations include subsetting vectors and data frames using indices or logical vectors, applying functions to entire vectors or selected elements, and reordering a data frame based on the values in one of its columns.
Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 The Binary Logistic...J. García - Verdugo
The document discusses binary logistic regression and provides an example. It analyzes data from a study of 100 men investigating the relationship between age and risk of coronary heart disease. Logistic regression is used to estimate the effect of age on the probability of disease. The analysis finds that for each one year increase in age, the odds of disease increase by 13% (odds ratio of 1.13).
The document provides important facts and formulae related to decimal fractions. Some key points:
- Decimal fractions have denominators that are powers of 10 (e.g. 1/10 = 0.1)
- To convert a decimal to a vulgar fraction, write the denominator as successive powers of 10 and remove the decimal point
- Operations like addition, subtraction, multiplication and division can be performed on decimal fractions by considering the number of decimal places
- A recurring decimal has a digit or set of digits that repeat continuously, and is expressed using a bar or dot above the recurring digits
- Pure and mixed recurring decimals can be converted to vulgar fractions using different methods outlined in the document
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
The document discusses using a normal distribution to analyze inventory data from multiple years to better understand inventory levels. It includes calculations of the mean, median, standard deviation, and other measures to compare the data sets and determine which metrics most accurately depict central tendency and dispersion. The analysis would help improve the company's inventory system by better modeling demand variability based on historical data.
Problem 1(a) Complete the following ANOVA table based on 20 obs.docxChantellPantoja184
Problem 1:
(a) Complete the following ANOVA table based on 20 observations for the regression equation
(a) Is the overall regression significant? Fill in the missing values in the table.
Source DF SS MS F
Regression ___ 350 ____ ____
Error ___ _____
Total 500
(b) Suppose that you have computed the following sequential sums of squares due to regression:
Regressor Variables in Model SS Regression
………………………………………. 300
……………………………………… 250
…………………………………….. 340
……………………………………. 325
Fill in the missing values in the following “computer output”:
Source DF Partial SS F-value Pr>F
……………………………………………………………………………………….. 0.1245
………………………………………………………………………………………. 0.3841
………………………………………………………………………………………. 0.0042
………………………………………………………………………………………. 0.0401
Problem 2:
The time required for a merchandise to stock a grocery store shelf with a soft drink product as well as the number of cases of product stocked are given below. Consider a linear regression of delivery time against number of cases.
X=number of cases
Y=delivery time
Delivery time number of cases Hat diagonals
1.41 4 0.5077
2.96 6 0.3907
6.04 14 0.2013
7.57 19 0.3092
9.38 24 0.5912
Observations used L.S. Model
4,6,14,19,24
6,14,19,24
4,14,19,24
4,14,19,24
4,6,14,24
4,6,14,19
(a)
Calculate the PRESS statistic for the model .
(b) Calculate the regular residual for the model above. Then, compare these residuals with the PRESS residuals for this model.
Exercises from the Text
Use SAS whenever possible to do these exercises:
# 3.4 on p 122
# 3.5
# 3.8
# 3.15
# 3.21
# 3.27
# 3.28
# 3.31
# 3.38
# 3.39
Example with SAS on Sequential and Partial Sum of Squares
Data Weather;
Title 'Lows and Highs from N&O Jan 28,29,30 1992';
Title2 'using actual numbers (yesterday values)';
input city $ hi2 lo2 yhi ylo thi tlo;
* Mon Tues Wed ;
cards;
seattle 51 44 52 44 59 47
.
.
.
;
proc reg; model thi = yhi hi2 tlo ylo lo2/ss1 ss2;
test tlo=0, ylo=0, lo2=0;
/*-----------------------------------------------
| Showing sequential and partial sums of squares|
| Note t**2 = F relationship for partial F. By |
| hand, construct F to leave out .
Problem 1(a) Complete the following ANOVA table based on 20 obs.docx
Bionomial Options Pricing
1. GROUP PROJECT 2
GROUP 28
Group Member Student No. Email
C Rohith Thatchan 3035324950 rohithchokka@gmail.com
GUO, Gunan 3035324106 ggn1994@sina.cn
HU, Zeyu 3035323932 ryan.hzy@outlook.com
JIANG, Wei 3035236335 samuel.w.jiang@gmail.com
YAN, Yu Sze 3035268455 yanyusze@gmail.com
OCTOBER 1, 2016
2. Part (a)
Use n = 1, 5, 10, 25, or 50 (correspondingly, h = 50/365, 10/365, 5/365, 2/365, or 1/365) in binomial option model
to calculate the option value. Take the risk-neutral pricing approach instead of constructing the complete
binomial trees.
The option value is calculated as follows using risk-neutral pricing approach:
Number of binomial steps (n) 1 5 10 25 50
Length of one period (h) 0.1370 0.0274 0.0137 0.0055 0.0027
Risk-neutral probability (p) 0.4539 0.4793 0.4854 0.4908 0.4935
Size of up move (u) 1.2116 1.0878 1.0610 1.0380 1.0267
Size of down move (d) 0.8368 0.9218 0.9438 0.9639 0.9743
Option Value (C) 1.4565 1.1540 1.1455 1.1083 1.1181
Part (b)
Use the Black-Scholes formula to calculate the option value.
Given Black-Scholes formula, the option value is calculated as follow:
. ∗ . ∗
.
0.13411 N(d ) = 0.44666
0.5 = -0.31917 N(d ) = 0.37480
, , . , . ,
,
. ∗
1.11609
Part (c)
Using the functions for option values and option Greeks available in the CD-ROM accompanying the textbook,
calculate the option value (to confirm the answer above) and four option Greeks (delta, gamma, theta and vega).
Given 6 parameters: 1) price of underlying asset; 2) strike price; 3) volatility; 4) continuously compounded risk-free
interest rate; 5) continuously compounded dividend yield; and 6) time to expiration. The option values and 4 option
Greeks are as follows:
Item Function (S, K, σ, r, T, δ) Outputs
Option Value BSCall (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 1.116093
Delta BSCallDelta (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.446659
Gamma BSCallGamma (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.106823
Theta BSCallTheta (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) -0.015704
Vega BSCallVega (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.029267
3. Part (d)
Generate a random sequence of stock prices for 50 days.
A set of 50 random numbers (z) following standard normal distribution (i.e. mean=0, standard deviation=1) is generated
with “Generate Random Number” function in Microsoft Excel.
50 random numbers (z) following standard normal distribution
1.168005 1.742483 -0.926711 0.449127 -0.330140
0.685358 -0.560838 -0.983819 0.248989 0.241342
-0.605913 -1.142803 -0.571795 -1.234511 -0.054072
-0.509372 0.378445 0.030412 -0.597661 -1.321650
0.292239 0.422042 -0.799669 0.565680 -0.239610
0.617906 0.014574 -2.168090 0.811629 -1.044061
0.947120 1.269937 0.512509 0.962325 0.518362
0.240318 0.958441 -1.682820 1.465792 1.624012
-1.200706 -0.153560 0.676869 0.902423 -0.900011
0.639250 1.140747 -0.638594 0.670534 1.754834
Given defined parameters = 20%, = 0, = 50%, = 1/365, a sequence of stock prices for 50 days can be
generated by inputting 50 random numbers (z) in the following equation:
Stock prices for 50 days with random numbers (z)
20.625039 22.267185 23.145855 20.255206 22.223524
21.002636 21.947248 22.562145 20.391816 22.368933
20.676462 21.304935 22.231594 19.747571 22.341891
20.406849 21.521417 22.253868 19.445088 21.586751
20.567749 21.764918 21.797450 19.739160 21.456215
20.907357 21.777695 20.599298 20.167073 20.882166
21.436474 22.518282 20.881747 20.685685 21.171737
21.576155 23.095009 19.986150 21.499054 22.095522
20.912986 23.007107 20.347529 22.017375 21.585592
21.270172 23.709205 20.014403 22.411765 22.604695
4. Part (e)
Based on the stock price series generated in (d), calculate the option value and the four option Greeks from day
1 to day 50. Note that as time goes by, the maturity of the option drops to zero. Plot the option value from now to
maturity.
The option values can be obtained with Black-Scholes option pricing models, given parameter S from Part (d), K = 21,
T = (50 − )/365, = 50%, r = 5% and = 0. Option values and 4 option Greeks are as follows:
Random
number z
Stock
price
Strike
price
Time to
maturity
Option
value
Delta Gamma Vega Theta
N/A 20 21.000000 0.136986 1.116093 0.446659 0.106823 0.029267 -0.015704
1.168005 20.625039 21.000000 0.134247 1.399480 0.511924 0.105536 0.030134 -0.016629
0.685358 21.002636 21.000000 0.131507 1.583263 0.550773 0.103910 0.030139 -0.017065
-0.605913 20.676462 21.000000 0.128767 1.392270 0.515578 0.107456 0.029577 -0.017002
-0.509372 20.406849 21.000000 0.126027 1.240388 0.485176 0.110061 0.028881 -0.016883
0.292239 20.567749 21.000000 0.123288 1.302721 0.501766 0.110481 0.028811 -0.017241
0.617906 20.907357 21.000000 0.120548 1.461829 0.538260 0.109410 0.028826 -0.017720
0.947120 21.436474 21.000000 0.117808 1.743732 0.594835 0.105364 0.028520 -0.018089
0.240318 21.576155 21.000000 0.115068 1.809666 0.609612 0.104874 0.028089 -0.018274
-1.200706 20.912986 21.000000 0.112329 1.410809 0.536860 0.113349 0.027843 -0.018322
0.639250 21.270172 21.000000 0.109589 1.591143 0.576558 0.111222 0.027572 -0.018694
1.742483 22.267185 21.000000 0.106849 2.200790 0.681858 0.098023 0.025966 -0.018423
-0.560838 21.947248 21.000000 0.104110 1.968920 0.650401 0.104567 0.026219 -0.018935
-1.142803 21.304935 21.000000 0.101370 1.554129 0.580039 0.115251 0.026515 -0.019395
0.378445 21.521417 21.000000 0.098630 1.662871 0.604923 0.113943 0.026026 -0.019629
0.422042 21.764918 21.000000 0.095890 1.793866 0.632866 0.111757 0.025382 -0.019771
0.014574 21.777695 21.000000 0.093151 1.782071 0.634995 0.113103 0.024983 -0.020020
1.269937 22.518282 21.000000 0.090411 2.262944 0.715508 0.100197 0.022968 -0.019297
0.958441 23.095009 21.000000 0.087671 2.673433 0.772154 0.088341 0.020655 -0.018213
-0.153560 23.007107 21.000000 0.084932 2.587383 0.766826 0.091269 0.020516 -0.018607
1.140747 23.709205 21.000000 0.082192 3.130237 0.828138 0.074979 0.017321 -0.016695
-0.926711 23.145855 21.000000 0.079452 2.657848 0.784941 0.089586 0.019066 -0.018561
-0.983819 22.562145 21.000000 0.076712 2.195519 0.730740 0.105677 0.020634 -0.020381
-0.571795 22.231594 21.000000 0.073973 1.938626 0.696473 0.115612 0.021134 -0.021424
0.030412 22.253868 21.000000 0.071233 1.932635 0.701247 0.116858 0.020612 -0.021692
-0.799669 21.797450 21.000000 0.068493 1.602428 0.646698 0.130299 0.021202 -0.022913
-2.168090 20.599298 21.000000 0.065753 0.903729 0.475873 0.150776 0.021034 -0.023130
0.512509 20.881747 21.000000 0.063014 1.020328 0.517096 0.152075 0.020893 -0.024049
-1.682820 19.986150 21.000000 0.060274 0.596879 0.375554 0.154632 0.018615 -0.022100
0.676869 20.347529 21.000000 0.057534 0.719110 0.428881 0.160876 0.019161 -0.023907
-0.638594 20.014403 21.000000 0.054795 0.562276 0.371160 0.161345 0.017707 -0.023074
0.449127 20.255206 21.000000 0.052055 0.632001 0.406449 0.167883 0.017927 -0.024630
0.248989 20.391816 21.000000 0.049315 0.663650 0.425848 0.173143 0.017753 -0.025755
-1.234511 19.747571 21.000000 0.046575 0.402028 0.310537 0.165686 0.015047 -0.022912
-0.597661 19.445088 21.000000 0.043836 0.294639 0.254122 0.157462 0.013049 -0.021026
0.565680 19.739160 21.000000 0.041096 0.352834 0.294628 0.172350 0.013799 -0.023746
0.811629 20.167073 21.000000 0.038356 0.468047 0.365142 0.190358 0.014848 -0.027459
6. Part (f)
Assume that you are the market maker who just issued 1 million units of call option. To hedge your short option
position, you take a long position on the synthetic call and rebalance your account daily (daily delta hedging).
Since hedging is not continuous but daily, there will be resulted hedging profit or loss every day. Prepare a
spreadsheet showing the following information over the entire period: stock price, option value, option delta,
your hedging position (stock position and money market position), and the hedging profit or loss. Compute the
cumulative hedging profit or loss (with interest) over the entire period and express it as a percentage of the
original call premium.
In this case, Delta is changing on a daily basis, and Delta-hedging position has to be rebalanced accordingly. Hedging
profit and loss may be a result of these adjustment. Given daily stock prices, call option value and Delta, we may
calculate the position of stock, amount of borrowed money, gain or loss from stock and option, overnight interest. The
overnight net profit can be calculated by subtracting interest payment from gain of stock and option position. By adding
up the future value of overnight net profit, we can get the cumulative net hedging profit.
Over 50-days period, the cumulative net hedging profit for the is $ 67,671.50, which is 6.06% of original call premium.
Day Gain/(loss)from
stock position
Gain/(loss) From
option position
Overnight
Interest paid
Over Night Net
Profit
Net Profit at
Maturity
1 279,179.50 (283,387.19) (1,070.91) (5,278.60) (5,314.15)
2 193,300.84 (183,783.23) (1,254.74) 8,262.88 8,317.39
3 (179,647.93) 190,993.45 (1,367.82) 9,977.70 10,042.14
4 (139,006.61) 151,881.82 (1,269.68) 11,605.53 11,678.89
5 78,065.18 (62,333.25) (1,186.46) 14,545.47 14,635.41
6 170,403.59 (159,108.22) (1,235.36) 10,060.01 10,120.83
7 284,802.61 (281,902.07) (1,341.43) 1,559.11 1,568.32
8 83,086.94 (65,934.33) (1,507.97) 15,644.63 15,734.90
9 (404,275.09) 398,857.19 (1,554.00) (6,971.89) (7,011.16)
10 191,758.98 (180,334.07) (1,344.82) 10,080.09 10,135.47
11 574,835.48 (609,647.29) (1,462.06) (36,273.87) (36,468.18)
12 (218,151.44) 231,870.12 (1,778.51) 11,940.16 12,002.48
13 (417,761.30) 414,790.65 (1,685.81) (4,656.46) (4,680.12)
14 125,568.07 (108,741.93) (1,480.04) 15,346.10 15,421.97
15 147,299.14 (130,995.05) (1,555.71) 14,748.38 14,819.26
16 8,086.30 11,795.36 (1,641.26) 18,240.40 18,325.55
17 470,269.22 (480,872.92) (1,650.34) (12,254.04) (12,309.56)
18 412,652.93 (410,489.65) (1,897.26) 266.02 267.19
19 (67,874.02) 86,050.16 (2,076.78) 16,099.35 16,167.86
20 538,386.98 (542,854.13) (2,062.48) (6,529.62) (6,556.51)
21 (466,532.04) 472,389.80 (2,261.01) 3,596.75 3,611.07
22 (458,178.42) 462,328.54 (2,124.84) 2,025.28 2,033.06
23 (241,547.00) 256,892.89 (1,957.88) 13,388.01 13,437.62
24 15,513.27 5,991.52 (1,855.62) 19,649.17 19,719.28
25 (320,061.34) 330,206.40 (1,873.12) 8,271.94 8,300.32
26 (774,842.76) 698,699.32 (1,711.62) (77,855.06) (78,111.44)
27 134,410.23 (116,598.58) (1,219.11) 16,592.54 16,644.90
28 (463,110.35) 423,448.96 (1,339.48) (41,000.87) (41,124.62)
29 135,717.21 (122,231.78) (946.50) 12,538.92 12,575.05
8. Part (g)
Repeat steps (d), (e), and (f) 50 times. Each time, you get the cumulative hedging profit or loss as a percentage of
the original call premium. Compile a list of these numbers and prepare a table of summary statistics (mean,
standard deviation, maximum, etc.). For this repetition job, show me the list and the table only (not the 50
spreadsheets). Discuss what you have learned about pricing and risk management from the simulation.
We have obtained the following Summary Statistics by repeating steps (d), (e), and (f) for 50 times.
Summary Statistics
Mean 0.036497
Standard Error 0.024155
Median 0.032494
Standard Deviation 0.170799
Sample Variance 0.029172
Kurtosis 0.437200
Skewness -0.076208
Range 0.803712
Minimum -0.406759
Maximum 0.396954
Sum 1.824844
Count 50
t-value 1.510971
t critical (two-tailed) 2.009575
Implications on option pricing
The calculation of option price with Black-Scholes option pricing model requires input of 6 parameters: 1) price of
underlying asset; 2) strike price; 3) volatility; 4) continuously compounded risk-free interest rate; 5) continuously
compounded dividend yield; and 6) time to expiration. Options prices are affected by the 6 parameters.
1) Price of underlying asset
The underlying asset is stock in this case. From the simulation in Part (e), we may find that as the price of stock
increases, call prices increase. Conversely, as the price of stock, call prices decrease. An increase in stock price
increases possibility of exercising the call option and also increase the profit of exercising it. This relationship can
also be demonstrated by the positive deltas in Part (e).
2) Strike price
The call option prices increase as strike price decrease. For otherwise identical call options, the lower strike price
call option has a higher option price since the call option are more likely to be exercised and the profit if exercised
is larger.
3) Volatility of underlying asset
Call option price increases as volatility increases. The call option eliminates downside risk of the stock price by
locking the buy price. In this case, a higher degree of volatility is favorable to investors since the potential profits
can be larger but potential loss remains the same. This relationship can also be demonstrated by the positive vegas
in Part (e).
9. 4) Continuously compounded risk-free interest rate
If interest rates rise, call option price will increase. Conversely, if interest rate drop, call option prices will increase.
We may calculate Rho to demonstrate this relationship. Rho should be positive in this case.
5) Continuously compounded dividend yield
If the underlying's dividend increases, call prices will decrease. Conversely, if the underlying's dividend decreases,
call prices will increase. Dividends can affect option prices because the underlying stock's price typically drops by
the amount of any cash dividend on the ex-dividend date. We may calculate Psi to demonstrate this relationship. Psi
should be negative in this case.
6) Time to expiration
The call option price decreases as expiration approaches. The shorter period of time an option has until expiration,
the less the chance that it will end up in-the-money, or profitable. This relationship can also be demonstrated by the
negative thetas in Part (e).
Implications on risk management
Market makers would hedge a short position in call options by using delta hedging strategy, aiming to hedge the risk
associated with price movements in stocks. The short call option position can be hedged, for one unit call option, by
longing delta units of stocks and borrowing money, amount of which is equal to the difference between one unit of
option and delta units of stocks. We notice that interest cost is important in delta hedging analysis. Stock purchase is
funded by borrowed money and interest expense incurred decreases overall profit. The magnitude depends on the
amount of money borrowed and interest rate.
We also run a t-test, setting the null hypothesis that the cumulative return is zero. The t value equals to 1.510971 and
critical value of 95% confidence level with 49 degrees of freedom in t-test is ± 2.009575, which means we cannot reject
the null hypothesis at 95% confidence level. This conclusion is consistent with the conception that market makers should
break even in average.
In addition, we notice that if stock price increases, the market maker, who takes a short position in call option and uses
delta hedging strategy in this case, will have loss on option position and gain on stock position. When stock price changes
significantly, gamma increase and delta hedging is not enough to hedge risks, therefore, the market maker might consider
gamma hedging by purchasing certain amount of deep-out-of money calls and puts.