FINC621 Winter 2018-19 Lab Worksheet 06

Distributions, Sampling Methods & Monte Carlo Simulation (finc621-lab06)

About

In this worksheet we look at different distribution functions, sampling methods, and probability calculations. Next we consider a calculation of a Europen option using Monte Carlo simulation, and compare results to calculation using Black-Scholes.

Setup

Remember to always set your working directory to the source file location. Go to ‘Session’, scroll down to ‘Set Working Directory’, and click ‘To Source File Location’. Read carefully the below and follow the instructions to complete the tasks and answer any questions. Submit your work to RPubs as detailed in previous notes.

Note

Always read carefully the instructions on Sakai. For clarity, tasks/questions to be completed/answered are highlighted in red color (visible in preview) and numbered according to their particular placement in the task section. Quite often you will need to add your own code chunk.

Execute all code chunks, preview, publish, and submit link on Sakai follwoing the naming convention. Make sure to add comments to your code where appropriate. Use own language!

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Task 1: Distribution, Sampling & Probability

#Install package quantmod 
if(!require("quantmod",quietly = TRUE))
  install.packages("quantmod",dependencies = TRUE, repos = "https://cloud.r-project.org")

Consider the free cash flow example with an estimated range set by a minimum of -$25M and maximum $275M. We start by generating a sample random set. Following is an example to generate a random set of 10 data points from a uniform distribution with probability \(\frac{1}{b-a}\), a minimum of a, and a maximum of b.

ud = runif(10, min=5, max=15) # runif is in reference to uniform distribution
# Other R functions needed for this task are hist(), mean(), sd(), punif(), runif(),  pnorm(), and rnorm(). Check the Help command in R for more details about these functions.

##### 1A) Assuming a uniform probability distribution, generate a sample of 100 random observations (or deviates) representing the cash flows

ud = runif(100, min=-25, max=275) # runif is in reference to uniform distribution

Next we plot a histogram to describe the generated sample data, and calculate the mean and standard deviation of the sample using the proper R-functions ##### 1B) Plot a histogram and calculate the mean and standard deviation of your sample. Compare the mean and standard deviation of the sample to the theoretical values obtained using the proper formulas.

hist(ud)

mean(ud)

[1] 131.8009

sd(ud)

[1] 76.92596

the mean and standard deviation calculated from the R language are 131.8 and 76.9 respectively. if we use the mathmatical formula, mean=(275+25)/2=150 and standard deviation=86.6 ,the are not same and not very close. ##### 1C) Repeat 1A & 1B above by increasing the sample size to 1000. Share your insights.

Given the characteristics of a probability distribution we should be able to compute various probability scenarios using the proper functions in R.

td = runif(1000, min=-25, max=275) # runif is in reference to uniform distribution,and td is another data set
hist(td)

mean(td)

[1] 123.8558

sd(td)

[1] 86.59113

if we increase the sample size up to 1000, the mean and standard deviation are 123.9 and 86.8 respectively, much closer to the theoretical value. ##### 1D) Calculate the probability that the free cash flow is negative prob(negative)=(0–25)/(275+25)=8.33% We will now repeat the above exercises 1A, 1B, 1C using instead a normal distribution.

##### 1E) Repeat steps A-C for the case of a portfolio daily returns with normal probability distribution,a mean=1.2% and a standard deviation= 3.7%

ld=rnorm(100, mean =0.012, sd = 0.037)
hist(ld)

mean(ld)

[1] 0.006744504

sd(ld)

[1] 0.04120928

ed=rnorm(1000, mean =0.012, sd = 0.037)
hist(ed)

mean(ed)

[1] 0.01281574

sd(ed)

[1] 0.03577836

when the sample size increases, the mean and standard deviation are closer to the results calculated from the math formula. Similarly we should be able to compute various probability senarios with our obtained normal distribution.

##### 1F) Calculate the probability that returns will be negative using the values for mean and standard deviation as in 1E From the graph, when the sample size is 100, the prob of negative values is 44/100=44% when the sample size is 1000, the prob of negative values is 360/1000=36%

##### 1G) Repeat the calculation in 1F using instead the standard Z-value. Share insights. if we use the standard z-value, the prob of negative values is prob(x<0)=prob (z<(0-12)/37)=prob(z<-0.324)=37.5% ### Task 2: MC Simulation & European Option Pricing

Follow the Algorithm 5.2 example on p 167 (*) to calculate the price of a European option using a MC simulation. Note that the code in the book example is missing one detail and a correction. Those are left for your investigation.

##### 2A) Identify and explain the nature of the missing detail and the correction needed.

Given the MC simulation we should be able to price a European option. Note the introduction of user defined function in the book. Below is an example of a user defined function and usage. The function returns the squared value.

myfunction <- function(x=2){
  y=x^2
return(y)}
myfunction(x=5)

[1] 25

S=S+rSdt+sigmaSsqrt(dt)*E E~N(0,1) in this equation ,the r=10% ,representing the risk-free rate, this is not correct, this r is actually the expected annual return.

##### 2B) Use MC simulation to price a European Call option on a stock with initial price of $155, strike price $140, a time to expiration equal to six months, a risk-free interest rate of 2.5% and a volatility of 23%. Consider a number of periods n=100 and a number of simulations m=1000

sum1=0
type="c"
 S = 155  
K = 140   
t=0.5
rf=0.025
r=0.025
sigma=0.23
n=100
dt=t/100
for(i in 1:1000) {
  for (j in 1:100) {
     E=rnorm(1,mean=0,sd=1)
    S=S+r*S*dt+sigma*S*sqrt(dt)*E 
     
  }
 payoff=max(S-K,0)
  sum1=sum1+payoff
}
 
 
Optionvalue=sum1*exp(-rf*t)/1000
Optionvalue

[1] 37.32323

##### 2C) Write the mathematical representation of the discrete pricing equation modeled in the MC simulation. Explain what each variable in the equation represents, and provide the associated numerical value. ##### 2D) Compare the price obtained from the MC simulation to the option price using the Black-Scholes function pricing GBSOption(). Share insights.

library(fOptions)
 GBSOption(TypeFlag = "c", S = 155, X = 140, Time = 0.5, r = 0.025, b = 0.025, sigma = 0.23)

Title:
 Black Scholes Option Valuation 

Call:
 GBSOption(TypeFlag = "c", S = 155, X = 140, Time = 0.5, r = 0.025, 
     b = 0.025, sigma = 0.23)

Parameters:
          Value:
 TypeFlag c     
 S        155   
 X        140   
 Time     0.5   
 r        0.025 
 b        0.025 
 sigma    0.23  

Option Price:
 20.11899 

Description:
 Sun Jan 27 19:37:27 2019 

when we use the GBSoption function in the Rstudio, the price of the option above is nearly 20.1 and the result is stable , when we calculate with the Mc simulation the result fluctuates significantly and can rise to a very high figure. The MC simulation indicates that the trend for the option is vulnerable to many other factors and intend to show various trends , indicating the risk for the option investment. *http://computationalfinance.lsi.upc.edu