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

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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=-25000000, max= 275000000)

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] 135924665

sd(ud)

[1] 85347216

Using the proper formula

mean=\(\frac{c+b}{2}\) mean = \(\frac{275000000+25000000}{2}\) = 125000000

standard Deviation(sd)= \(\sqrt{\sigma^{2}}\)

standard deviation = \(\sqrt{\frac{275000000-25000000^{2}}{12}}\) = 86602540.38

##### 1C) Repeat 1A & 1B above by increasing the sample size to 1000. Share your insights.

ud1= runif(1000, min=-25000000, max= 275000000)
hist(ud1)

mean(ud1)

[1] 126871456

sd(ud1)

[1] 87424381

We can observe that the higher sample we choose, the closer the mean and standard deviation move to the Uniform distribution’s mean and standard deviation.

The difference is only because of the difference in sample size.

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

##### 1D) Calculate the probability that the free cash flow is negative

punif(0,-25000000,275000000)

[1] 0.08333333

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%

rd<- rnorm(100, mean= 0.012 , sd= 0.037)
hist(rd)

mean(rd)

[1] 0.009748704

sd(rd)

[1] 0.03730686

rd1<- rnorm(1000, mean= 0.012 , sd= 0.037)
hist(rd)

mean(rd)

[1] 0.009748704

sd(rd)

[1] 0.03730686

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

pnorm(0, 0.012, 0.037)

[1] 0.3728463

##### 1G) Repeat the calculation in 1F using instead the standard Z-value. Share insights.

z= \(\frac{x-mean}{sigma}\)

which means z= \(\frac{0-0.012}{0.037}\)

therefore z= -0.324

refering to z table z=0.3745

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.

The loop misses the part where we make each simulation start from the first share price i.e 155. We add the code to start each simulation with S stating baseS = S and then S=baseS. We also add the number of random outcomes we want that is 1 in each simulation.

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)

##### 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

EulerMC= function(Type="c",S=155,K=140,t=1/2,r=0.025,sigma=0.23,n=100,m=1000,dt=t/n){
  baseS=S
  sum=0;
  for(i in 1:m) {#number of simulation paths}
    S=baseS
    for(j in 1:n){
    E=rnorm(1,0,1);
    S=S+r*S*dt+sigma*S*sqrt(dt)*E;}
  if(Type=="c"){payoff=max(S-K,0)}
  else if (Type=="p"){payoff=max(K-S,0)}
  else{payoff=max(S-K,0)} #default
  sum=sum+payoff}
OptionValue=(sum*exp(-r*t))/m;
return(OptionValue)}
x=EulerMC(Type="c",S=155,K=140,t=1/2,r=0.025,sigma=0.23,n=100,m=1000,dt=0.5/100)
x

[1] 18.90753

##### 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. \(\hat{C}_{m}^{n}\) = \(\frac{1}{m}\) \(\sum_{i=1}^{m}\) f( \(\hat{S}\) _{i,n}$ ) \(e^{-rT}\)

The above formula represents the algorithm used, where, we first run the share price and then we discount it back with \(e^{-rT}\) and then take an average by dividing it by the number of simulations m.

In the formula m=1000, S=100, r=0.025, t=6/12, n=100

##### 2D) Compare the price obtained from the MC simulation to the option price using the Black-Scholes function pricing GBSOption(). Share insights.

if(!require("fOptions",quietly = TRUE))
  install.packages("fOptions",dependencies = TRUE, repos = "https://cloud.r-project.org")
GBSOption(TypeFlag = "c", S = 155, X = 140,
Time = 1/2, r = 0.025, b = 0, sigma = 0.23)

Title:
 Black Scholes Option Valuation 

Call:
 GBSOption(TypeFlag = "c", S = 155, X = 140, Time = 1/2, r = 0.025, 
     b = 0, sigma = 0.23)

Parameters:
          Value:
 TypeFlag c     
 S        155   
 X        140   
 Time     0.5   
 r        0.025 
 b        0     
 sigma    0.23  

Option Price:
 18.63254 

Description:
 Wed Jan 30 13:19:06 2019 

Both the simulations give approximately the same result. Athough the MC simulation spreads through a range due to multiple simulations. Black-scholes does just one simulation and does not select random values for white noise which in turn gives just one option price.

*http://computationalfinance.lsi.upc.edu