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

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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] 135963694

sd(ud)

[1] 78985765

Theoretical Mean = \(\frac{a+b}{2}\) = \(\frac{-25,000,000 + 275,000,000}{2}\) = 125,000,000

Theoretical Standard Deviation = \(\sqrt{\frac{(b-a)^{2}}{12}}\) = \(\sqrt{\frac{(275,000,000-(-25,000,000))^{2}}{12}}\) = 86,602,540.38

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

ud2 = runif(1000, min=-25000000, max=275000000)

hist(ud2)

mean(ud)

[1] 113606160

sd(ud)

[1] 89388960

Theoretical Mean = \(\frac{a+b}{2}\) = \(\frac{-25,000,000 + 275,000,000}{2}\) = 125,000,000

Theoretical Standard Deviation = \(\sqrt{\frac{(b-a)^{2}}{12}}\) = \(\sqrt{\frac{(275,000,000-(-25,000,000))^{2}}{12}}\) = 86,602,540.38

Since the sample size has increased, the RStudio calculated values are relatively closer to the calculated theoretical values.

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 pnorm(q = 0, mean = 113606160, sd = 89388960, lower.tail = TRUE)

pnorm(q = 0, mean = 113606160, sd = 89388960, lower.tail = TRUE)

[1] 0.1018787

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%

nd = rnorm(100, mean =0.012, sd = 0.037)

hist(nd)

mean(nd)

[1] 0.0132127

sd(nd)

[1] 0.03661379

nd2 = rnorm(1000, mean =0.012, sd = 0.037)

hist(nd2)

mean(nd2)

[1] 0.01237582

sd(nd2)

[1] 0.03703619

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(q=0, mean = 0.012, sd=0.037)

[1] 0.3728463

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

z = \(\frac{(0 - 0.012)}{0.037}\) = -0.3243

pnorm(q=-0.3243, mean = 0, sd = 1)

[1] 0.3728555

This is significanly close to the value calculated in #1F. This shows that standardizing values is just as accurate as using the raw numbers given.

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 correction is that in order to calculate the option value, you need to divide the sum of the option prices by the number of simulations rather than (sumexp(-rfT)/m

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, So, K, T, r, sigma, n, m)
  {sum=0
  for (i in 1 : m){ 
    S=So
    for(j in 1:n){
      E <- rnorm(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/m) * exp(-rf*T))
  return (OptionValue)}
EulerMC("c", 155, 140, 0.5, 0.025, 0.23, 100, 1000)

Error in r * S * dt : non-numeric argument to binary operator

##### 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. \(P_{t} = P_{0}*e^{(\mu-\frac{\sigma^{2}}{2})}T+\sigma\epsilon_{t}\sqrt{\Delta T}\)

\(\mu\) = risk free rate at 2.5% \(\sigma\) = volatility at 23% T = time to expiration at 6 months at 0.5 \(P_{0}\) = current stock price at $155

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

require(fOptions)

Loading required package: fOptions
Loading required package: timeDate
unknown timezone 'default/America/Chicago'Loading required package: timeSeries
Loading required package: fBasics

GBSOption(TypeFlag = "c", S = 155, X = 140, Time = 6/12, r = 0.025, b = 0.025, sigma = 0.25)

Title:
 Black Scholes Option Valuation 

Call:
 GBSOption(TypeFlag = "c", S = 155, X = 140, Time = 6/12, r = 0.025, 
     b = 0.025, sigma = 0.25)

Parameters:
          Value:
 TypeFlag c     
 S        155   
 X        140   
 Time     0.5   
 r        0.025 
 b        0.025 
 sigma    0.25  

Option Price:
 20.77477 

Description:
 Mon Jan 28 23:04:13 2019 

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

plot(cars)

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