FINC621 Winter 2018-19 Extra Credit (10pts)

Asian Option Pricing & MC Simulation

About

This assignment is an extra credit opportunity worth 10 points. The goal is to demonstrate your understanding of the topic in the different areas of problem statement through a flowchart diagram propely marked, the analytics through a clear representation of the mathematical equations, the coding through the clear and properly commented execution steps, and finally results evaluation through the assessment of quality and integrity.

The topic of interest is the pricing of an Asian option. Asian options are a form of exotic options unlike the more vanilla European options. The case considered here is a special type of Asian options where the payoff is given by the average price of the underline asset over some predetermined period until expiration time. Other variations of Asian options (outside our scope) are possible and left for the reader’s investigation. The case in point is described on p.169 . Note that Asian options are characterized as path dependent options. By this we mean that the payoff of the option is dependent on the intermediate values of the underline asset along the path. In contrast a European option is path independent because the payoff depends only on the final value of the underline asset, and not on the intermediate or historical values. The case of American options is slightly tricky. Because of the early exercise opportunity one may be inclined to view American options as path dependendent. However at any exercise point the payoff is only dependent on the current asset price and not historical. As such it is not really path dependent.

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!

Any sign of plagiarism, will result in dissmissal of work!

---

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

Attaching package: ‘zoo’

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

Version 0.4-0 included new data defaults. See ?getSymbols.
Learn from a quantmod author: https://www.datacamp.com/courses/importing-and-managing-financial-data-in-r

Question 1 : Flowchart Diagram (2pts)

Inspired by the flowchart diagram of the MC simulation for European option, create an equivalent flowchart diagram for the considered case of Asian option. Describe the variables, label clearly, and write the discrete mathematical representation to be coded later. You can create the flowchart using a separate tool and include an image capture here. A hand drawn image is acceptable only if well executed.

knitr::include_graphics("/Users/chrisfrancis/Downloads/LOYOLA/FINC_621/MC_Flow.png")

Call = Max(Mean(S1,…,Sn) - K,0)

Put = Max(K- Mean(S1,…,Sn), 0)

PV_{CallorPut} = AverageOptionPrice∗exp(−rf∗T)

S_{t+dt} = S_{t}+r∗S_{t}∗dt+AnnualVolatility∗S∗sqrt(dt)∗Rnd (Discrete)

S_{t+dt} = S_{t}∗exp((r−q−Annualvolatility2/2)∗dt+AnnualVolatility∗sqrt(dt)∗Rnd) (Continuous)

Where: R_{f} = risk free rate

r = expected annual return

T = time to maturity

dt = T/n; n =the number of periods

q = dividend yield

Rnd = random number from standard normal distribution N(0,1)

Question 2: MC Simulation (2pts)

Implement the R-code for the MC simulation to price an Asian Call option with input characteristics similar to the European Call option considered in Lab 6, task 2B.

MCAsian <- function(Type="c",So=100,K=100,T=1/12,r=0.1,sigma=0.25,n=300,m=100){
  dt=T/n;
  sum=0;
  for(i in 1:m) {# number of simulated paths
    S=So
    Csum=0; # cumulative sum along the path
    for(j in 1:n){ #length of path
      E=rnorm(1,0,1);
      S=S+r*S*dt + sigma*S*sqrt(dt)*E;
      Csum=Csum+S}
    if(Type=="c"){payoff=max((1/n)*Csum - K,0)}
    else if(Type=="p"){payoff=max(K-(1/n)*Csum,0)}
    else{payoff=max((1/n)*Csum - K,0)} # Default
    sum=sum+payoff}
  OptionValue=(sum*exp(-r*T))/m;
  return(OptionValue)}
AsianOptionPrice=MCAsian(Type = "c",So=155, K=140,T=1/2,r=0.025,sigma = 0.23,n=10,m=10000)
print(AsianOptionPrice)

[1] 16.95241

Question 3: Function Calculation (2pts)

For this question the below package is needed. Follow the example at bottom of p.171 to price the same Asian Call option using instead the approximate function referenced in the example. Compare to the MC simulation.

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

TW=TurnbullWakemanAsianApproxOption(TypeFlag = "c",S=155,SA=155,X=140,Time=1/2,time=1/2, tau=0,r=0.025,b=0.025,sigma=0.23)
print(TW)

Title:
 Turnbull Wakeman Asian Approximated Option 

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

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

Option Price:
 16.63123 

Description:
 Sat Feb  9 17:58:03 2019 

The MC Simulation and the prebuilt function are very close to being equal. The difference is only 0.32 cents.

Question 4: Greeks Calculation (2pts)

Calculating the Greeks for the Asian Option is not as straightforward using the approximate function. Instead we can approximate using numerical differentiation. Describe how would you implement a numerical differentiation to calculate the Delta. Show the code execution and the intermediate values to calculate the Delta and the Theta for the considered Asian Call option.

Using the MCAsian function and changes of 1 dollar on top and bottom of the underlying (154 and 156), I can calculate delta. By having two option prices, delta can be obtained through differentiation.

OptionPrice1=MCAsian(Type = "c",So=154, K=140,T=1/2,r=0.025,sigma = 0.23,n=10,m=10000)
print(OptionPrice1)

[1] 16.07694

OptionPrice2=MCAsian(Type = "c",So=156, K=140,T=1/2,r=0.025,sigma = 0.23,n=10,m=10000)
print(OptionPrice2)

[1] 17.61195

Delta=(OptionPrice2 - OptionPrice1)/(2*1)
print(Delta)

[1] 0.7675083

OptionPrice3=MCAsian(Type = "c",So=155, K=140,T=0.5+0.001,r=0.025,sigma = 0.23,n=10,m=10000)
print(OptionPrice3)

[1] 16.75141

OptionPrice4=MCAsian(Type = "c",So=155, K=140,T=0.5-0.001,r=0.025,sigma = 0.23,n=10,m=10000)
print(OptionPrice4)

[1] 17.05379

Theta=(OptionPrice3 - OptionPrice4)/(2*0.01)
print(Theta)

[1] -15.11903

Question 5: Evaluation of Results (2pts)

Explain how the price of the Asian Call option compares to the equivalent European Call option. Is it an expected behavior? What about the Delta comparison? Poorly stated insights will receive poor grade!

The Asian call price is less than the Equivalent Europea. This makes sense based on the behavior of the Asian Option. The payoff is based on the period average and typically the average price for the period is less than the price at expiration.

Computing Delta for the European Call option:

GBSGreeks(Selection ="Delta",TypeFlag ="c",S=155,X=140,Time=1/2,r=0.025,b=0.025,sigma = 0.23)

[1] 0.783484

Observing here the deltas of the two options are pretty close. This depicts that both exhibit similar changes in the underlying.

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