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Aðalbjörn Þórólfsson ath@alit.is

Aðalbjörn Þórólfsson ath@alit.is. Financial derivatives. Stock price models and parameters ( hlutabréfalíkön og kennistærðir ) Approx. 2 weeks Simple models Stochastic behavior ( slembiferli ) Financial derivatives ( fjármálaafleiður ) Approx. 2 weeks Introduction The Black-Scholes model

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Aðalbjörn Þórólfsson ath@alit.is

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  1. Aðalbjörn Þórólfssonath@alit.is Líkön og mælingar – Fjármálaafleiður

  2. Financial derivatives • Stock price models and parameters (hlutabréfalíkön og kennistærðir) • Approx. 2 weeks • Simple models • Stochastic behavior (slembiferli) • Financial derivatives (fjármálaafleiður) • Approx. 2 weeks • Introduction • The Black-Scholes model • Bank visit and other material • Approx. 1 week Líkön og mælingar – Fjármálaafleiður

  3. Simple stock price models Líkön og mælingar – Fjármálaafleiður

  4. Price model 1 • The value of a stock at a given time t [t0,T] is supposed to be a continuous real function: S :[t0,T] R • The value of S(t0) is known. • The change in value with time is constant: dS = dt or S(t) = S(t0) + (t- t0) • Limitation: Doesn’t allow fluctuations with time. Líkön og mælingar – Fjármálaafleiður

  5. Stochastic processes • A variable whose value changes over time in an uncertain way is said to follow a stochastic process (slembiferli). • In a Markov process,future movements of a variable depend only on where we are, not the history of how we got there. • Stock prices are usually assumed to follow Markov processes. Líkön og mælingar – Fjármálaafleiður

  6. Basic Wiener process • A variable Z follows a basic Wiener process if it has the following two properties: • The change Z during a small time period t is Z =  t where  is a random drawing from a standardized normal distribution (0,1). • The values of Z for any two different short intervals of time t are independent. Líkön og mælingar – Fjármálaafleiður

  7. The normal distribution • Standardized form: • Mean: = 0. Variance  = 1. • Why the normal distribution? If X is the mean of N independent measurements of the same phenomena, then the distribution of X becomes normal as N (central limit theorem). • Examples: • Independent measurements of your height. • Stock price with no drift in the mean with time. Líkön og mælingar – Fjármálaafleiður

  8. Basic Wiener processes • Advantage: Allows fluctuations with time. • Limitation: Doesn’t show an average drift with time. Líkön og mælingar – Fjármálaafleiður

  9. Generalized Wiener process • A generalized Wiener process for a variable S is defined by: dS = a dt + b dZ where dZ (= dt) is defined as before. • The discrete form is: S = a t + b t where  is a random drawing from a standardized normal distribution (0,1). • A special case is b = 0, and we find model 1 for stock price: S = a t Líkön og mælingar – Fjármálaafleiður

  10. Price model 2 • The change in value with time is based on the general Wiener process: dS =  dt +  dZ or, in a discrete form: S = t +  t where  is defined as before. • Solution (T=t-t0, N=T/t): • The change in S after a period Tis normally distributed, has a mean =Tand a variance =2 T. Líkön og mælingar – Fjármálaafleiður

  11. Model comparisons • Model 1 - Linear: S(t) = 1.0 * t • Basic Wiener (no average drift with time): S(t) =  t • Model 2 - General Wiener: S(t) = 1.0 * t + 1.0 *  t Líkön og mælingar – Fjármálaafleiður

  12. The factor  • The factor  describes the amplitude of the stochastic behavior of stocks (volatility (flökt)), and thus how high the associated investment risk is: dS =  dt +  dZ • The factor  actually depends on time, but can be taken as a constant when considering relatively short periods. • The volatility can be estimated by: • The naked eye • Calculating the standard deviation of past data (see later) sdev =  = T Líkön og mælingar – Fjármálaafleiður

  13. More realistic considerations • In model 2, the drift and variability terms are independent of S(t). • However, investors require a certain percentage return: dS ~ S’ dt or S(t)=S(t0)e’Twith ’= /S(t0). • Also, the volatility term is, to a good appoximation, a percentage of the price: dS ~ S    dt Líkön og mælingar – Fjármálaafleiður

  14. Price model 3 (1) • The change in value is proportional the value: dS = S’ dt + S dZ • Result of Ito’s lemma: • The change in ln(S) after a period Tis normally distributed, has a mean =(’-2/2)Tand a variance =2 T. Líkön og mælingar – Fjármálaafleiður

  15. Price model 3 (2) • S is log-normally distributed, with a mean and a variance • How to calculate S: • Note: ’=/S(t0 ) gives a higher change than wished for. A better result is gotten with ’=ln(/S(t0 )) Líkön og mælingar – Fjármálaafleiður

  16. Historic volatility • Let ui=ln(Si /Si-1), i=0,1,...,N. This is the daily return in interval i, since Si=Si-1eui. • An estimate of the variance of the ui ‘s (one interval) is: • The variance (one interval) is =2t, so an estimate of the volatility for a period T=Ntis: Líkön og mælingar – Fjármálaafleiður

  17. Assignment 1 • Write programs that calculate and display the values of S(t) according to stock price models 1, 2 and 3. • Get real one-year data from the web and determine S(t0) and  (linear fit). Try to determine  by the naked eye and then by using the variance formula. Note that calculated  is only comparable to real data in model 3. • Compare the values /S(t0),  and the return/risk ratio: (S(t)-S(t0))/(S(t0)) for two different companies. Are the /S(t0) and the returns comparable? • Write and return a short report, including graphics and code. Líkön og mælingar – Fjármálaafleiður

  18. Tips (1) • Data on bi.is or financialweb.com • Use for example Excel, Perl and Gnuplot • One year = 1. • Linear fit: y = ax + b Líkön og mælingar – Fjármálaafleiður

  19. Tips (2) • Standardized normal distribution: • If x1 and x2 are random variables between 0 and 1, then y1 and y2 are standardized and normally distributed if • In model 3, the rise/fall tends to be overestimated if we use the obtained from linear fitting. Líkön og mælingar – Fjármálaafleiður

  20. Perl #!/usr/bin/perl #Launches the PERL compilator $file_in=$ARGV[0]; #The first argument passed to the program $file_out = "data.dat"; #Assign a filename open (INFILE,$file_in); #Relate INFILE with filename @ALL = <INFILE>; #Read all the data into a vector close (INFILE); #Close the infile open (OUTFILE,">" . $file_out); #Open for output for($k=1;$k<=$#ALL;$k++) #Loop over vector content { ($date,$day,$price,@rest) = split(/\s/,$ALL[$k]); #Split the fields on spaces $price =~ s/,/./; #Substitute . for , $day_norm = $day/365; #Normalize the time print (OUTFILE "$day_norm\t$price\n"); #Output data } close (OUTFILE); #Close the outfile --- To debug code, type: perl –wc program.pl To run code, type: ./program.pl infile.dat Líkön og mælingar – Fjármálaafleiður

  21. Gnuplot set terminal postscript landscape "Times-Roman" 22 set output “plot.ps“ set title “Title“ set ylabel "Value (kr/share)“ set xlabel "Time (years)" 0,-1 #set data style linespoints set data style lines #set nokey #set yrange [100:102.3] #set xrange [100:102.3] plot "data.dat“ using 1:2, 4.15 + 0.17*x --- To make a fit, start gnuplot, then type: f(x)=a*x+b fit f(x) ‘data.dat’ via a,b Líkön og mælingar – Fjármálaafleiður

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