Steven R. DunbarDepartment of Mathematics203 Avery HallUniversity of Nebraska-LincolnLincoln, NE 68588-0130http://www.math.unl.edu

Voice: 402-472-3731Fax: 402-472-8466

Stochastic Processes and

Advanced Mathematical Finance

A Coin Tossing Experiment

Rating

Student: contains scenes of mild algebra or calculus that may require guid-ance.

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Section Starter Question

Suppose you start with a fortune of $10, and you want to gamble to achieve$20 before you go broke. Your gambling game is to flip a fair coin successively,and gain $1 if the coin comes up “Heads” and lose $1 if the coin comes up“Tails”. What do you estimate is the probability of achieving $20 before goingbroke? How long do you estimate it takes before one of the two outcomesoccurs? How do you estimate each of these quantities?

Key Concepts

1. Performing an experiment to gain intuition about coin-tossing games.

Vocabulary

1. We call victory the outcome of reaching a fortune goal by chancebefore going broke.

2. We call ruin the outcome of going broke by chance before reaching afortune goal.

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Mathematical Ideas

Pure Risk Modeling

We need a better understanding of the paths that risky securities take. Weshall make and investigate a greatly simplified model of randomness and risk.For our model, we assume:

1. Time is discrete, occurring at t0 = 0, t1, t2, . . ..

2. No risk-free investments are available, (i.e. no interest-bearing bonds).

3. No options, and no financial derivatives are available.

4. The only investments are risk-only, that is, our net fortune at the nthtime is a random variable:

Tn+1 = Tn + Yn+1

where T0 = 0 is our given initial fortune, and for simplicity,

Yn =

{+1 probability 1/2

−1 probability 1/2.

Our model is commonly called “gambling” and we will investigate the prob-abilities of making a fortune by gambling.

Some Humor

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Figure 1: Welcome to my casino!

Figure 2: Where not to lose at the casino!

An Experiment

The reader should perform the following experiment as a “gambler” to gainintuition about the coin-tossing game. We call victory the outcome of reach-ing a fortune goal by chance before going broke. We call ruin the outcome

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of going broke by chance before reaching a fortune goal.

1. Each “gambler” has a chart for recording the outcomes of each game(see below) and a sheet of graph paper.

2. Each “gambler” has a fair coin to flip, say a penny.

3. Each “gambler” flips the coin, and records a +1 (gains $1) if the coincomes up “Heads” and records −1 (loses $1) if the coin comes up“Tails”. On the chart, the player records the outcome of each flipwith the flip number, the outcome as “H” or “T” and keeps track ofthe cumulative fortune of the gambler so far. Keep these records in aneat chart, since some problems refer to them later.

4. Each “gambler” should record 100 flips, which takes about 10 to 20minutes.

Toss n 1 2 3 4 5 6 7 8 9 10H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 11 12 13 14 15 16 17 18 19 20H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 21 22 23 24 25 26 27 28 29 30H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 31 32 33 34 35 36 37 38 39 40H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 41 42 43 44 45 46 47 48 49 50H or TYn = +1,−1Tn =

∑ni=1 Yi

5

Toss n 51 52 53 54 55 56 57 58 59 60H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 61 62 63 64 65 66 67 68 69 70H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 71 72 73 74 75 76 77 78 79 80H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 81 82 83 84 85 86 87 88 89 90H or TYn = +1,−1Tn =

∑ni=1 Yi

Toss n 91 92 93 94 95 96 97 98 99 100H or TYn = +1,−1Tn =

∑ni=1 Yi

Some Outcomes

A simulation with 30 “gamblers” obtained the following results:

1. 14 gamblers reached a net loss of −10 before reaching a net gain of+10, that is, were “ruined”.

2. 9 gamblers reached a net gain of +10 before reaching a net loss of −10,that is, achieved “victory”.

3. 7 gamblers still had not reached a net loss of +10 or −10 yet.

This roughly matches the predicted outcomes of 1/2 the gamblers beingruined, not as close for the predicted proportion of 1/2 for victory. Themean duration until victory or ruin was 41.6. Compare this to your results.

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Sources

This section is adapted from ideas in William Feller’s classic text, An Intro-duction to Probability Theory and Its Applications, Volume I, Third Edition.

The GeoGebra script is adapted from a demonstration and discussionamong “carlosgomes” of Amarante, Portugal; Noel Lambert of Vosges, France;and Juan Vicente Snchez Gaitero of Cadiz, Spain on geogebra.org

Algorithms, Scripts, Simulations

Algorithm

The probability of Heads is p. In matrix-oriented languages, use k columnsof the matrix for the iterations of an experiment each with n trials. This ismore efficient than using for-loops or the equivalent.

Then each n× 1 column vector contains the outcomes of the experiment.Use the random number generator to create a random number in (0, 1). Tosimulate the flip of a coin use logical functions to score a 1, also counted asa Head, if the value is less than p; a 0, also counted as a Tail, if the valueis greater than p. Column sums then give a 1 × k vector of total numbersof Heads. Statistical measures can then be applied to this vector. For winsand losses, linearly transform the coin flips to 1 if Heads, a −1 if Tails.Cumulatively sum the entries to create the running total. Apply statisticalmeasures to the last row of the totals.

Finally, set values for victory and ruin. Check when each column firsthits victory or ruin.

Experiment Scripts

Geogebra GeoGebra script for coin flipping experiment

R R script for coin flipping experiment.

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1 p <- 0.5

2 n <- 100

3 k <- 30

4 coinFlips <- array( 0+( runif(n*k) <= p), dim=c(n,k))

5 # 0+ coerces Boolean to numeric

6 headsTotal <- colSums(coinFlips) # 0..n binomial rv

sample , size k

7 muHeads <- mean(headsTotal) # Expected value is n

/2

8 sigmaSquaredHeads <- var(headsTotal) # Theoretical

value is np(1-p)

9 cat(sprintf("Empirical Mean of Heads: %f \n", muHeads ))

10 cat(sprintf("Empirical Variance of Heads: %f \n",

sigmaSquaredHeads ))

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12 winLose <- 2*coinFlips - 1 # -1 for Tails , 1 for

Heads

13 totals <- apply( winLose , 2, cumsum)

14 # -n..n (every other integer) binomial rv sample

15 # the second argument ‘‘2’’ means column -wise

16 muWinLose <- mean( totals[n,]) # Expected value is 0

17 sigmaSquaredWinLose <- var( totals[n,]) # Theoretical

value is 4np(1-p)

18 cat(sprintf("Empirical Mean of Wins minus Losses: %f \n",

muWinLose ))

19 cat(sprintf("Empirical Variance of Wins minus Losses: %f

\n", sigmaSquaredWinLose ))

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Octave Octave script for coin flipping experiment

1 p = 0.5;

2 n = 100;

3 k = 30;

4

5 coinFlips = rand(n,k) <= p;

6 headsTotal = sum(coinFlips); # 0..n binomial rv sample ,

size k

7 muHeads = mean(headsTotal); # Expected value is n/2

8 sigmaSquaredHeads = var(headsTotal); # Theoretical value

is np(1-p)

9 disp("Empirical Mean of Heads:"), disp(muHeads)

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10 disp("Empirical Variance of Heads:"), disp(

sigmaSquaredHeads)

11

12 winLose = 2* coinFlips - 1; # -1 for Tails , 1 for Heads

13 totals = cumsum(winLose); # -n..n (every other integer)

binomial rv sample

14 muWinLose = mean( totals(n,:) ); # Expected value is 0

15 sigmaSquaredWinLose = var( totals(n,:) ); # Theoretical

value is 4np(1-p)

16 disp("Empirical Mean of Wins minus Losses:"), disp(

muWinLose)

17 disp("Empirical Variance of Wins minus Losses:"), \

18 disp(sigmaSquaredWinLose)

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Perl Perl PDL script for coin flipping experiment

1 $p = 0.5;

2 $n = 100;

3 $k = 30;

4

5 $coinFlips = random( $k, $n ) <= $p; #note order of

dims!!

6 $headsTotal =

7 $coinFlips ->transpose ->sumover; # 0..n binomial r

.v. sample , size k

8

9 #note transpose , PDL likes x (row) direction for

implicitly threaded operations

10

11 ( $muHeads , $prms , $median , $min , $max , $adev , $rmsHeads

) =

12 $headsTotal ->stats;

13

14 # easiest way to get the descriptive statistics

15

16 printf "Empirical Mean of Heads: %6.3f \n", $muHeads;

17

18 # Theoretical value is np

19 $varHeads = $rmsHeads * $rmsHeads;

20 printf "Emprical Variance of Heads: %6.3f \n", $varHeads;

21

22 # Theoretical value is np(1-p)

23

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24 $winLose = 2 * $coinFlips - 1; # note use of threading

25 $totals = ( $winLose ->xchg( 0, 1 )->cumusumover )->

transpose;

26

27 # note xchg transpose , PDL likes x (row) direction for

implicity threaded operations

28

29 ( $muWinLose , $prms , $median , $min , $max , $adev ,

$rmsWinLose ) =

30 $totals ( 0 : $k - 1, $n - 1 )->stats; # Expected

value is 0

31

32 # pdl are indexed from 0 like C and Perl , so that’s why

$k -1, $n -1

33

34 printf "Empirical Mean number of Wins over Losses: %6.3f

\n", $muWinLose;

35

36 # Theoretical value is 0

37 $varWinLose = $rmsWinLose * $rmsWinLose; #

Theoretical value is 4np(1-p)

38 printf "Empirical Variance of Wins over Losses: %6.3f \n

", $varWinLose;

39

SciPy Scientific Python script for coin flipping experiment

1 import scipy

2

3 p = 0.5

4 n = 100

5 k = 30

6

7 coinFlips = scipy.random.random ((n,k)) <= p

8 # Note Booleans True for Heads and False for Tails

9 headsTotal = scipy.sum(coinFlips , axis = 0)

10 # Note how Booleans act as 0 (False) and 1 (True)

11 muHeads = scipy.mean(headsTotal)

12 stddevHeads = scipy.std(headsTotal)

13 sigmaSquaredHeads = stddevHeads * stddevHeads

14 print "Empirical Mean of Heads:", muHeads

15 print "Empirical Variance of Heads:", sigmaSquaredHeads

16

17 winLose = 2* coinFlips - 1

10

18 totals = scipy.cumsum(winLose , axis = 0)

19 muWinLose = scipy.mean(totals[n-1,range(k)])

20 stddevWinLose = scipy.std(totals[n-1, range(k)])

21 sigmaSquaredWinLose = stddevWinLose * stddevWinLose

22 print "Empirical Mean of Wins minus Losses:", muWinLose

23 print "Empirical Variance of Wins minus Losses:",

sigmaSquaredWinLose

Victory or Ruin Scripts

R R script for victory or ruin.

1 victory <- 10 # top boundary for random walk

2 ruin <- -10 # bottom boundary for random walk

3

4 victoryOrRuin <- array(0, dim = c(k))

5

6 hitVictory <- apply( (totals >= victory), 2, which)

7 hitRuin <- apply( (totals <= ruin), 2, which)

8

9 for (j in 1:k) {

10 if ( length(hitVictory [[j]]) == 0 && length(

hitRuin [[j]]) == 0 ) {

11 # no victory , no ruin

12 # do nothing

13 }

14 else if ( length(hitVictory [[j]]) > 0 && length(

hitRuin [[j]]) == 0 ) {

15 # victory , no ruin

16 victoryOrRuin[j] <- min(hitVictory [[j]])

17 }

18 else if ( length(hitVictory [[j]]) == 0 && length(

hitRuin [[j]]) > 0 ) {

19 # no victory , ruin

20 victoryOrRuin[j] <- -min(hitRuin [[j]])

21 }

22 else # ( length(hitVictory [[j]]) > 0 && length(

hitRuin [[j]]) > 0 )

23 # victory and ruin

24 if ( min(hitVictory [[j]]) < min(hitRuin [[j]]) ) { #

victory first

25 victoryOrRuin[j] <- min(hitVictory [[j]]) #

code hitting victory

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26 }

27 else { # ruin first

28 victoryOrRuin[j] <- -min(hitRuin [[j]]) # code

hitting ruin as negative

29 }

30 }

31

32 victoryBeforeRuin <- sum (0+( victoryOrRuin > 0)) # count

exits through top

33 ruinBeforeVictory <- sum (0+( victoryOrRuin < 0)) # count

exits through bottom

34 noRuinOrVictory <- sum (0+( victoryOrRuin == 0))

35

36 cat(sprintf("Victories: %i Ruins: %i No Ruin or Victory:

%i \n",

37 victoryBeforeRuin , ruinBeforeVictory , noRuinOrVictory

))

38

39 avgTimeVictoryOrRuin <- mean( abs(victoryOrRuin) )

40 varTimeVictoryOrRuin <- var( abs(victoryOrRuin) )

41

42 cat(sprintf("Average Time to Victory or Ruin: %f \n",

43 avgTimeVictoryOrRuin))

44 cat(sprintf("Variance of Time to Victory or Ruin: %f \n",

45 varTimeVictoryOrRuin))

46

47 hist( victoryOrRuin , nclass = 2*max(abs(victoryOrRuin))+1

) # sample exit time distribution

48

Octave Octave script for victory or ruin,

1 victory = 10; # top boundary for random walk

2 ruin = -10; # bottom boundary for random walk

3

4 # Octave find treats a whole matrix as a single column ,

5 # but I need to treat experiment column by column , so

here is a place

6 # where for clarity , better to loop over columns , using

find on each

7 victoryOrRuin = zeros(1,k);

8 for j = 1:k

9 hitVictory = find(totals(:,j) >= victory);

10 hitRuin = find(totals(:,j) <= ruin);

12

11 if ( !rows(hitVictory) && !rows(hitRuin) )

12 # no victory , no ruin

13 # do nothing

14 elseif ( rows(hitVictory) && !rows(hitRuin) )

15 # victory , no ruin

16 victoryOrRuin(j) = hitVictory (1);

17 elseif ( !rows(hitVictory) && rows(hitRuin) )

18 # no victory , but hit ruin

19 victoryOrRuin(j) = -hitRuin (1);

20 else # ( rows(hitvictory) && rows(hitruin) )

21 # victory and ruin

22 if ( hitVictory (1) < hitRuin (1) )

23 victoryOrRuin(j) = hitVictory (1); # code hitting

victory

24 else

25 victoryOrRuin(j) = -hitRuin (1); # code hitting ruin as

negative

26 endif

27 endif

28 endfor

29

30 victoryOrRuin # display vector of k hitting times

31

32 victoryBeforeRuin = sum( victoryOrRuin > 0 ) # count

exits through top

33 ruinBeforeVictory = sum( victoryOrRuin < 0 ) # count

exits through bottom

34 noRuinOrVictory = sum( victoryOrRuin == 0 )

35

36 avgTimeVictoryOrRuin = mean( abs(victoryOrRuin) )

37 varTimeVictoryOrRuin = var( abs(victoryOrRuin) )

38

39 max(totals)

40 hist( victoryOrRuin , max(max(victoryOrRuin))) # sample

exit time distribution

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Perl Perl PDL script for victory or ruin

1 use PDL:: NiceSlice;

2

3 $victory = 10; # top boundary for random walk

4 $ruin = -10; # bottom boundary for random walk

5

13

6 $victoryOrRuin = zeros($k);

7

8 $hitVictory = ( $totals >= $victory );

9 $hitRuin = ( $totals <= $ruin );

10

11 $whenVictory = $hitVictory ( 0 : $k - 1, : )->xchg( 0, 1

)->maximum_ind;

12 $whenRuin = $hitRuin ( 0 : $k - 1, : )->xchg( 0, 1

)->maximum_ind;

13 foreach $i ( 0 .. $k - 1 ) {

14 if ( $whenVictory ($i) == 0 and $whenRuin ($i) == 0 )

{

15

16 # no victory , no ruin , do nothing

17 $victoryOrRuin ($i) = 0;

18 }

19 elsif ( $whenVictory ($i) > 0 and $whenRuin ($i) == 0

) {

20 $victoryOrRuin ($i) .= $whenVictory ($i);

21 } # victory , no ruin

22 elsif ( $whenVictory ($i) == 0 and $whenRuin ($i) > 0

) {

23 $victoryOrRuin ($i) .= -$whenRuin ($i);

24 } # no victory , ruin

25 else {

26 if ( $whenVictory ($i) < $whenRuin ($i) ) { #

victory first

27 $victoryOrRuin ($i) .= $whenVictory ($i); #

code hitting victory

28 }

29 else {

30 $victoryOrRuin ($i)

31 .= -$whenRuin ($i); # code hitting

ruin as negative

32 }

33 }

34 }

35

36 $victoryBeforeRuin =

37 ( $victoryOrRuin > 0 )->sumover; # count exits

through top

38 $ruinBeforeVictory =

39 ( $victoryOrRuin < 0 )->sumover; # count exits

through bottom

40

14

41 ( $muTimeVictoryOrRuin , $prms , $median , $min , $max ,

$adev ,

42 $rmsTimeVictoryOrRuin

43 ) = $victoryOrRuin ->abs ->stats;

44

45 print "Mean time to victory or ruin",

$muTimeVictoryOrRuin , "\n";

46 print "Standard deviation of time to victory or ruin",

$rmsTimeVictoryOrRuin ,

47 "\n";

48

49 $histLimit = $victoryOrRuin ->abs ->max;

50 print hist( $victoryOrRuin , -$histLimit , $histLimit , 1 ),

"\n";

51

SciPy Scientific Python script for victory or ruin.

1 victory = 10; # top boundary for random

walk

2 ruin = -10; # bottom boundary of

random walk

3

4 victoryOrRuin = scipy.zeros(k,int);

5 hitVictory = scipy.argmax( (totals >= victory), axis =0)

6 hitRuin = scipy.argmax( (totals <= ruin), axis =0)

7 for i in range(k):

8 if hitVictory[i] == 0 and hitRuin[i] == 0:

9 # no victory , no ruin , do nothing

10 victoryOrRuin[i] = 0

11 elif hitVictory[i] > 0 and hitRuin[i] == 0:

12 victoryOrRuin[i] = hitVictory[i]

13 elif hitVictory[i] == 0 and hitRuin[i] > 0:

14 victoryOrRuin[i] = -hitRuin[i]

15 else:

16 if hitVictory[i] < hitRuin[i]:

17 victoryOrRuin[i] = hitVictory[i]

18 else:

19 victoryOrRuin[i] = -hitRuin[i]

20

21 victoryBeforeRuin = sum( victoryOrRuin > 0 ) # count

exits through top

22 ruinBeforeVictory = sum( victoryOrRuin < 0 ) # count

exits through bottom

15

23 noRuinOrVictory = sum( victoryOrRuin == 0 )

24

25 print "Victories:", victoryBeforeRuin , "Ruins:",

ruinBeforeVictory ,

26 print "No Ruin or Victory:", noRuinOrVictory

27

28 avgTimeVictoryOrRuin = scipy.mean( abs(victoryOrRuin) )

29 stdTimeVictoryOrRuin = scipy.std( abs(victoryOrRuin) )

30 varTimeVictoryOrRuin = stdTimeVictoryOrRuin *

stdTimeVictoryOrRuin

31

32 print "Average Time to Victory or Ruin:",

avgTimeVictoryOrRuin

33 print "Variance of Time to Victory or Ruin:",

varTimeVictoryOrRuin

Problems to Work for Understanding

1. How many heads were obtained in your sequence of 100 flips? Whatis the class average of the number of heads in 100 flips? What is thevariance of the number of heads obtained by the class members?

2. What is the net win-loss total obtained in your sequence of 100 flips?What is the class average of the net win-loss total in 100 flips? What isthe variance of the net win-loss total obtained by the class members?

3. How many of the class reached a net gain of +10 (call it victory) beforereaching a net loss of −10 (call it ruin)?

4. How many flips did it take before you reached a net gain of +10 (vic-tory) or a net loss of −10 (ruin)? What is the class average of thenumber of flips before reaching victory before ruin at +10 or a net lossof −10?

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5. What is the maximum net value achieved in your sequence of flips?What is the class distribution of maximum values achieved in the se-quence of flips?

6. Perform some simulations of the coin-flipping game, varying p. Howdoes the value of p affect the experimental probability of victory andruin?

Reading Suggestion:

References

[1] William Feller. An Introduction to Probability Theory and Its Applica-tions, Volume I, volume I. John Wiley and Sons, third edition, 1973. QA273 F3712.

[2] Emmanuel Lesigne. Heads or Tails: An Introduction to Limit Theoremsin Probability, volume 28 of Student Mathematical Library. AmericanMathematical Society, 2005.

Outside Readings and Links:

1. Virtual Laboratories in Probability and Statistics Red and Black GameSearch for Red and Black Game, both the explanation and the simula-tion.

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2. University of California, San Diego, Department of Mathematics, A.M.Garsia A java applet that simulates how long it takes for a gambler togo broke. You can control how much money you and the casino startwith, the house odds, and the maximum number of games. Results area graph and a summary table. Submitted by Matt Odell, September8, 2003.

I check all the information on each page for correctness and typographicalerrors. Nevertheless, some errors may occur and I would be grateful if you wouldalert me to such errors. I make every reasonable effort to present current andaccurate information for public use, however I do not guarantee the accuracy ortimeliness of information on this website. Your use of the information from thiswebsite is strictly voluntary and at your risk.

I have checked the links to external sites for usefulness. Links to externalwebsites are provided as a convenience. I do not endorse, control, monitor, orguarantee the information contained in any external website. I don’t guaranteethat the links are active at all times. Use the links here with the same caution asyou would all information on the Internet. This website reflects the thoughts, in-terests and opinions of its author. They do not explicitly represent official positionsor policies of my employer.

Information on this website is subject to change without notice.

Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1Email to Steve Dunbar, sdunbar1 at unl dot edu

Last modified: Processed from LATEX source on July 18, 2016

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