Stochastic mathematics in application to finance

[WMDhamnekar]

Get your friends and family to play this simple little game that illustrates a key aspect of stochastic mathematics in application to finance.

1. Draw on a big sheet of paper a sequence of 30 squares and label them consecutively 0 (bankrupt), 1, 1, 2, 2, 3, 3, 4, . . . , 14, 14, 15 (millionairedom). Each of these squares represents one state in a 30-state Markov chain. Imagine each state represents the value of some asset such as the value of a small business that each player is managing.
2. Give each player a token and a six-sided die.
3. At the start place each token on the second “2”, the fifth state. Imagine this corresponds to the small business having an initial value of $200,000.
4. Each turn in the game corresponds to, say, one year in time. In each year the business may be poor or may grow. Thus in each turn each player rolls his/her die and moves according to the following rules:
   • If a player rolls a 1 or 2, then he/she moves down some states;
   • if a player rolls a 3, he/she stays in the same state;
   • if a player rolls a 4, 5, or 6, he/she moves up some states.

But the number of states (squares) a player moves is given by the number written in each square. Thus in the first move, because the fifth square/state is a “2,” a player moving up moves from the fifth square to the seventh square, and a player moving down moves to the third square.

That the number written in each square is (roughly) proportional to the position of the square in the sequence corresponds to the financial reality that small businesses usually grow/shrink by small amounts, whereas large companies grow/shrink by large amounts. Investors expect returns in proportion to their investments.

1. Each player continues to role his/her die and move until reaching 0 or 15. That is, players continue to operate their businesses until they either go bankrupt or reach millionairedom.

Questions:

1. Why do you expect each business to grow? That is, why do you expect each player to reach the “millionairedom” state?
2. When you play the game, roughly what proportion of players reach millionairedom? What proportion go bankrupt?
3. How do you explain the actual results?

How would you answer all these three questions?

How to answer these three questions?

I know wiener process/Exponential Brownian motion. I know how to plot exponetial brownian motions in octave. But I don't know how to use that knowledge here?

How can I simulate this problem in 'Octave' or in 'r'?

[Mentor Note -- Adding attribution to avoid copyright violation: Exercise 1.1 from page 33 of Elementary Calculus of Financial Mathematics by A. J. Roberts https://tuck.adelaide.edu.au/ecfm.php ]

 

[mathman]

Moves are ambiguous. roll 1 or 2 means down 1 or 2? Roll 4, 5., or 6 means move up 4,5, or 6?

 

[FactChecker]

Moves are ambiguous. roll 1 or 2 means down 1 or 2? Roll 4, 5., or 6 means move up 4,5, or 6?

The die roll only tells whether to go down (1,2) or up (4,5,6), not how much. So it is more likely to go up than down. The amount is determined by the number of the current position. So the amount moved is roughly proportional to the current position.

 

[andrewkirk]

Question 1 is easy. At each step a player has a 3/6 chance of growing by a certain amount K>0 and a 2/6 chance of shrinking by the same amount. So the expected growth in any move is K/6, which is > 0.
K equals the number on the square, multiplied by $100,000.

I don't know what they mean by question 3 but to answer question 2, a simulation is the easiest approach.

In R, define a vector giving the size of the move a player makes from each square:

move_size = c(0, 1, 1, 2, 2, ...., 14, 14, 0) # there's a more compact way to write that, but I'll leave that to you

Define a function that gives a n-vector of -1, 0, 1 values with probabilities 2/6, 1/6, 3/6 respectively, eg

direction <- function(n){
res <- runif(n) # generate vec of random uniforms
return(ifelse(res < 2/6, -1, ifelse(res < 3/6, 0, +1) )
}

Set the initial state to all 5s (5th square):

state <- rep(5, n) # this specifies the state for all n sims, by specifying the index of the current square for each

Then set n to the number of simulations you want and T to the number of time steps, and loop through T time steps, in each step updating the state by:

state <- state + move_size[state] * direction(n)

At the end:
- the simulated prob of bankruptcy is (number of 1s in state) / n
- the simulated prob of millionairedom is (number of 30s in state) / n

If those two don't add to very close to 1, you need to increase T, the number of time steps, so that almost all sims end up as 1 or 30.

You probably want n at least 1000 to have some confidence in your estimates.

 

[FactChecker]

Questions:

1. Why do you expect each business to grow? That is, why do you expect each player to reach the “millionairedom” state?

Why would you expect this? This game has two termination states, "bankruptcy", and "millionaire". The player starts very close to the "bankruptcy" termination state and I would expect many to go bankrupt.

1. When you play the game, roughly what proportion of players reach millionairedom? What proportion go bankrupt?

I am not expert enough to know if there is an easy analytical answer to this.
CORRECTION: It is possible to use the state transition matrix for the Markov process and solve it analytically. (thanks, @pbuk )

1. How do you explain the actual results?

I guess that depends on what the results are. I am not able to guess.

How would you answer all these three questions?
How to answer these three questions?

How is these different from each other?

I know wiener process/Exponential Brownian motion. I know how to plot exponetial brownian motions in octave. But I don't know how to use that knowledge here?

How can I simulate this problem in 'Octave' or in 'r'?

Without knowledge of an analytical approach, I would simulate it but I wouldn't know how to use those languages to do it.

 

[Office_Shredder]

The return statement is missing a parenthesis at the end.

 

[pbuk]

Get your friends and family to play this simple little game that illustrates a key aspect of stochastic mathematics in application to finance.
...

This is Excercise 1.1 from page 33 of Elementary Calculus of Financial Mathematics by A. J. Roberts https://tuck.adelaide.edu.au/ecfm.php.

• Posting here is a violation of the author's intellectual property rights.
• Questions about exercises in text books belong in a homework forum, not here.

 

[WMDhamnekar]

Question 1 is easy. At each step a player has a 3/6 chance of growing by a certain amount K>0 and a 2/6 chance of shrinking by the same amount. So the expected growth in any move is K/6, which is > 0.
K equals the number on the square, multiplied by $100,000.

I don't know what they mean by question 3 but to answer question 2, a simulation is the easiest approach.

In R, define a vector giving the size of the move a player makes from each square:

move_size = c(0, 1, 1, 2, 2, ...., 14, 14, 0) # there's a more compact way to write that, but I'll leave that to you

Define a function that gives a n-vector of -1, 0, 1 values with probabilities 2/6, 1/6, 3/6 respectively, eg

direction <- function(n){
res <- runif(n) # generate vec of random uniforms
return(ifelse(res < 2/6, -1, ifelse(res < 3/6, 0, +1) )
}

Set the initial state to all 5s (5th square):

state <- rep(5, n) # this specifies the state for all n sims, by specifying the index of the current square for each

Then set n to the number of simulations you want and T to the number of time steps, and loop through T time steps, in each step updating the state by:

state <- state + move_size[state] * direction(n)

At the end:
- the simulated prob of bankruptcy is (number of 1s in state) / n
- the simulated prob of millionairedom is (number of 30s in state) / n

If those two don't add to very close to 1, you need to increase T, the number of time steps, so that almost all sims end up as 1 or 30.

You probably want n at least 1000 to have some confidence in your estimates.

I tried your program in 'r'.

What should be my next command in this program? How shall I draw the conclusion now?

Please look below program.

> move_size=c(0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,0)
> direction <- function(n){
+   res <- runif(n) # generate vec of random uniforms
+   return(ifelse(res < 2/6, -1, ifelse(res < 3/6, 0, +1) ))
+ }
> n=1000;
> state <- rep(5, n) # this specifies the state for all n sims, by specifying the index of the current square for each
> state <- state + move_size[state] * direction(n)
> T<-runif(n,min=0,max=1)
> state
   [1] 7 3 3 7 3 7 7 3 7 5 7 3 7 3 5 7 7 3 7 7 5 7 3 5 5 3 7 7 7 7 7 7 3 5 7 7 7 7 3
  [40] 7 7 3 7 7 3 3 7 7 7 7 5 3 3 7 5 7 3 7 5 3 3 3 3 5 7 7 3 5 5 3 7 7 3 7 3 5 5 3
  [79] 5 7 7 5 7 7 7 3 5 7 7 5 7 3 3 7 7 7 5 7 7 5 3 3 7 7 7 5 7 7 7 7 7 7 3 7 7 7 7
 [118] 7 7 7 7 7 3 3 7 7 3 5 3 7 7 3 3 5 7 5 7 7 3 3 7 3 7 3 7 7 3 3 3 7 5 5 3 3 3 7
 [157] 7 7 7 7 7 7 7 5 7 7 7 7 7 3 5 3 7 3 3 7 3 5 7 3 3 7 7 7 3 7 3 7 7 7 3 3 7 3 7
 [196] 7 7 3 7 7 7 3 7 7 7 7 3 5 3 7 7 7 3 7 7 5 7 7 7 5 3 3 3 3 7 7 7 3 5 7 7 7 7 5
 [235] 7 7 7 3 3 3 5 7 3 7 7 5 3 7 3 5 7 7 3 3 3 7 5 7 7 7 7 7 7 7 3 3 7 7 7 7 7 7 7
 [274] 7 3 7 5 7 7 7 3 5 3 3 7 3 5 7 3 7 3 5 5 3 7 7 7 3 7 3 5 7 7 3 7 7 3 5 7 7 3 7
 [313] 3 7 7 7 3 5 3 3 5 3 3 3 3 5 7 7 3 7 5 7 5 5 3 7 7 7 7 3 3 7 3 7 7 7 3 7 7 7 3
 [352] 3 7 7 7 3 7 3 7 3 7 7 5 3 5 3 3 7 3 3 3 7 5 7 3 7 7 5 7 3 7 3 7 3 7 3 7 7 3 5
 [391] 3 3 3 5 3 5 7 7 3 7 7 7 5 7 7 3 3 7 3 5 7 3 5 7 5 3 7 3 3 3 3 3 7 7 7 7 5 3 7
 [430] 5 7 7 7 3 3 7 7 3 7 7 5 3 3 7 7 7 3 7 5 7 7 5 5 7 3 3 7 3 7 7 7 5 7 7 3 7 7 3
 [469] 7 7 3 7 3 5 7 3 7 3 7 7 7 3 7 3 7 7 3 5 3 7 7 5 3 3 7 7 3 3 3 7 7 5 3 3 3 7 7
 [508] 7 7 7 5 5 5 3 7 7 5 3 7 5 3 7 7 3 5 7 7 7 5 7 7 7 3 5 7 5 3 7 7 3 7 7 3 7 3 5
 [547] 7 7 3 3 3 3 7 3 7 7 7 7 5 5 3 7 7 5 7 3 7 7 5 3 5 5 3 7 3 7 7 3 3 3 3 3 7 7 3
 [586] 3 5 7 7 7 7 7 7 7 5 7 3 3 7 3 7 7 7 7 3 3 7 3 7 7 7 7 3 7 5 7 7 7 7 7 5 7 7 7
 [625] 7 7 7 7 7 7 7 3 3 7 3 7 5 5 5 3 7 5 7 3 5 7 3 3 7 7 7 5 3 3 7 7 3 7 3 3 7 7 5
 [664] 7 3 7 5 7 3 7 3 5 3 3 3 7 7 3 7 3 5 3 7 3 7 7 5 3 3 5 7 7 7 3 5 7 7 5 5 3 7 7
 [703] 7 3 7 7 3 3 7 7 7 3 7 7 3 7 3 5 5 7 5 3 7 7 7 3 3 3 3 3 7 7 7 7 7 3 7 5 7 7 3
 [742] 7 7 7 7 7 3 5 7 7 3 7 3 3 3 3 7 5 7 7 7 7 5 3 7 5 3 3 3 5 7 7 3 7 7 7 5 3 3 3
 [781] 3 7 7 3 7 7 3 3 7 7 3 3 3 3 7 7 7 7 3 3 5 3 7 3 7 3 7 7 5 7 3 3 7 3 5 7 3 5 3
 [820] 7 5 3 7 5 3 5 7 7 7 3 3 7 5 3 7 7 7 3 7 5 7 3 7 3 3 7 7 7 3 5 3 7 7 7 7 7 7 3
 [859] 3 7 7 5 7 7 3 3 3 3 7 7 7 7 7 7 7 7 7 7 7 5 7 7 5 3 7 7 5 7 5 3 7 7 7 7 3 7 3
 [898] 7 3 3 3 5 5 3 7 7 7 3 7 7 7 3 3 5 5 3 5 3 7 7 5 3 3 7 7 7 7 7 7 7 3 7 7 3 5 7
 [937] 3 3 3 7 7 7 7 3 3 5 7 3 7 5 7 7 7 5 7 5 5 7 3 3 7 7 3 7 3 3 7 7 5 7 7 7 7 7 3
 [976] 7 5 7 3 3 5 7 7 3 5 7 3 5 7 3 5 3 3 7 7 3 7 3 5 5
>

 

[WMDhamnekar]

This is Excercise 1.1 from page 33 of Elementary Calculus of Financial Mathematics by A. J. Roberts https://tuck.adelaide.edu.au/ecfm.php.

• Posting here is a violation of the author's intellectual property rights.
• Questions about exercises in text books belong in a homework forum, not here.

So, moderator can move this exercise to homework forum.

 

[pbuk]

Before this thread is locked/deleted I think I should point out that everyone has missed the vital hint in the question:

Each of these squares represents one state in a 30-state Markov chain.

 

[berkeman]

So, moderator can move this exercise to homework forum.

Done.

 

[FactChecker]

Before this thread is locked/deleted I think I should point out that everyone has missed the vital hint in the question:

Yes. It is possible to use the state transition matrix for the Markov process and solve it analytically.

 

[BWV]

really an awkward model, most texts would use a binomial tree for a beginning example

try calculating the value of a call option on the business with, say X=8

 

[FactChecker]

really an awkward model, most texts would use a binomial tree for a beginning example

try calculating the value of a call option on the business with, say X=8

A Markov transition matrix would be large (30x30) but sparce (each non-terminal state can only transition to two other states). I would think that it could be solved on a computer fairly easily.

 

[pbuk]

A Markov transition matrix would be large (30x30) but sparce (each non-terminal state can only transition to two other states). I would think that it could be solved on a computer fairly easily.

In particular, it could be solved in Octave very easily: symbolic math packages are really good for calculating eigenvectors. They are really bad for writing simulations.

 

[pbuk]

In particular, it could be solved in Octave very easily: symbolic math packages are really good for calculating eigenvectors. They are really bad for writing simulations.

However I don't think the author is expecting you to do that, I think he is expecting you to actually play the game. The results are surprising.

 

[andrewkirk]

I tried your program in 'r'.

What should be my next command in this program? How shall I draw the conclusion now?

Your code only performs one time step. You need to write a loop to evolve the state through T time steps, where T is a fairly large number, as I indicated above.

As @pbuk points out above, this problem can also be solved by creating a transition matrix and then finding its eigenvectors. That's more elegant. Although I expect that will take less computer time, it probably takes less person time to just run the simulation. I coded it just now in about 5 mins and it runs in a fraction of a second for 1000 sims over 1000 time steps.

In doing so I realised the problem spec is faulty, as a person on 14 that rolls a 5 should move up 14 squares but at most three steps takes them off the board. So it should say that for up moves the number of squares one moves is limited to what takes them to the 15 tile. Easily coded in R by enclosing the expression for the post-step vector in pmin(30, ...).

My sim got a surprising proportion of bankruptcies. I'll check that with the eigenvector approach when I get a mo.

 

[WMDhamnekar]

Author and Professor A. J. Roberts (University of Adelaide,Australia) provided tha answer to question 1 and 2 as follows:

1. On average, a player moves up 16% of the time. That is, we could make a business case that shows 16% growth per year.

Does your simulations give the same results?

2. Only about 1 in 3 reach millionairedom; the rest go bankrupt.

3. Stochastic fluctuations ruin the expected growth.

Do you agree with the answers given by Author Prof. A. J. Roberts(Adelaide University,Australia) for question 2 and 3?

Would you write your program using </> button with general code?

 

[FactChecker]

IMO, question 1 is poorly worded and the answer is also poorly worded.

1. Why do you expect each business to grow? That is, why do you expect each player to reach the “millionairedom” state?

I expected many to go bankrupt because they started so close to that termination state. The question does not say to ignore the termination states.

Author and Professor A. J. Roberts (University of Adelaide,Australia) provided tha answer to question 1 and 2 as follows:

1. On average, a player moves up 16% of the time. That is, we could make a business case that shows 16% growth per year.

On average, a player moves up 50% of the die rolls. He moves up with a die roll of 4, 5, or 6, which is half of the rolls. There is a difference between the expected value of a roll versus the percentage of time a particular result occurs.

 

[BWV]

But a process can have a positive expectation, but the skewness results in near certainty of bankruptcy- take a binomial process of gaining 40% or losing 35% with equal prob starting with $100 and going bankrupt if < 1. With an even number of wins and losses you are bankrupt before 100 plays.

 

[FactChecker]

2. Only about 1 in 3 reach millionairedom; the rest go bankrupt.

3. Stochastic fluctuations ruin the expected growth.

I wonder if the culprit is that they start dangerously close to the bankrupt state? If they are started in the middle would the majority become millionaires?

 

[pbuk]

My sim got a surprising proportion of bankruptcies.

Yes, this is correct

I don't think the author is expecting you to do that, I think he is expecting you to actually play the game. The results are surprising.

and is I assume the point of the question (as indicated by the author's answer to Q3).

1. On average, a player moves up 16% of the time. That is, we could make a business case that shows 16% growth per year.

That is not correct, a player moves up 50% of the time. 16% is (approximately) the expected value of the move.

Does your simulations give the same results?

Mine do (and @andrewkirk's too I suspect).

2. Only about 1 in 3 reach millionairedom; the rest go bankrupt.

Sure do.

3. Stochastic fluctuations ruin the expected growth.

Sure do.

Do you agree with the answers given by Author Prof. A. J. Roberts(Adelaide University,Australia) for question 2 and 3?

Yes, but not for question 1. Also this question was poorly worded as pointed out by @FactChecker: I would have asked "Why do you expect these businesses mostly to grow? That is, why do you expect most players to reach the “millionairedom” state?"

Would you write your program using </> button with general code?

I have no idea what this means: I wrote mine in JavaScript but I am not going to share it with you becasue this is your homework.

 

[pbuk]

I wonder if the culprit is that they start dangerously close to the bankrupt state?

Good point. In particular if you start at or above square 20* there is a 50% chance you will become a millionaire in the first year.

Because of this I suspect that trying to calculate eigenvectors will not help (can you guess what they will look like?)

* it might be square 19 or 21

 

[FactChecker]

Because of this I suspect that trying to calculate eigenvectors will not help (can you guess what they will look like?)

I think that it can still be used to determine the terminal state probabilities for any initial distribution.

 

[BWV]

I wonder if the culprit is that they start dangerously close to the bankrupt state? If they are started in the middle would the majority become millionaires?

The average up move, with 1/2 probability, is a gain of 50%, there is a 1/6 probability of no gain or loss and a 1/3 probability of a 50% loss, the expected % gain is therefore ~ +7%

the variance is [.25/2+0+.25/3] -(.07)^2 ~ 20%

If you approximate this with a lognormal, the median result will be negative if the expectation is less than half the variance

 

[pbuk]

I think that it can still be used to determine the terminal state probabilities for any initial distribution.

Do you mean for any given initial position (distribution)? How would knowledge of an eigenvector help with that (unless the initial position is that of the eigenvector of course)?

Or do you mean the terminal state probabilities are unique for all initial positions? We already know that this is not the case because of the >50% example in #23.

I encourage you again to think about what the eigenvectors must look like. Hint: what are the stable states?

 

[FactChecker]

Do you mean for any given initial position (distribution)? How would knowledge of an eigenvector help with that (unless the initial position is that of the eigenvector of course)?

My memory of this subject is too old to answer this with confidence, but I thought that a decomposition of the Markov state transition matrix allowed us to determine the limit transition after infinite time steps. This could be applied to any vector of a prior distribution to determine its associated terminal probabilities.

Or do you mean the terminal state probabilities are unique for all initial positions? We already know that this is not the case because of the >50% example in #23.

The terminal probabilities are a function of the initial distribution.

I encourage you again to think about what the eigenvectors must look like. Hint: what are the stable states?

Certainly the terminal states are stable, but I am not sure that there is a guarantee in general that there are no stable cycles of states. (Maybe there are no cycles in this case, but I can certainly come up with examples.)

 

[andrewkirk]

We can see that a state vector consisting of zeros except for 1 on one of the two terminal squares (1 and 30) is stable because both such vectors have eigenvalue 1. Call those two states S01 and S30. A state is stable (meaning more time steps don't change it) iff it equals an eigenvector with eigenvalue 1. Stable cycles are another question.

From eigen decomposition of the transition matrix we can learn that S01 and S30 are the only stable eigenvectors, and that the system must converge to a state consisting of a linear combo of S01 and S30. We know that because the eigenvalues of all other 28 eigenvectors have complex modulus less than 1, so repeated multiplication by the trans mtx sends the initial state's projection onto that eigenvector to zero.

We can;t tell the ultimate split between the S01 and S30 eigenvectors just by looking at the matrix though. To work that out, we need to decompose the initial state vector V0 over the eigenvectors as a basis. The components of that decomposition corresponding to S01 and S30 (ie the projections of V0 on S01 and S30) give the relative proportions for the terminal state. Since those components depend on V0, the ultimate split between S01 and S30 also depends on the initial state. That makes sense, because we expect a higher ultimate proportion of S30 (millionairedom) if the starting wealth is higher (less likely to go bankrupt before becoming a millionaire).

 

[WMDhamnekar]

My answer:
As each of 30 squares represents one state in the 30-squares Markov chain, I prepared the transition matrix 'x' in octave. Here is the matrix 'x'

>> x
x =

 Columns 1 through 14:

   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0
   0.3333   0.1667   0.5000        0        0        0        0        0        0        0        0        0        0        0
        0   0.3333   0.1667   0.5000        0        0        0        0        0        0        0        0        0        0
        0   0.3333        0   0.1667        0   0.5000        0        0        0        0        0        0        0        0
        0        0   0.3333        0   0.1667        0   0.5000        0        0        0        0        0        0        0
        0        0   0.3333        0        0   0.1667        0        0   0.5000        0        0        0        0        0
        0        0        0   0.3333        0        0   0.1667        0        0   0.5000        0        0        0        0
        0        0        0   0.3333        0        0        0   0.1667        0        0        0   0.5000        0        0
        0        0        0        0   0.3333        0        0        0   0.1667        0        0        0   0.5000        0
        0        0        0        0   0.3333        0        0        0        0   0.1667        0        0        0        0
        0        0        0        0        0   0.3333        0        0        0        0   0.1667        0        0        0
        0        0        0        0        0   0.3333        0        0        0        0        0   0.1667        0        0
        0        0        0        0        0        0   0.3333        0        0        0        0        0   0.1667        0
        0        0        0        0        0        0   0.3333        0        0        0        0        0        0   0.1667
        0        0        0        0        0        0        0   0.3333        0        0        0        0        0        0
        0        0        0        0        0        0        0   0.3333        0        0        0        0        0        0
        0        0        0        0        0        0        0        0   0.3333        0        0        0        0        0
        0        0        0        0        0        0        0        0   0.3333        0        0        0        0        0
        0        0        0        0        0        0        0        0        0   0.3333        0        0        0        0
        0        0        0        0        0        0        0        0        0   0.3333        0        0        0        0
        0        0        0        0        0        0        0        0        0        0   0.3333        0        0        0
        0        0        0        0        0        0        0        0        0        0   0.3333        0        0        0
        0        0        0        0        0        0        0        0        0        0        0   0.3333        0        0
        0        0        0        0        0        0        0        0        0        0        0   0.3333        0        0
        0        0        0        0        0        0        0        0        0        0        0        0   0.3333        0
        0        0        0        0        0        0        0        0        0        0        0        0   0.3333        0
        0        0        0        0        0        0        0        0        0        0        0        0        0   0.3333
        0        0        0        0        0        0        0        0        0        0        0        0        0   0.3333
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0

 Columns 15 through 28:
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
   0.5000        0        0        0        0        0        0        0        0        0        0        0        0        0
        0   0.5000        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0   0.5000        0        0        0        0        0        0        0        0        0        0
        0        0        0        0   0.5000        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0   0.5000        0        0        0        0        0        0        0
   0.1667        0        0        0        0        0        0   0.5000        0        0        0        0        0        0
        0   0.1667        0        0        0        0        0        0        0   0.5000        0        0        0        0
        0        0   0.1667        0        0        0        0        0        0        0   0.5000        0        0        0
        0        0        0   0.1667        0        0        0        0        0        0        0        0   0.5000        0
        0        0        0        0   0.1667        0        0        0        0        0        0        0        0   0.5000
        0        0        0        0        0   0.1667        0        0        0        0        0        0        0        0
        0        0        0        0        0        0   0.1667        0        0        0        0        0        0        0
        0        0        0        0        0        0        0   0.1667        0        0        0        0        0        0
        0        0        0        0        0        0        0        0   0.1667        0        0        0        0        0
        0        0        0        0        0        0        0        0        0   0.1667        0        0        0        0
        0        0        0        0        0        0        0        0        0        0   0.1667        0        0        0
        0        0        0        0        0        0        0        0        0        0        0   0.1667        0        0
        0        0        0        0        0        0        0        0        0        0        0        0   0.1667        0
        0        0        0        0        0        0        0        0        0        0        0        0        0   0.1667
   0.3333        0        0        0        0        0        0        0        0        0        0        0        0        0
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
Columns 29 and 30:

        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0        0
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
        0   0.5000
   0.1667   0.5000
        0   1.0000        >> x^1000
ans =

 Columns 1 through 14:

   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0
   0.8536   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.7561   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.6910   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.6235   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.5826   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.5351   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.5105   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.4670   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.4312   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.4117   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.3902   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.3626   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.3129   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.3030   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.2978   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.2738   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.2619   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.2476   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1725   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1647   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1647   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1561   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1561   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1450   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1450   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1252   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1252   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
   0.1212   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
 Columns 15 through 28:

        0        0        0        0        0        0        0        0        0        0        0        0        0        0
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
   0.0000   0.0000        0   0.0000   0.0000        0   0.0000   0.0000        0   0.0000        0        0   0.0000   0.0000
        0        0        0        0        0        0        0        0        0        0        0        0        0        0
 Columns 29 and 30:

        0        0
        0   0.1464
        0   0.2439
        0   0.3090
        0   0.3765
        0   0.4174
        0   0.4649
        0   0.4895
        0   0.5330
        0   0.5688
        0   0.5883
        0   0.6098
        0   0.6374
        0   0.6871
        0   0.6970
        0   0.7022
        0   0.7262
        0   0.7381
        0   0.7524
        0   0.8275
        0   0.8353
        0   0.8353
        0   0.8439
        0   0.8439
        0   0.8550
        0   0.8550
        0   0.8748
        0   0.8748
        0   0.8788
        0   1.0000

x(5,30)=0.3765 imply that if any player starts from the x(5,5)=$200000 state, he/she has only 37.65% chance to get millionairedom status.I think my answer is correct. Isn't it?

 

[FactChecker]

@WMDhamnekar , I like your analysis, except at the end. There I have some doubts. Both $x$ and $x^{1000}$ are transition matrices from one state to another, so to say that a player "starts from $x(5,5)$=$200000" bothers me. The starting position should be in a vector, $Initial$, not in the transition matrix. The vector represents the probabilities of starting in each state. So saying that a player starts at $200000 should be saying that $Initial[5]=1$ and all other $Initial$ values are zero. Then the transition matrix should be applied to $Initial$ to determine the probability distribution of the resulting transition. So, starting from those initial values, what is $Final=x^{1000}Initial$? $Final[30]$ might be exactly what you said, but I am not sure.

Also, in a general problem, there might be cycles. In that case, $Final=x^{1000}Initial$ would be the probabilities after exactly 1000 steps. The result of the cycling for 1000 steps would just be a snapshot and might not be representative of any limiting probability.