Problem Solving Assignment: Mathematics Contingent Claims

Mathematics of Contingent Claims
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Institution
Date
1] How many rolls of a k- side dice should you expect to make before all numbers 1,2…k have been seen at least once
Assuming that the die is 6-sided, then if we roll once, then we could roll and get 5 different sides when we roll
As such we get1/ (5/6) being 6/5 rolls, until the last success being 6/1
The probabilities of the geometric variables are then 6/6, 6/5,6/4,6/3,6/2 and 6/1
For the k-sided dice then the number of rolls is
1+k/ (k-1) + k/ (k-2) +…+k/1=
k
k
Σ k/1 =
k Σ 1/i
i=1
i=1
The dice problem follows the Coupon Collector’s problem, where the success in each of the trails does not change, but then the probability of success reduces after each success (Tijms, 2012, pp 83-84). The Coupon Collector’s problem is also applicable using the Monte Carlo Simulation, where the number of rolls and sides per dice is considered.
2] Consider a dice game in which you are given at most n rolls to get the highest score possible. You final score is the number showing on the last dice rolled, when you either run out or you choose to stop. Find the optimal strategy for deciding when to stop rolling and take the current dice score. Compute the expected value of your score when this strategy is followed for all values n.
Assumption
Assuming that the dice has 6 sides
In this case f (1) = (k+1), which is (6+1)/2=7/2
Then if n=1 for a six sided dice then the average score is 7/2
When n=2, then the score should be > 3.5
Then f (2) = (6/6) + (5/6) + (4/6) + (3/6) (7/2) =4.25
Expected value
For the dice game with n rolls, with k sides, then the expected value is
f(n-1)
f(n-1)
 + Σ
j
k
j=f(n-1)+1
k
F (n) =
3] A knight moves on a 5x5 chessboard, making any of the allowed moves that keep it on the board with equal probability. (Note- a knight’s move in chess is two steps in one direction, followed by one in the other direction. Find the along-time average fraction of the time the knight spends in the centre of the board.
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In the chessboard if the ...