Answer to Question #190292 in Financial Math for Mathu

Question

Question 2 [9 marks]. This question is about the Arbitrage Theorem.

(a) State the Arbitrage Theorem. [3]

(b) A 6-sided die is rolled. The 6 possible outcomes are 1, 2, 3, 4, 5, 6. You can bet on any outcome. If you bet £1 on i and the outcome is j 6= i, then you lose your pound. But if you bet £1 on i and the outcome is i, then you get back your pound and a reward of £u. For what value of u will this game be arbitrage-free? Justify your answer.

Answer 1

(a)

 The theorem states that either the expected value of all possible investments is the same and equal to the risk-free-rate of return.

(b)

Arbitrage is betting on all possible outcomes ,So certain expected profit is received irrespective of the outcome. To make a gambling game arbitrage free ,the price for winning will be decided such that A gambler cannot do arbitrage gambling. So now lets look at our case. If we bet on all outcomes ,1 pound for each outcome we bet on ,so there are 6 possible outcomes with equal chances. Total amount of betting will be 6 pounds and so if the winning price is greater than 6 pounds every one will do that Since there is a sure profit So winning price should not be greater than 6 Now the winning price is u pounds in this game ,And a return of 1 pound on winning bet.

So total winning will be µ+1

µ+1 should be less than or equal to 6

µ + 1 < = 6

µ <=5

So the winning price should be less than or equal to 5 pounds

Lets see for example one can maximum bet on all 6 outcomes to gain advantage

So he placed 6 pounds on betting. Since only 1 outcome will come in rolling a dice He will lose 5 pounds invested on remaining 5 possible outcomes and the 1 pound on winning outcome is returned any way.

Now he has a loss of 5 pounds ,Since our price money µ is less than 5 pounds. At maximum he will just balance the loss but will never gain profit.

So answer is µ<=5 pounds to make game arbitrage free