rev_probs - the amounts paid o² the principal do add up to...

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Revision for Mathematical Finance 1. Show that (1 + a/n ) n e a as n → ∞ , where a 0 is constant. Hint: Analyse the behaviour of (1 + a/x ) x as x → ∞ by taking log and applying Taylor’s theorem (treating the remainder with professional care!). Alternatively show directly from the definition of log that a/ ( n + a ) < log(1 + a/n ) < a/n for n > a and deduce the result from this. 2. You take out a loan for the amount L . For the next n months you will pay back amount a at the end of each month. Interest accrues at rate r per month, that is if you owe R j after the payment at the end of month j you will owe (1+ r ) R j at the end of the next month (before your next payment is made). Express the amount a that leads to the loan being paid back exactly in terms of L , n and r . Hint: let R j be the amount owed at the end of month j (after the payment, so R 0 = L and R n = 0) and relate R j to R j +1 . Check that a > rL . Why is this essential? How much of the payment at the end of month j was for interest and how much for paying off the loan principal? Check that
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Unformatted text preview: the amounts paid o² the principal do add up to L . 3. Suppose X ∼ N ( μ, σ 2 ) and Y = a + bX ( b n = 0). ±ind E ( Y ), Var( Y ) and Cov( X, Y ). What is the distribution of Y ? ±ind E ( e X ). 4. In each round of a game you win 0 . 15 with chance 0 . 55 or lose 0 . 1 with chance 0 . 45. Successive rounds are independent and you play 1,000 times. Use the Central Limit Theorem to estimate the chance that your total winnings exceed 35. 5. Suppose that over a unit period the price of a stock changes from s to either us with chance p or ds with chance 1 − p . Successive price changes are independent. Estimate the probability that the stock price will increase by at least 30% over the next 100 time periods when u = 1 . 02, d = 0 . 98 and p = 0 . 52. Hint: switch to log scale and compare with the previous problem. 1...
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