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probs20_39 - Michlmas 2010 Mathematical Finance Questions 4...

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Michælmas 2010 Mathematical Finance: Questions 4 20. A random trial has three possible outcomes with odds o 1 = 1, o 2 = 2, o 3 = 5. Is there a betting scheme based on which outcome occurs that results in a sure win? Suppose that bets on pairs of the outcomes are allowed i.e. it’s possible to bet that either 1 or 2 will occur. What should the odds be on these bets to eliminate any arbitrage opportunity? 21. Consider a binomial price tree two periods long with S 0 = 100, u = 2, d = 1 / 2. Find the price of a European call option with expiry 2 and strike price 150 as a function of r , the interest rate per period. 22. Suppose that in Example 1.1 the stock price after one period is one of 50, 100 or 200 (so the price can remain unchanged). Show there is no longer a unique no-arbitrage price for a (150 , 1) call option and identify a range of values for C which ensure no arbitrage is possible. 23. In a weak arbitrage no outcome leads to a return worse than the risk free interest rate, some returns equal this rate but the return is strictly better for at least one outcome. Show that at either endpoint of the interval for C found in the previous question a weak arbitrage is possible. 24. Suppose that f : R 2 R and define F by F ( u ) = integraltext -∞ f ( x, u ) dx . Suppose that for every fixed x , f ( x, · ) is (i) increasing, (ii) convex. Show that in each case F has the same properties.
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