Arbitrage Theory in Continuous Time by Tomas Björk

By Tomas Björk

Combining sound mathematical rules with the mandatory monetary concentration, Arbitrage conception in non-stop Time is particularly designed for graduate scholars, and comprises solved examples for each new method offered, a number of routines, and instructed analyzing lists for every bankruptcy.

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Om these conditions, and from the risk neutral valuation formula, it follows 1 . E~ V ~= h (1 + R)T [v-]> 0, which shows that h is not an arbitrage portfolio. 6. Show, in the one period binomial model, that if lI(1;X) probability 1, then you can make a riskless profit. 21. 3 Consider the multiperiod example in the text. 0. Show explictly how you can make an arbitrage profit. 24, by using induction on the time horizon T. For the origins of the binomial model, see Cox, Ross and Rubinstein (1979), and Rendleman and Bartter (1979).

The proof is left to the reader. Fh. 21 Suppose that X is reachable using the portfolio h. Then it is possible to make an arbitrage profit. 1 1 We now turn to the completeness of the model. e. every clairiz can be replicated by a self-financing portfolio. It is possible, and not very hard, to give a formal proof of the proposition, using mathematical induction. The formal proof will, however, look rather messy with lots of indices, so instead we prove the proposition for a concrete example, I1 1 THE MULTIPERIOD MODEL 19 using a binomial tree.

We now consider a particular contingent claim, namely a European call on the underlying stock. The date of expiration of the option is T = 3, and the strike price is chosen to be K = 80. Formally this claim can be described as X = max [ST- K, 01. We will now show that this particular claim can be replicated, and it will be obvious from the argument that the result can be generalized to any binomial model and any claim. The idea is to use induction on the time variable and to work backwards in the tree from the leaves at t = T to the root at t = 0.

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