Special Topics in Economics
Special Topics in Economics ECN 510
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This 2 page Class Notes was uploaded by Hannah Hahn on Wednesday October 21, 2015. The Class Notes belongs to ECN 510 at Syracuse University taught by Staff in Fall. Since its upload, it has received 61 views. For similar materials see /class/225650/ecn-510-syracuse-university in Economcs at Syracuse University.
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Date Created: 10/21/15
Arbitrage Consider a market with three assets cash bond B15 stock St and derivative E The price vector P at time t 0 and payoff matrix D at time i 1 are up down B0 301R B01R P 50 D Su 8d F0 fa fd Again Map 1 6d0wn p with 0 lt p lt 1 The fact that the payoff of F depends only on whether the stock goes up or down and no other uncertain factors de nes it to be a derivative Notation R is the oneperiod interest rate We also use the continuously compounded rate r de ned by er 6 1 R or r 1111 R6t where 61 is the time step taken as 1 here We also de ne the real numbers u suSO and d sdSO Note that u gt 1 R gt d why Aportfolio is de ned by a vector of weights 6 16 gb x It costs 9P Bo30XF0i and its payoffis with 31 BUl R 6D 21pB1ltigtsu xfu wB1ltb8dxfdl An arbitrage is a portfolio such that either 6PltO and 6DZO or 6Pgo and 6DgtO In words with an arbitrage one either receives a positive amount today and a nonnegative amount tomorrow or a nonnegative amount today and a positive amount tomorrow both with probability 1 Suppose that we know 7quot SO and the payoff matrix Can we obtain from this the no arbitrage price F0 To do so we construct aportfolio 6 Md 1 such that 6 D 00 the absence of arbitrage requires 6 P 0 This choice of w 6 de nes a portfolio of only the stock and the bond that replicates the payoff of the derivative and therefore should have de ne the value and hence price F0 and this distance will go to zero as m n gt 00 because both Ct and Cl converge to Ct Because the distance between Xi and XXquot converges to zero there must be some limit point X where both X and Xfm converge to We ll call this limit point Xt f3 Csst Thus it is de ned by the limit in mean square of T n 03st Z CtBt Ban 0 z391 This de nition also gives the integral at time t instead of T by replacing C5 by Caliojt s This general Ito integral inherits the most important properties of the integral of a simple process Xt is a martingale with quadratic variation process ltXgtt f0tdX5 2 Cfds Thus its variance is also varXt ngs and if Ct is nonstochastic then t t Xt Cst N Cfds 0 0 r l It Processes and Ito s Lemma We now have given meaning to the integral equation X asdBS where 05 satis es the same properties as Ci be fore Now let at be another adapted process a function of BS 8 g t and de ne t t Xt X0 usds USdBS 0 0 Then Xt is called an Ito process We call ut the drift of the process and at the volatility Note that the rst integral is an ordinary Riemann integral whereas the second is a stochastic integral The integral equation may be reexpressed in differential form dXt UtdBt This is just another way of writing the integral equation the meaning of a differential term like atdB stems from the corresponding stochastic integral