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Unformatted text preview: Stochastic Calculus for Finance, Volume I and II by Yan Zeng Last updated: August 20, 2007 This is a solution manual for the two-volume textbook Stochastic calculus for finance , by Steven Shreve. If you have any comments or find any typos/errors, please email me at [email protected] The current version omits the following problems. Volume I: 1.5, 3.3, 3.4, 5.7; Volume II: 3.9, 7.1, 7.2, 7.5–7.9, 10.8, 10.9, 10.10. Acknowledgment I thank Hua Li (a graduate student at Brown University) for reading through this solution manual and communicating to me several mistakes/typos. 1 Stochastic Calculus for Finance I: The Binomial Asset Pricing Model 1. The Binomial No-Arbitrage Pricing Model 1.1. Proof. If we get the up sate, then X 1 = X 1 ( H ) = Δ uS + (1 + r )( X- Δ S ); if we get the down state, then X 1 = X 1 ( T ) = Δ dS + (1 + r )( X- Δ S ). If X 1 has a positive probability of being strictly positive, then we must either have X 1 ( H ) > 0 or X 1 ( T ) > 0. (i) If X 1 ( H ) > 0, then Δ uS + (1 + r )( X- Δ S ) > 0. Plug in X = 0, we get u Δ > (1 + r )Δ . By condition d < 1 + r < u , we conclude Δ > 0. In this case, X 1 ( T ) = Δ dS + (1 + r )( X- Δ S ) = Δ S [ d- (1 + r )] < 0. (ii) If X 1 ( T ) > 0, then we can similarly deduce Δ < 0 and hence X 1 ( H ) < 0. So we cannot have X 1 strictly positive with positive probability unless X 1 is strictly negative with positive probability as well, regardless the choice of the number Δ . Remark: Here the condition X = 0 is not essential, as far as a property definition of arbitrage for arbitrary X can be given. Indeed, for the one-period binomial model, we can define arbitrage as a trading strategy such that P ( X 1 ≥ X (1 + r )) = 1 and P ( X 1 > X (1 + r )) > 0. First, this is a generalization of the case X = 0; second, it is “proper” because it is comparing the result of an arbitrary investment involving money and stock markets with that of a safe investment involving only money market. This can also be seen by regarding X as borrowed from money market account. Then at time 1, we have to pay back X (1 + r ) to the money market account. In summary, arbitrage is a trading strategy that beats “safe” investment. Accordingly, we revise the proof of Exercise 1.1. as follows. If X 1 has a positive probability of being strictly larger than X (1 + r ), the either X 1 ( H ) > X (1 + r ) or X 1 ( T ) > X (1 + r ). The first case yields Δ S ( u- 1- r ) > 0, i.e. Δ > 0. So X 1 ( T ) = (1+ r ) X +Δ S ( d- 1- r ) < (1+ r ) X . The second case can be similarly analyzed. Hence we cannot have X 1 strictly greater than X (1 + r ) with positive probability unless X 1 is strictly smaller than X (1 + r ) with positive probability as well....
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