Stochastic Calculus Solution Manual

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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 yz44@cornell.edu. 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
1.1.

Stochastic Calculus for Finance I: The Binomial Asset Pricing Model

1. The Binomial No-Arbitrage Pricing Model

Proof. If we get the up sate, then X1 = X1 (H) =
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Remark: This illustrates an important point. The “fair price” of a stock cannot be determined by the risk-neutral pricing, as seen below. Suppose S1 (H) and S1 (T ) are given, we could have two current prices, S0 and S0 . Correspondingly, we can get u, d and u , d . Because they are determined by S0 and S0 , respectively, it’s not surprising that risk-neutral pricing formula always holds, in both cases. That is, 1+r−d u−d S1 (H)

S0 =

+

u−1−r u−d S1 (T )

1+r

, S0 =

1+r−d u −d

S1 (H) +

u −1−r u −d S1 (T )

1+r

.

Essentially, this is because risk-neutral pricing relies on fair price=replication cost. Stock as a replicating component cannot determine its own “fair” price via the risk-neutral pricing formula. 1.4. Proof. Xn+1 (T ) = = ∆n dSn + (1 + r)(Xn − ∆n Sn )

∆n Sn (d − 1 − r) + (1 + r)Vn pVn+1 (H) + q Vn+1 (T ) ˜ ˜ Vn+1 (H) − Vn+1 (T ) (d − 1 − r) + (1 + r) = u−d 1+r = p(Vn+1 (T ) − Vn+1 (H)) + pVn+1 (H) + q Vn+1 (T ) ˜ ˜ ˜ = pVn+1 (T ) + q Vn+1 (T ) ˜ ˜ = Vn+1 (T ).

1.6.

2

Proof. The bank’s trader should set up a replicating portfolio whose payoff is the opposite of the option’s payoff. More precisely, we solve the equation (1 + r)(X0 − ∆0 S0 ) + ∆0 S1 = −(S1 − K)+ .
1 Then X0 = −1.20 and ∆0 = − 2 . This means the trader should sell short 0.5 share of stock, put the income 2 into a money market account, and then transfer 1.20 into a separate money market account. At time one, the portfolio consisting

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