Stochastic Calculus Solution Manual
This is a solution manual for the twovolume 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 NoArbitrage Pricing Model
Proof. If we get the up sate, then X1 = X1 (H) = …show more content…
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 riskneutral pricing relies on fair price=replication cost. Stock as a replicating component cannot determine its own “fair” price via the riskneutral 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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