By Andrea Pascucci, Wolfgang J. Runggaldier

With the Bologna Accords a bachelor-master-doctor curriculum has been brought in numerous international locations to ensure that scholars could input the activity marketplace already on the bachelor point. considering monetary associations offer non negligible activity possibilities additionally for mathematicians, and scientists generally, it seemed to be applicable to have a monetary arithmetic direction already on the bachelor point in arithmetic. such a lot mathematical innovations in use in monetary arithmetic are relating to non-stop time versions and require therefore notions from stochastic research that bachelor scholars do as a rule no longer own. uncomplicated notions and methodologies in use in monetary arithmetic can besides the fact that be transmitted to scholars additionally with out the technicalities from stochastic research by utilizing discrete time (multi-period) versions for which common notions from chance suffice and those are more often than not customary to scholars not just from technological know-how classes, but additionally from economics with quantitative curricula. There don't exists many textbooks for multi-period versions and the current quantity is meant to fill during this hole. It bargains with the elemental subject matters in monetary arithmetic and, for every subject, there's a theoretical part and an issue part. The latter features a nice number of attainable issues of whole answer

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9 3 81 Lastly, in the scenario h1 = 3 we have ⎧ 7 1 22 2 9 5 ⎪ ⎨ 6 α2 + 27 α2 + 4 β2 = 27 , 1 1 1 2 9 2 2 α2 + 3 α2 + 4 β2 = 3 , ⎪ ⎩1 1 1 2 9 8 4 α2 + 9 α2 + 4 β2 = 9 , α21 = from which α21 = 0, α22 = −1, β2 = 4 . 46. 42) for which i Sni = Sn−1 (1 + μi (hn )), n = 1, . . d. with values in {1, 2, 3} and ⎧ ⎪ ⎨ui if h = 1, 1 + μi (h) = mi if h = 2, ⎪ ⎩ di if h = 3. 1 1 1 Choosing u1 = 73 , u2 = 22 9 , m1 = m2 = 1, d1 = 2 , d2 = 3 and r = 2 , we have that there exists a unique equivalent martingale measure Q for which, deﬁning qi := Q(hn = i), i = 1, 2, 3, it holds that q1 = 1 , 2 q2 = 1 , 6 q3 = 1 .

D. with values in {1, 2, 3} and ⎧ ⎪ ⎨ui 1 + μi (h) = mi ⎪ ⎩ di if h = 1, if h = 2, if h = 3. Choosing u1 = 2, m1 = 1, d1 = 12 , u2 = 73 , m2 = have that the martingale measure is deﬁned by Q (hn = 1) = q1 = 3 , 8 Q (hn = 2) = q2 = 3 , 8 7 9, d2 = 1 3 and r = Q (hn = 3) = q3 = 1 4 we 1 . 4 Consider a two-period evolution (N = 2) and an option of the type “collar” with underlying S 1 , whose payoﬀ is given by H2 = min{max{S21 , K1 }, K2 } with K1 = 1, K2 = 2. Determine: i) the initial price H0 of the option; ii) the hedging strategy (α1 , α2 , β).

23), we ﬁrst compute the prices in the ﬁrst period by the 1 risk-neutral formula H1 = 1+r E Q H2 | S12 . We have H1u := 1 1 E Q H2 | S11 = u1 = 1+r 1+ 1 4 (2q1 + 2q2 + q3 ) = 7 . 7 Solved problems 43 Payoﬀ “collar” 1 4 2 2 2 2 1 1 1 1/2 1/2 1 1/4 1 Fig. 6. Two-period trinomial tree: price of the asset S 1 and payoﬀ of a “collar” option Analogously we have H1m = 11 , 10 H1d = 4 . 5 Now we can compute the initial price of the option H0 = 1 1 E Q [H1 ] = 1+r 1+ 1 4 3 u 3 m 1 d H + H1 + H1 8 1 8 4 ii) As regards the ﬁrst period, the replication condition ⎧ 1 1 2 2 u ⎪ ⎨α1 u1 S0 + α1 u2 S0 + β1 (1 + r) = H1 , α11 m1 S01 + α12 m2 S02 + β1 (1 + r) = H1m , ⎪ ⎩ 1 α1 d1 S01 + α12 d2 S02 + β1 (1 + r) = H1d , yields the system ⎧ 7 2 5 7 1 ⎪ ⎨2α1 + 3 α1 + 4 β1 = 5 , 7 2 5 11 1 α1 + 9 α1 + 4 β1 = 10 , ⎪ ⎩1 1 1 2 5 4 2 α1 + 3 α1 + 4 β1 = 5 , with solution α11 = 1, α12 = − 9 , 20 β1 = 9 .