By Andrea Pascucci, Wolfgang J. Runggaldier
Pricing and hedging -- Portfolio optimization -- American thoughts -- rates of interest
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Additional info for Financial mathematics : theory and problems for multi-period models
26, the discounted price S i is a martingale in the measure deﬁned by Q(h = 1) = 1 + r − di , ui − d i i = 1, 2. With the given numerical data, we have 1− 1 + r − d1 = u1 − d 1 3− 1 3 1 3 = 1 , 4 and 1− 1 + r − d2 = u2 − d 2 2− 2 3 2 3 = 1 , 4 and therefore the martingale measure is uniquely determined by q := Q(h = 1) = 14 . ii) Since it will be useful for the third point below, we compute by a backward procedure the arbitrage prices in the ﬁrst period, in the two possible scenarios h = −1, 1 (cases of increase and decrease of the underlying assets): we denote the arbitrage prices by H1u and H1d respectively.
8. In the columns on the right, the values of the European Call and the ﬁnal values of the shortfall strategy with initial value V0 = 13 are shown. Notice that the strategy replicates the payoﬀ in all cases except for the trajectory S0 = S1 = S2 = 1 at which it superreplicates: the ﬁnal value of the shortfall strategy V2 = 13 is strictly greater than the payoﬀ of the Call option that is null. 48 (Hedging in a complete market with insuﬃcient initial capital). 1): S0 = 1, u = 2, d = 1/2, r = 0, K = 1 and N = 2.
60). 10), the proportion of risky asset is equal to αu S u π2u = 2 u1 = 2; H1 in other words, starting from a wealth V1u = 1, the strategy consists of borrowing a unit of bond to buy one unit of risky asset whose value is S1u = 2. 61) is equivalent to α2 + β2 = 12 , α2 4 + β2 = 0, with solution α2d = 2 , 3 1 β2d = − . 60). Therefore π2d = α2d S1d = 2; H1d in other words, starting from an initial wealth V1d = 12 , the strategy consists of borrowing 16 of the riskless asset to buy 23 of risky asset whose value is S1d = 12 .
Financial mathematics : theory and problems for multi-period models by Andrea Pascucci, Wolfgang J. Runggaldier