108156.pdf

70 Mathematics for Finance 5.W(t) has a normal distribution with mean 0 and variancet,thatis,with density√ 21 πte−x 2 2 t. This ...

Risky Assets 71 normal distribution.Thenumberσis called thevolatilityof the priceS(t). The density of the distribution ofS(t) ...

This page intentionally left blank ...

4. Discrete Time Market Models.............................. Having discussed a number of different models of stock price dynami ...

74 Mathematics for Finance respectively, and the risk-free position byy. The wealth of an investor holding such positions at tim ...

Discrete Time Market Models 75 4.1.1 Investment Strategies The positions held by an investor in the risky and risk-free assets ...

76 Mathematics for Finance Definition 4.1 Aportfoliois a vector (x 1 (n),...,xm(n),y(n)) indicating the number of shares and bon ...

Discrete Time Market Models 77 short position is taken in stockj), negativey(n) corresponds to borrowing cash (taking a short ...

78 Mathematics for Finance always determined by the current wealth and the positions in risky assets. Proposition 4.1 Given the ...

Discrete Time Market Models 79 x 1 (1) =−12,x 2 (1) = 31 andy(1) = 2, then it would lead to insolvency, since the time 1 value ...

80 Mathematics for Finance Exercise 4.3 Show that the No-Arbitrage Principle would be violated if there was a self-financing pre ...

Discrete Time Market Models 81 Exercise 4.6 Given the bond and stock prices in Exercise 4.5, is there an arbitrage strategy if ...

82 Mathematics for Finance following three cases: 1)a= 0 (a trivial portfolio consisting of no cash and no stock). ThenV(1) = 0 ...

Discrete Time Market Models 83 Proof This is an immediate consequence of Exercise 3.18 and Proposition 4.2. 4.1.4 Fundamental ...

84 Mathematics for Finance Example 4.5 LetA(0) = 100,A(1) = 110,A(2) = 121 and suppose that stock prices can follow four possibl ...

Discrete Time Market Models 85 By Theorem 4.4 the existence of a risk-neutral probability implies that there is no arbitrage. ...

86 Mathematics for Finance future prices of which may be random (except, of course, at maturity). We shall also relax the assump ...

Discrete Time Market Models 87 Assumption 4.2a (Positivity of Prices) The prices of primary securities, including the money ma ...

88 Mathematics for Finance Definition 4.2a An investment strategy is calledself-financingif the portfolio constructed at timen≥1 ...

Discrete Time Market Models 89 for anyj=1,...,m,anyi=1,...,kand anyn=0, 1 , 2 ,...,whereE∗(·|S(n)) denotes the conditional exp ...