108156.pdf

280 Mathematics for Finance

Chapter 4

4.1We can use the formulae in the proof of Proposition 4.1 to find

y(1) =^200 −^35.^24 × 10060 −^24.^18 ×^20 ∼=− 23. 98 ,

V(1)∼= 35. 24 ×65 + 24. 18 × 15 − 23. 98 × 110 ∼= 15. 50 ,

y(2) =^15 .50 + 40.^50 × 11065 −^10.^13 ×^15 ∼= 22. 69 ,

V(2)∼=− 40. 50 ×75 + 10. 13 ×25 + 22. 69 × 121 ∼=− 38. 60.

4.2For a one-step strategy admissibility reduces to a couple of inequalities,V(0)≥

0andV(1)≥0. The first inequality can be written as

10 x+10y≥ 0.

The second inequality means that both values of the random variableV(1)

should be non-negative, which gives two more inequalities to be satisfied byx

andy,

13 x+11y≥ 0 ,

9 x+11y≥ 0.

The set of portfolios (x, y) satisfying all these inequalities is shown in Fig-

ure S.7.

Figure S.7 Admissible portfolios in Exercise 4.2

4.3Suppose that there is a self-financing predictable strategy with initial value

V(0) = 0 and final value 0
=V(2)≥0, such thatV(1)<0 with positive

probability. The last inequality means that this strategy is not admissible, but

we shall construct an admissible one that violates the No-Arbitrage Principle.

Here is how to proceed to achieve arbitrage:

• Donotinvestatallattime0.


• At time 1 check whether the valueV(1) of the non-admissible strategy is
  negative or not. IfV(1)≥0, then refrain from investing at all once again.
  However, ifV(1)<0, then take the same position in stock as in the non-
  admissible strategy and a risk-free position that is lower by−V(1) than
  that in the non-admissible strategy.