The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 413 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 413

With the above equations we computed prices from payoffs and state-

price deflators. If prices and payoffs were given, we could compute state-

price deflators from the homogeneous system for state prices established

above. Suppose that the following price processes were given:

{ S^1 () ω}=

t

{ S^2 () ω}=

t

{ S^3 () ω}=

t

68.75 68.75 0

68.75 68.75 0

68.75 78.05 0

68.75 78.05 0

73.2 69.37 0

73.2 69.37 0

73.2 78.88 0

73.2 78.88 0

67.125 69.12 0

67.125 69.12 0

67.125 75.27 0

67.125 75.27 0

We could then write the following system of equations to compute state-

price deflators:

0.25 × 50 × π 2 () 1 + 0.25 × 100 × π 2 () 2 – 68.75 ×0.5 × πA 11 , = 0

0.25 × 70 × π 2 () 1 + 0.25 × 110 × π 2 () 2 – 78.05 ×0.5 × πA 11 , = 0

( 55 ×0.5 + 0.5 × 15 ) × πA 11 , + (70.25 ×0.5 + 0.5 × 20 ) × πA 21 ,

• 68.75 × πA 10 , = 0

0.25 × 30 × π 2 () 1 + 0.25 × 120 × π 2 () 2 – 69.37 ×0.5 × πA 11 , = 0

0.25 × 40 × π 2 () 1 + 0.25 × 140 × π 2 () 2 – 78.88 ×0.5 × πA

11

= 0

,

(55.5 ×0.5 + 0.5 × 8 ) × πA 11 , + ( 71 ×0.5 + 0.5 × 15 ) × πA 21 ,

• 73.2 × πA
  10
  = 0
  ,