e62: lecture 2
22/09/10
Mathematical Problems of E62
1
Algebraic
Geometric
linear algebra
intersecting hyperplanes
linear combinations
linear programs
polyhedra
optima
duality
constraints versus
objective
quadratic programs
gradients
local minima
max
cx
s
.
t
.
Ax
≤
b
x
≥
0
Ax
=
b
min
yb
s
.
t
.
yA
≥
c
y
≥
0
max
cx

1
2
x
T
Hx
s
.
t
.
Ax
≤
b
x
≥
0
e62: lecture 2
22/09/10
•
A context and application area for LA and LP
•
A contingent claim is a contract
•
Receive a payoff depending on an uncertain outcome
•
May pay a price to purchase the contract
•
Examples
•
Lottery
•
Insurance
•
Negotiated contract with contingencies
•
Stocks, bonds, options, and other derivatives
•
Mathematical representation
•
Enumerate possible outcomes 1, 2
, .
.., M
•
Specify outcomecontingent payoffs
Contingent Claims
2
a
∈±
M
e62: lecture 2
22/09/10
Stocks and Bonds
•
Assume one year holding period
3
stock
zerocoupon bond
price
p
1
p
2
outcomes
future stock price = 1
, .
.., M
future stock price = 1
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 '10
 VANROY
 Linear Programming, payoﬀ, future stock price, payoff vector

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