The Arbitrage Theorem

1
The Arbitrage Theorem

• Henrik Jönsson
• Mälardalen University
• Sweden

2
Contents

• Necessary conditions
• European Call Option
• Arbitrage
• Arbitrage Pricing
• Risk-neutral valuation
• The Arbitrage Theorem

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Necessary conditions

• No transaction costs
• Same risk-free interest rate r for borrowing
  lending
• Short positions possible in all instruments
• Same taxes
• Momentary transactions between different assets
  possible

4
European Call Option

• C - Option Price
• K - Strike price
• T - Expiration day
• Exercise only at T
• Payoff function, e.g.

5
Arbitrage
The Law of One Price In a competitive market,
if two assets are equivalent, they will tend to
have the same market price.
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Arbitrage

• Definition
• A trading strategy that takes advantage of two or
  more securities being mispriced relative to each
  other.
• The purchase and immediate sale of equivalent
  assets in order to earn a sure profit from a
  difference in their prices.

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Arbitrage

• Two portfolios A B have the same value at tT
• No risk-less arbitrage opportunity ?
• They have the same value at any time t?T

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Arbitrage Pricing

• The Binomial price model

0?d?u
0?q?1
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Arbitrage Pricing
r risk-free interest rate
d lt (1r) lt u

• 1r lt d

• 1r gt u

Action at time 0 t0 tT
Borrow S -(1r)S
Buy stock -S dS
Return 0 gt0
Action at time 0 t0 tT
Lend -S (1r)S
Sell stock S -uS
Return 0 gt0
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Arbitrage Pricing

• Equivalence portfolio

• Call option

(tT)
(t0)
r risk-free interest rate
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Arbitrage Pricing

• Choose ? and B such that

12
Arbitrage Pricing
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Risk-neutral valuation
Expected rate of return (1r)
( p equivalent martingale probability )

• p risk-neutral probability

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Risk-neutral valuation

• Expected present value of the return 0

Price of option today Expected present value of
option at time T
C (1r)-1pCu (1-p)Cd
Risk-neutral probability p
( p equivalent martingale probability )
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The Arbitrage Theorem

• Let X?1,2,,m be the outcome of an experiment
• Let p (p1,,pm), pj PXj, for all j1,,m
• Let there be n different investment opportunities
• Let ? (?1,, ?n) be an investment strategy (?i
  pos., neg. or zero for all i)

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The Arbitrage Theorem

• Let ri(j) be the return function for a unit
  investment on investment opportunity i

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The Arbitrage Theorem

• If the outcome Xj then

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The Arbitrage Theorem

• Exactly one of the following is true Either
• there exists a probability vector p (p1,,pm)
  for which
• or
• b) there is an investment strategy ? (?1,,
  ?m) for which

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The Arbitrage Theorem

• Proof Use the Duality Theorem of Linear
  Programming

Primal problem
Dual problem

• If x primal feasible y dual feasible then
• cTx bTy
• x primal optimum y dual optimum
• If either problem is infeasible, then the other
  does not have an optimal solution.

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The Arbitrage Theorem

• Proof (cont.)

Primal problem
Dual problem
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The Arbitrage Theorem
Proof (cont.)

• Dual feasible iff y probability vector
  under which all investments have the expected
  return 0

• Primal feasible when ?i 0, i1,, n,

cT? bTy 0 ? Optimum! No sure win is
possible!
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The Arbitrage Theorem

• Example
• Stock (S0) with two outcomes
• Two investment opportunities
• i1 Buy or sell the stock
• i2 Buy or sell a call option (C)

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The Arbitrage Theorem

• Return functions
• i1
• i2

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The Arbitrage Theorem

• Expected return
• i1
• i2

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The Arbitrage Theorem

• (1) and the Arbitrage theorem gives
• (2), (3) the Arbitrage theorem gives the
  non-arbitrage option price

(3)