Chapter 2 A primer on the arbitrage theorem

Introduction(5)
4. Comparing arbitrage-free prices with arbitrageactual prices, traders can find excess profit opportunities.
Notation(1)
Asset Prices
S1 (t ) S (t ) 2 St = M S N (t )
Relevance of the Arbitrage Theorem
Consider again the representation(2-1), Multiplying representation(2the first row of the dividend matrix by the vector of ψ 1 ,ψ 2, we get: 1 = (1 + r )ψ + (1 + r )ψ 1 2 % % define: P1 = (1 + r )ψ 1 , P2 = (1 + r )ψ 2 . Because of the 2 % % positive of state prices, we get: 0 < Pi < 1 and ∑ Pi = 1 i =1 % ' s can be interpreted as two probabilities with the Pi two states under consideration. They are different from the true probabilities that govern the occurrence of the two states of the world. So they are called risk-adjusted synthetic probabilities. risk-
Returns and payoffs
d11 d12 L d1K d d L d 2K 21 22 D= LLLL d N1 d N 2 L d NK
where dij denote the number of units of account paid by one unit of security i in state j. dij includes two components-capital gains or losses and componentspayoutspayouts- dividends or coupon interest payments.
A Basic Example of Asset Pricing(1)
Considering a case with time assumingly consisting of “now” and a “next period” which are separated by an interval of length ∆ . Suppose that market participant is interested only in three assets: 1. A risk-free asset such as Treasury-bill whose gross riskTreasuryreturn until next period is (1 + r ∆) 2. A underlying asset, for example, a stock S (t ) 3. A derivative asset, a call option with premium C (t ) and strike price C0
Introduction(2)
In the first way, one can make a series of investment with no current net commitment, yet expect to make a positive profit. For example, one can short-sell a stock and use shortthe proceeds to buy call options written on the same security…… In the second kind, a portfolio can ensure a negative net commitment today, while yielding nonnegative profits in the future.
(1 + r ) 1 (1 + r ) S (t ) = S (t + 1) S (t + 1) ψ 1 , 2 ψ 1 C (t ) C1 (t + 1) C2 (t + 1) 2 (2 − 1)
then there are no arbitrage possibilities; and if there are no arbitrage opportunities, then positive constants ψ 1 ,ψ 2 satisfying can be found.
where the index t will represent time. St may be vector of asset prices at time of t.
Notation(2)
States of the World
ω1 ω W = 2 M ωK
The Use of Synthetic Probabilities(1)
The representation (2-1) implies three separate equalities: (21 = (1 + r )ψ 1 + (1 + r )ψ 2 S (t ) = ψ 1S1 (t + 1) +ψ 2 S2 (t + 1) C (t ) = ψ 1C1 (t + 1) +ψ 2C2 (t + 1)
Notation(4) Each row represents
d11 d12 L d1K d d L d 2K 21 22 D= LLLL d 1 d N 2 L d NK
payoffs to one unit of a given security in different states of the word.
A Basic Example of Asset Pricing(2)
Asset prices will form a vector St of only three elements, B(t ) St = S (t ) C (t ) Payoffs will be grouped in a matrix Dt
CHAPTER 2
A Primer on the Arbitrage Theorem
Introduction(1)
In its simplest form, arbitrage means taking simultaneous position in different assets so that one guarantees a riskless profit higher than the riskless return given by U.S Treasury bills. If such profits exits, we say that there is an arbitrage opportunity. Arbitrage opportunities can arise in two different fashions.
Introduction(4)
1.One case may be when a derivatives house decides to engineer a new financial product whose actual price process can not be observed. 2.In risk management, future uncertainties are a series of hypothetical evens that has not been observed. 3.In the process of marking to market. When a treasurer want to know the current value of a nonliquid asset.
(1 + r ∆ ) B (t ) Dt = S1 (t + ∆) C1 (t + ∆ ) (1 + r ∆) B (t ) S2 (t + ∆ ) C2 (t + ∆ )
A First Glance at the Arbitrage Theorem
Let B(t)=1, ∆ = 1 . Given St and Dt ,and that the two states have positive probabilities of occurrence. THEOREM: THEOREM: if positive constants ψ 1 ,ψ 2 (state price) can be found such that asset prices satisfy :
where each ωi represents a distinct outcome that may occur. These states are mutually exclusive, and at least one of them is guaranteed to occur.
Notation(3)