Download Arbitrage. Risk neutral valuation relationship

Session VI:
Arbitrage. Risk neutral valuation relationship
Lecturer: Dr. Jose Olmo
Module: Economics of Financial Markets
MSc. Financial Economics
Department of Economics, City University, London
The absence of arbitrage in investment decisions occurs when strategies entailing zero
risk have the risk-free rate of return. Therefore zero payoffs are result of zero initial investments. This definition of arbitrage is a generalization of the law of one price (LoOP)
that asserts that two assets traded in the same market under the same conditions should
have the same price. Hence if two assets have different prices there exists an arbitrage
opportunity by buying the asset at the lower price and selling it at the higher price. Individuals adopting this strategy make a profit out of nothing (without taking any risk). This
strategy cannot persist long due to the forces driving financial markets.
The absence of arbitrage itself does not provide any criteria for investors to choose
their portfolios and in turn cannot determine asset prices. It simply places restrictions
on asset prices in order to be consistent with market equilibrium. This assumption on
market behavior just rules out the existence of prices incompatible with equilibrium market
conditions.
Arbitrage is defined by actions seeking secure profits in investment decisions without
bearing risk. An arbitrage portfolio satisfies the following conditions.
1
2
• Wealth of the initial portfolio is zero. Assets can be held in positive and negative
amounts but the net value of the portfolio is zero.
• It is risk-free and provides positive or at least null payoffs in every state of the nature.
An arbitrage opportunity is given by the exploitation of an arbitrage portfolio. The
arbitrage principle is defined by the absence of arbitrage opportunities. Thus, the return
on an initial investment A under no arbitrage is either R0 (return on a risk-free asset) or
positive for some states of nature and negative for others. The arbitrage principle does
not place restrictions either on investor’s behavior or preferences. The only assumption
on their behaviour is that they prefer more wealth to less and that they try to achieve
an optimal portfolio. On the other hand if there are arbitrage opportunities this optimal
portfolio does not exist because investors can always exploit these opportunities to obtain
unbounded profits. Therefore if there exist investors with an optimal portfolio arbitrage
opportunities are absent.
Consider the example introduced in Bailey (2005, section 7.2.1.)
Suppose three assets A, B, C trading in an economy where the uncertainty is modelled
by two states of nature: State 1, 2. The sketch of prices and payoffs is in the following
table:
Payoff A
Payoff B
Payoff C
State 1
10
8
9
State 2
8
0
12
Price
3
2
pc
Denote xA , xB and xC for the share of each asset invested in a portfolio P. Then
wA + wB + wC = 1.
We further assume this portfolio P have zero initial outlay. Then, it satisfies
3wA + 2wB + pc wC = 0.
(1)
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The arbitrage principle ensures that the following restrictions will hold for this portfolio
10wA + 8wB + 9wC = 0,
8wA + 0wB + 12wC = 0.
These conditions imply that wC = − 32 wA , and wB = − 12 wA .
Substituting in equation (1) we obtain
3wA − wA − 32 pc wA = 0.
Then (2 − 32 pc )wA = 0. The solutions for this equation are either wA = 0 or pC = 3 with
wA > 0. Note that wA = 0 does not make sense when considering a portfolio of three
assets. Any other price for C yields arbitrage opportunities.
Consider an arbitrage strategy given by pC = 1. A portfolio with wA = −1, wB =
wC = 1 has zero outlay, that is, it is free to buy this portfolio. However the payoffs of this
portfolio are
In state 1:
−1 × 10 + 1 × 8 + 1 × 9 = 7.
In state 2:
−1 × 8 + 1 × 0 + 1 × 12 = 4.
This portfolio provides strictly positive payoffs for both states of nature with a zero initial
outlay. In case this arbitrage opportunity existed the own market conditions would make
it disappear by lowering the price of A and rising the prices of B and C.
This example illustrates the following characteristics of the arbitrage principle.
• The arbitrage principle is an extension of the law of one price. There exists a link
between prices and future payoffs to prevent arbitrage opportunities.
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• The arbitrage principle only rules out the existence of portfolios with zero initial
investment obtaining a secure profit. This principle does not prevent portfolios with
zero initial outlays from obtaining profits for some states of nature and losses for
others.
• Market frictions and transaction costs must be absent. Otherwise there can exist
arbitrage opportunities that are not eliminated by the laws of demand and supply.
Market frictions and transaction costs can make this freelunch worthless.
• The arbitrage principle does not provide itself rules to derive assets prices in market
equilibrium. It only guarantees the existence of relation between prices and payoffs.
Replication of Portfolios
The absence of arbitrage in financial markets provides a very powerful tool for pricing
derivatives. Derivatives are financial instruments whose price is contingent upon the price of
another underlying asset. The valuation strategy consists on replicating the future payoffs
of the derivative as a linear combination of the future payoffs of some linearly independent
assets trading in the market. If it is possible to do so the arbitrage principle is sufficient to
find the price of the derivative.
Ex.- An investor is interested in buying a petrol station. Its future payoffs depend on
future prices of oil. The current price of oil is $12 per gallon. The future can be divided
in two states of nature: s=1,2. At t = 1 oil prices for state s = 1 is $20 per gallon; and
for state s = 2 is $10 per gallon. The payoffs of the petrol station are $20M for s = 1
and $5M for s = 2 respectively. The risk-free interest rate for the period outstanding from
t = 0 to t = 1 is R0 = 10%.
We can construct a portfolio that replicates the payoffs of the petrol station with the
risk-free asset and oil (underlying asset).
1xR0 + 20xoil = 20M ,
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1xR0 + 10xoil = 5M .
The portfolio is xoil = 1.5M of oil gallons, and xR0 = −10M . If there are no arbitrage
opportunities the price of the petrol station should be
−10M × p1 + 1.5M × p2 = x.
Today’s price of the risk-free asset is
1
1.10
= 0.91 and of oil is p2 = $8. Then
−9.1M + 18 = x. ⇒ x = 8.9M .
The no-arbitrage price of the petrol station is $8.9M .
Note that if there were four states of nature we would need four linearly independent
assets to price the petrol station.
State Prices
The strategy of replicating portfolios can be used to assign prices to each state of nature.
These are called Arrow-Debreu prices. The Arrow-Debreu price for state s is constructed
by replicating a portfolio that provides a payoff of 1 for state s and zero for the rest of
states of nature.
In the former example the Arrow-Debreu payoffs corresponding to state 1 are given by
20 × 1 + 10 × 0 = 20,
and for state 2,
20 × 0 + 10 × 1 = 10.
Under no arbitrage opportunities the cost of this portfolio is
6
20φ1 + 10φ2 = 12.
On the other hand the risk free asset provides the same payoffs in every state of nature.
That is
1 × 1 + 1 × 0 = 1,
1 × 0 + 1 × 1 = 1.
The no-arbitrage price should be
1φ1 + 1φ2 = 0.91.
The solution of this system of equations is φ1 = 0.29 and φ2 = 0.62. These values are the
prices assigned by the market to each state. For the risk-free asset we have
1φ1 + 1φ2 =
1
.
1+R0
The parameters (φ1 , . . . , φS ) allow one to calculate the value of any asset traded in the
market. These prices reflect
• investors’ willingness to pay per one unit of future consumption in each state of
nature.
• Uncertainty associated with each state.
• Temporal value of money.
For S states of nature the price of an asset j is given by the portfolio of Arrow-Debreu
assets (vj1 , . . . , vjS ) with prices (φ1 , . . . , φS ). Note that vjs in this context denotes the share
of asset j invested in state s.
pj =
S
P
φs vjs .
s=1
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This equation is called the fundamental valuation relationship. It reads in matrix form as

 
v
 11
p1

   ·
  
· 
  
 = ·
  
· 
   ·

pn
vn1

 
 φ
  1
· 




·
 


·  
 · 


·  
 φ
. . . . . . v1S
... ...
... ...
... ...
. . . . . . vnS
s
The no arbitrage condition can be formally restated using Arrow-Debreu prices and the
fundamental valuation relationship as follows.
Fundamental Theorem of No Arbitrage
Consider an economy with n financial assets and two periods t and T where the uncertainty is modelled by S states of nature. Asset prices are denoted p1 , . . . , pn ; and vjs
denote the payoff of one unit of asset j in state s. A portfolio is characterized by a vector
x = (x1 , . . . , xn ) with xj the number of shares of asset j. The price of this portfolio at t = 0
n
P
xj pj .
is
j=1
Its payoffs are
n
P
xj vjs , s = 1, . . . , S.
j=1
More compactly,

n
P

 j=1xj vj1 




 ... 






 ... 

 n

P
xj vjS
j=1
Suppose that investors’ aim is to minimize the cost of a portfolio giving positive payoffs
in every state of nature. Mathematically, the problem is expressed as
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M = min
n
P
xj p j
j=1
s.t.
n
P
xj vjs ≥ 0, ∀ s = 1, . . . , S.
j=1
If there exists a solution to this problem this must be M = 0. Otherwise for M < 0,
M = xp0 , we can always find the portfolio with cost 2M and payoffs 2xV 0 with


v
 11

 ·


V=
 ·


 ·

v1S
. . . . . . vn1
... ...
... ...
... ...


· 


· 



· 

. . . . . . vnS
The payoffs of this portfolio are greater than those of the portfolio with value M for every
state of nature. Thus Portfolio M cannot be optimal.
Definition 1. A financial market is free of arbitrage opportunities if and only if the cost
of the optimal portfolio with payoffs higher or equal than zero for every state of nature is
zero. That is
xV 0 ≥ 0 if and only if xP 0 ≥ 0.
Hence the theorem of no arbitrage can be stated as follows
Theorem 1. A portfolio with prices P = (p1 , . . . , pn ) is free of arbitrage opportunities if
and only if there exists a vector Φ ∈ <S with φi ≥ 0, ∀ i = 1, . . . , S and such that p = ΦV .
Proof 1. ⇐ (Necessary condition). Suppose a portfolio x = (x1 , . . . , xn ) with positive
payoffs for every state of nature: xV 0 ≥ 0. If there exists a vector Φ = (φ1 , . . . , φS ) with
φi ≥ 0, ∀ i = 1, . . . , S satisfying P = ΦV , then by multiplying by x we obtain xΦV ≥ 0.
This expression amounts to xP ≥ 0.
The cost of any feasible portfolio yielding positive payoffs must be greater or equal than
zero. The minimum value is attained for x = 0. This is the definition of absence of
arbitrage.
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⇒ (Sufficient condition). Suppose a portfolio represented by P = (p1 , . . . , pn ) and
(x1 , . . . , xn ) and a market free of arbitrage opportunities. This means that the solution
to the minimization problem stated before exists and is zero. The lagrangian of the problem
is as follows
L=
n
P
n
P
xj p j − φ 1
j=1
xj vj1 − . . . − φS
j=1
n
P
xj vjS ,
j=1
with φi ≥ 0. (by definition of lagrange multipliers).
The solutions to this problem are
pj = φ1 vj1 + . . . + φS vjS , j = 1, . . . , n.
In matrix form P = ΦV .
Risk neutral probabilities
Under no arbitrage opportunities asset prices are related to the matrix of future payoffs
by the Arrow-Debreu prices. Dividing each component of P = ΦV 0 by the price of each asset
pj we obtain the fundamental valuation equation in terms of asset returns. Transposing
the matrices this equation reads as

 
1 + R11

1

  
   1 + R21
· 
  
 = ·
· 
  
   ·

1
1 + Rn1

 
 φ
  1
1 + R2S 


· 


 
·
 
 · 
 


·
 φ
S
1 + RnS
. . . . . . 1 + R1S
... ...
... ...
... ...
... ...
with vjs = (1 + Rjs )pj .
If one of the assets in the market is risk-free the first row satisfies
1=
S
P
φs (1 + R0 ).
s=1
10
Hence we can define
πs∗ ≡ (1 + R0 )φs , s = 1, . . . , S.
These quantities can be interpreted as probabilities since 0 ≤ πs∗ ≤ 1, and
S
P
πs∗ = 1.
s=1
These probabilities are associated to the states of nature. Note that these probabilities
do not correspond to the true likelihood of occurrence of the state. We will use πs to denote
the original probabilities; these probabilities in general differ from the probabilities derived
from the Arrow-Debreu prices. They are denominated risk neutral probabilities. The
existence of these probabilities implies absence of arbitrage opportunities and viceversa;
the arbitrage principle implies risk neutral probabilities.
Risk neutral valuation relationship
The fundamental valuation equation can be written in terms of risk neutral probabilities
as
 
1
 
 p2 
 
 =
·
 
 
pn

1
1+R0

1


 v21


 ·



 ·

vn1
... ...
... ...
... ...
... ...
... ...
 
 π∗
  1
v2S 




·
 



· 

·


·  
 π∗
S
vnS
1
The price of each asset is
pj =
1
1+R0
S
P
πs∗ vjs .
s=1
The value of asset j today can be interpreted as the expected value under risk neutral
probabilities of the future cash flows of the asset across different states of nature discounted
by the risk-free interest rate.
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pj =
1
E ∗ [vj ].
1+R0
This expression permits one to calculate the price of any asset traded in the market as the
expected value of the future cash flows discounted at the risk-free interest rate rather than
at an uncertain risky rate of return.
Risk premium
Risk neutral probabilities internalize the risk implicit in each state of nature. Consider
an economy with two assets A and B; the future is divided in two possible states (u) or
(l). There are two different dates, today (t) and tomorrow (T). The expected value of the
price at T under the original probabilities is
E[AT ] = πu Au + πl Al .
E[BT ] = πu Bu + πl Bl .
Suppose the corresponding asset prices for today are calculated by discounting the risk-free
interest rate. Then
At =
1
E[AT ].
1+R0
Bt =
1
E[BT ].
1+R0
This implies the expected return of each asset is certain and equals the risk-free interest
rate.
E[AT ]
At
− 1 = R0 .
E[BT ]
Bt
− 1 = R0 .
This assumption cannot be true if we assume that investors are risk averse. Recall these
investors require a risk premium for the risk they assume by investing in risky positions
(E[RA ] − R0 > 0). It is immediate to see that
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At <
E[AT ]
,
1+R0
Bt <
E[BT ]
.
1+R0
Risk averse investors require lower prices for risky assets than those implied by discounting
the future cash flows at the risk-free interest rate. This valuation equation is only true for
risk averse investors if the expected value is derived from risk neutral probabilities.
At =
1
E ∗ [AT ]
1+R0
=
1
1+R0
(πu∗ Au + πl∗ Al ),
Bt =
1
E ∗ [BT ]
1+R0
=
1
1+R0
(πu∗ Bu + πl∗ Bl ).
These probabilities are called risk neutral probabilities because the risk premium is zero
under π ∗ .
E ∗ [AT ]
At
− 1 = R0 .
E ∗ [BT ]
Bt
− 1 = R0 .
In other words, under risk neutral probabilities any risky asset traded in the market has
the same expected return, that is equal to the return on the risk-free investment.
Martingale Property
The fundamental valuation relationship asserts that
pj =
S
P
φs vjs .
i=1
This equation may be re-written in terms of the original probabilities πs as
pj =
S
P
φs vjs =
i=1
S
P
³ ´
πs
i=1
φs
πs
vjs .
³ ´
Denoting Hs =
value.
φs
πs
we find a new valuation relationship in terms of the true expected
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pj =
S
P
πs Hs vjs = E[Hvj ], ∀ j.
i=1
The price of each asset is the expected value of future cash flows discounted by an stochastic
factor H that depends only on the states. The preceding expression reads in terms of returns
as
S
P
πs Hs (1 + Rjs ) = E[HRj ] = 1, ∀ j.
i=1
Note that a similar expression was found for the fundamental valuation relationship when
maximizing individuals’ utility function.
If prices are computed by the risk neutral measure we have pj =
1
E ∗ [vj ],
1+R0
∀ j. This
formula may be extended to incorporate information available up to time t. Then asset j
price at time t can be written as
pj,t =
1
E ∗ [pj,T |=t ],
(1+R0 )T −t
with =t reflecting the set of available information. Note that vj = (1 + R0 )pj,T .
This property is denominated martingale condition. More generally,
Definition 2. A sequence pt is called martingale sequence if
pt = E[pt+1 |=t ],
for E the expectations operator under some random measure.
From the previous definition is immediate to define martingale sequences in differences
(msd). The sequence εt+1 = pt+1 − pt is a msd if it satisfies the following
E[εt+1 |=t ] = 0.
Therefore the sequence of prices is a martingale under the risk neutral measure. Note
that this property does not hold under the expected value given by the original probabilities.
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pj = E[Hvj ] = E[H]E[vj ] + Cov(H, vj ).
The expected value of the stochastic discount factor H is
E[H] =
S
P
πs H s =
i=1
S
P
³ ´
πs
i=1
φs
πs
=
S
P
φs =
i=1
1
.
1+R0
Then
pj =
1
E[vj ]
1+R0
+ Cov(H, vj ).
The latter term Cov(H, vj ) describes the risk premium on the risky asset.
Example: Pricing a call option
Calculate the price today (t) of an European call option on IBM stocks with current
price Pt = 45. The strike price of the stock is k = 40. There are two states of nature at
time T. In bull markets (u) the stock price will be Pu,T = 60 and in bear markets (l) the
price will be Pl,T = 30. The payoffs of the call option are cT = max(0, PT − k). Then
cu,T = 20, and cl,T = 0. Assume the risk-free asset is R0 = 0.05.
The price of the option can be calculated by replicating the payoffs of the call option,
or alternatively by computing the Arrow-Debreu prices. We will use the second strategy. If
there are two states of nature there must be two assets linearly independent in order to price
the option. These are the underlying IBM asset and the risk-free asset with R0 = 0.05.
Under no-arbitrage opportunities their prices will satisfy






 0.95   1 1 
 Φu 
 

 45  =  60 30  


 


 

Φl
20 0
ct
Then
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1 × φu + 1 × φl = 0.95,
60 × φu + 30 × φl = 45,
20 × φu + 0 × φl = cT .
From the first two equations of the system we obtain
φu = 0.55, and φl = 0.40.
The price of the call option is given by the third equation.
cT = 11.
This price can be also obtained by replicating payoffs of the call option. Construct a
portfolio with shares (x1 , x2 ) such that
x1 × 60 + x2 × 1 = 20,
x1 × 30 + x2 × 1 = 0.
The solution to this system is
x1 = 2/3, and x2 = −20.
The portfolio has 2/3 of IBM stock and borrows 20 units of the risk-free asset. Under
no-arbitrage opportunities the price of the call option is
ct = (2/3) × 45 − 20 × 0.95 = 11.
Risk neutral probabilities are determined by
πu∗ = 0.57, and πl∗ = 0.43.
REFERENCES
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This information is sufficient to price any other asset trading in the market. For example
consider an asset with terminal payoffs Qu,T = 50 and Ql,T = 42. The no-arbitrage price
of this asset is
Qt = 0.95 [0.57 × 50 + 0.43 × 42].
Under no arbitrage opportunities Qt = 44.23.
Completeness of the markets
In the former example if there were three states of nature there would be infinite ArrowDebreu prices. The system would require another asset linearly independent with the others
to enable derivative pricing. Without this it would not be possible to find a unique vector
of Arrow-Debreu prices. Hence it is a necessary condition to have as many states of nature
as linearly independent assets trading in the market. This condition defines a complete
market.
Definition 3. A financial market is complete if for every possible structure of future payoffs
there exists a portfolio of assets trading in the market that replicates the payoffs.
This definition boils down to assert that the vector of Arrow-Debreu prices is unique.
Moreover, if these prices are positive the existence of complete markets implies no arbitrage
opportunities.
References
[1] Bailey, R.E., (2005). The Economics of Financial Markets. Ed. Cambridge University
Press, New York (Chapter 7).
[2] Marin, J.M., Rubio, G., (2001). Economia Financiera. Ed. Antoni Bosch, Spain (Chapter 4).