Valuation Problem: “Fair Price” via Arbitrage Argument
Introduction to Mathematical Finance:
Part I: Discrete-Time Models
AIMS and Stellenbosch University
April-May 2012
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State-Prices aka Arrow-Debreu prices or prices of pure
securities
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State-Prices aka Arrow-Debreu prices or prices of pure
securities
Definition (Arrow-Debreu Securities)
Arrow-Debreu Securities: Consider two fictitious assets which pay
exactly 1 in one of the two states of the world and zero in the other.
In actual financial markets, Arrow-Debreu securities do not trade
directly, even if they can be constructed indirectly using a portfolio of
securities.
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Question: What is a fair price for theses assets?
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State-Prices aka Arrow-Debreu prices or prices of pure
securities
Big Breakthrough: Valuation by replication
Question: What is a fair price for theses assets?
The big breakthrough came when two economists (Fischer Black and
Myron Scholes in 1973) recognized that arbitrage was the secret to
unlocking the pricing formula.
Their big insight was that the payoff structure of an option can be
replicated by a portfolio of market traded assets. Since the cash
payoffs to the portfolio and the option are identical, it must be the case
that the price of the option equals the value of the portfolio; otherwise,
an arbitrage opportunity would exist.
No Arbitrage
=⇒ Law of One Price
=⇒ Price via Replication
Idea: Make a portfolio of Arrow-Debreu AD securities which
generate the payoffs of the existing claims: We call it Replication.
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More on the “Law of One Price”
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Solving the linear system gives the fair prices of the AD securities (also
seen as the forward price a1 for state up (resp. a2 for state down)):
Emanuel Derman: “The law of one price is not a law of nature. It’s a
reflection on the practices of human beings, who, when they have enough
time and information, will grab a bargain when they see one”.
Reading Material: “The boy’s guide to pricing and hedging” by
Emanuel Derman (online document).
R. Ghomrasni
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State-Prices aka Arrow-Debreu prices or prices of pure
securities
Financial economists refer to their essential principle as the law of one
price, which states that: “Any two securities with identical future
payouts, no matter how the future turns out, should have identical
current prices.”
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a1
=
a2
=
1 (1 + r )S0 − Sd
1+r
Su − Sd
1 Su − (1 + r )S0
1+r
Su − Sd
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Arrow-Debreu Securities
Arrow-Debreu Securities
These securities are fundamental.
a1 = α1 S0 + β 1
Homework: Do the same computation for aII
What is the fair price of the Arrow-Debreu (AD) security aI ? Set the fair
price of the AD security aI equal to the cost of the replicating
portfolio:
1
1
1
1
* α S1 (up) + β (1 + r ) = α Su + β (1 + r ) = 1
H
HH
j α1 S1 (down) + β 1 (1 + r ) = α1 Sd + β 1 (1 + r ) = 0
aI
=
=
=
Lemma
The replicating portfolio for the Arrow-Debreu security aI is given by
(α1 , β 1 ) =
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with
1
Sd 1
,−
×
Su − Sd
1+r
Su − Sd
R. Ghomrasni
1
Sd
S0
−
×
Su − Sd
1+r
Su − Sd
1 n (1 + r )S0 − Sd o
a little algebra
1+r
Su − Sd
1
A1 (up)
×q =
×q
1+r
B1 (up)
q :=
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Conclusion
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(1 + r )S0 − Sd
Su − Sd
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Conclusion Continued
The same (HWK: check this)
aII =
It follows clearly:
0<q<1
 (q =
(1+r )S0 −Sd
Su −Sd
⇐⇒
Sd < (1 + r )S0 < Su
⇐⇒
@ arbitrage
1
A2 (down)
× (1 − q) =
× (1 − q)
1+r
B1 (down)
, 1 − q) is a probability on Ω = {up, down}, where
P(S1 = Su ) = p real world probability does not matter. Only the list
of possible scenarios matters.
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Conclusion
General Case of a Contingent Claim
Remark
I
The design of an Arrow-Debreu Security is such that once its price
is available, it provides the answer to key valuation question:
what is a unit of the future state contingent numeraire worth today.
As a result, it constitutes the essential piece of information necessary
to price arbitrary cash flow.
I
If Arrow-Debreu Securities are traded, their prices constitute the
essential building blocks for valuing any arbitrary risky cash-flow:
Indeed, any contingent claim/ derivative, with any payoff
profile in the two possible states of the world, can be obtained
as a linear combination of the two Arrow-Debreu securities
that have just been described.
C = C0
This is true for any contingent claim C1 = F (S1 ) / any derivatives!
C0 =
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Key Result: Probabilistic formulation
C0 =
= Cu aI + Cd aII
o
1 n
=
q Cu + (1 − q) Cd
1+r
1
=
EQ [C1 ]
1+r
1
C1
C1
EQ [C1 ] = EQ [
] = EQ [ ]
1+r
1+r
B1
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Back to the Optimality: Seller/Buyer Point of View
1
EQ [C1 ]
1+r
where EQ [·] denotes expected value with respect to the new probabilities
qu = q and qd = 1 − q. Notice that the only unknown argument is q.
Remark
We observe that
Key result: (Fair) Prices are discounted expectation under risk
neutral probability measure
C0 =
1
EQ [C1 ] = hup = hlow
1+r
More precisely Key result: (Fair) Prices are the discounted
in other words, Replication coincides with the Optimality Criterion.
Moreover, it is much easier to compute expectation than to solve an
optimization problem, provided we know the risk-neutral measure Q .
expected values of future payoffs (under risk neutral probability
measure)
Remark
We can use a mathematical device, change of probability measure
(risk-neutral probabilities), to compute the replication cost more
directly. That is useful when we only need to know the price, not the
other details of the replicating portfolio.
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Why Risk Neutral?
Why Risk Neutral?
Definition (Arithmetic Return)
The (arithmetic) return RS of an asset S (stock, bond, portfolio...)
during the period t = 0 and t = 1 is defined by
Remark
RS =
EQ [C1 ] = (1 + r ) C0
Any derivatives growth as the risk-less asset under Q. Useful to
compute q just by remembering the above expression:
e the expected return of the riskfree asset B
For any probability measure P,
during the period t = 0 and t = 1 is
EQ [S1 ] = (1 + r ) S0
⇐⇒
q Su + (1 − q) Sd = (1 + r ) S0
(1 + r )S0 − Sd
q=
Su − Sd
⇐⇒
S1 − S0
S0
EeP [RB ] =
X B1 (ω) − B0 (ω)
e
P(ω)
B0 (ω)
ω∈Ω
=
=
X (1 + r ) − 1
e
P(ω)
1
ω∈Ω
X
e
r P(ω)
=r
ω∈Ω
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Why Risk Neutral?
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Comparison with True Probabilities
e the expected return of the risky asset S
For any probability measure P,
during the period t = 0 and t = 1 is
EeP [RS ]
=
X S1 (ω) − S0 (ω)
e
P(ω)
S0 (ω)
ω∈Ω
=
=
S1 (up) − S0 e
S1 (down) − S0 e
P(up) +
P(down)
S0
S0
Su − S0 e
Sd − S0 e
P(up) +
P(down)
S0
S0
The risk-neutral probabilities Q generally do not equal the true
probabilities P.
I
Usually
EP [RS ] = E
In particular, under the real world probability P:
EP [RS ]
I
S1 − S0
>r
S0
in order to compensate for risk.
Sd − S0
Sd − Su
−
×p
S0
S0
6= r in general!
=
e = Q, then
Only if P
EQ [RS ] = r
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Risk Neutral Valuation: Summary
I
I
The risk-neutral probability Q thus defined, is a mathematical
construct, a Change of Measure , which is useful to price
derivatives.
We have seen that
Note that in this risk-neutral world i.e. under Q all securities
have the same expected return (equal to the risk-free rate):
EQ [RS ]
S0 =
= r
Q
E [RC ]
I
Equivalent Martingale Measure
)S0 −Sd
The probability Q = (q = qu = (1+r
, 1 − q) has a nice
Su −Sd
interpretation as the unique probability making the discounted stock
price S̃ = { BS00 , BS11 } move in a fair way: the expectation (under Q ) is
constant.
= r
under Q, any portfolio/trading strategy (a, b) has the same
expected return (equal to the risk-free rate):
EQ [V1 ]
1
S1
S1
S0
=
EQ [S1 ] = EQ [
] = EQ [ ]
B0
1+r
1+r
B1
The probability measure Q = (q, 1 − q) = (qu , qd = 1 − qu ) is called an
equivalent martingale measure or equivalent risk neutral measure.
= EQ [aS1 + b(1 + r )] = a(1 + r )S0 + b(1 + r )
= (1 + r )V0
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Market Completeness
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First Fundamental Theorem of Asset Pricing
Theorem (First Fundamental Theorem of Asset Pricing)
There do not exist arbitrage opportunities if and only if there exists a
probability Q , called an equivalent martingale measure , such that
In this case, the One-Period Binomial Model is also complete (i.e. any
contingent claim is replicable): Any risk-neutral measure P? must
satisfy
?
?
Q ≈P
S0 (1 + r ) = E [S1 ] = p Su + (1 − p )Sd ,
and
and this condition uniquely determines the parameter p ? = P? (up) as
P? (up) =
i.e. they are equivalent
?
(1 + r )S0 − Sd
∈]0, 1[
Su − Sd
S1 S1 1
S0
= EQ
=
E Q S1
= EQ
B0
B1
1+r
1+r
In other words, S̃ =
S
B
= { BS00 , BS11 } is a martingale under the measure Q .
Definition
On Ω = {up, down}, P̃ is equivalent to P, written as P̃ ≈ P, means that
P̃(up) = p̃ ∈ (0, 1) and P̃(down) = 1 − p̃.
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Second Fundamental Theorem of Asset Pricing
Comments:
Definition (Complete Market)
A financial market is complete if and only if every contingent claim is
replicable (or attainable).
Theorem (Second Fundamental Theorem of Asset Pricing)
Assuming absence of arbitrage, there exists a unique Equivalent
Martingale Measure (EMM) if and only if the market is complete.
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