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Introduction to Mathematical Finance:Part I: Discrete-Time Models
AIMS and Stellenbosch UniversityApril-May 2012
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Valuation Problem: Fair Price via Arbitrage Argument
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State-Prices aka Arrow-Debreu prices or prices of puresecurities
Definition (Arrow-Debreu Securities)Arrow-Debreu Securities: Consider twofictitiousassets which payexactly 1 in one of the two states of the world and zero in the other.
In actual financial markets, Arrow-Debreu securitiesdo not tradedirectly, even if they can be constructed indirectly using a portfolio ofsecurities.
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State-Prices aka Arrow-Debreu prices or prices of puresecurities
Question: What is a fair price for theses assets?
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State-Prices aka Arrow-Debreu prices or prices of puresecurities
Question: What is a fair price for theses assets?
Idea: Make a portfolio of Arrow-Debreu AD securities whichgenerate the payoffs of the existing claims: We call it Replication.
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Big Breakthrough: Valuation by replication
The big breakthrough came when two economists (Fischer Black andMyron Scholes in 1973) recognized thatarbitrage was the secret tounlocking the pricing formula.
Theirbig insight was that the payoff structure of an option can bereplicated by a portfolio of market traded assets. Since the cash
payoffs to the portfolio and the option are identical, it must be the casethat 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
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More on the Law of One Price
Financial economists refer to their essential principle as thelaw of oneprice, which states that: Any two securities with identical futurepayouts, no matter how the future turns out, should have identicalcurrent prices.
Emanuel Derman: The law of one price is not a law of nature. Its areflection on the practices of human beings, who, when they have enoughtime and information, will grab a bargain when they see one.
Reading Material: The boys guide to pricing and hedgingbyEmanuel Derman (online document).
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State-Prices aka Arrow-Debreu prices or prices of puresecurities
Solving the linear system gives the fair prices of the AD securities (alsoseen as theforward price a1 for state up (resp. a2 for state down)):
a1 = 1
1+r
(1+r)S0 SdSu Sd
a2 = 1
1+r
Su (1+r)S0Su Sd
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Arrow-Debreu Securities
These securities are fundamental.
a1 = 1S0+
1
1S1(up) + 1(1+r) = 1Su+
1(1+r) = 1
1S1(down) + 1(1+r) = 1Sd+
1(1+r) = 0
LemmaThe replicating portfolio for the Arrow-Debreu security aIis given by
(1, 1) = 1
Su Sd,
1
1+r
SdSu Sd
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Arrow-Debreu Securities
Homework: Do the same computation foraII
What is the fair price of the Arrow-Debreu (AD) securityaI? Set the fairprice of the AD securityaIequal to the cost of the replicatingportfolio:
aI = S0
Su Sd
1
1+r
Sd
Su Sd
= 1
1+r
(1+r)S0 SdSu Sd
a little algebra
= 1
1+r q=
A 1(up)
B1(up) q
with
q:=(1+r)S0 Sd
Su Sd
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Conclusion
It follows clearly:
0< q< 1 Sd< (1+r)S0 < Su
arbitrage
(q= (1+r)S0SdSuSd
, 1 q) is a probability on ={up,down}, where
P(S1 = Su) = preal world probability does not matter. Only the listof possible scenarios matters.
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Conclusion Continued
The same (HWK: check this)
aII= 1
1+r (1 q) =A
2(down)
B1(down) (1 q)
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Conclusion
Remark
The design of an Arrow-Debreu Security is such that once its priceis available, it provides theanswer 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. 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 payoffprofile in the two possible states of the world, can be obtainedas a linear combination of the two Arrow-Debreu securitiesthat have just been described.
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General Case of a Contingent Claim
C= C0 = Cu aI+ Cd aII
= 1
1+r
q Cu+ (1 q) Cd
= 1
1+r EQ[C1]
This is true for any contingent claim C1 = F(S1) / any derivatives!
C0 = 1
1+rEQ[C1] = E
Q[ C11+r
] = EQ[C1B1
]
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Key Result: Probabilistic formulation
C0 = 1
1+r EQ[C1]
where EQ[] denotes expected value with respect to thenew probabilitiesqu= qand qd= 1 q. Notice that the only unknown argumentis q.
Key result: (Fair) Prices are discounted expectation under riskneutral probability measure
More preciselyKey result: (Fair) Prices are the discountedexpected values of future payoffs (under risk neutral probabilitymeasure)
RemarkWe can use a mathematical device, change of probability measure(risk-neutral probabilities), tocompute the replication cost moredirectly. That is useful when we only need to know the price, not theother details of the replicating portfolio.
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Back to the Optimality: Seller/Buyer Point of View
RemarkWe observe that
C0 =
1
1+r EQ
[C1] = hup= hlow
in other words, Replication coincides with the Optimality Criterion .Moreover, it ismuch easier to compute expectation than to solve anoptimization problem, provided we know the risk-neutral measureQ .
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Why Risk Neutral?
Remark
EQ[C1] = (1+r) C0
Any derivatives growth as the risk-less asset under Q. Useful tocomputeq just by remembering the above expression:
EQ[S1] = (1+r) S0
q Su+ (1 q) Sd= (1+r) S0
q=(1+r)S0 Sd
Su Sd
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Why Risk Neutral?
Definition (Arithmetic Return)The (arithmetic) returnRSof an assetS (stock, bond, portfolio...)during the periodt= 0 and t= 1 is defined by
RS=S1 S0
S0
For any probability measureP, the expected return of the riskfree assetBduring the periodt= 0 and t= 1 isEP[RB] =
B1() B0()
B0()P()
=
(1+r) 1
1P()
=
rP() = r
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Why Risk Neutral?
For any probability measureP, the expected return of the risky assetSduring the periodt= 0 and t= 1 is
EP[RS] =
S1() S0()
S0()P()
= S1(up) S0
S0P(up) + S1(down) S0
S0P(down)
= Su S0
S0 P(up) +Sd S0
S0 P(down)In particular, under the real world probability P:
EP[RS] = Sd S0
S0
Sd SuS0
p
= r in general!
Only ifP=Q, thenEQ[RS] = r
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Comparison with True Probabilities
The risk-neutral probabilitiesQ generally do not equal the trueprobabilitiesP.
Usually
EP[RS] = ES1 S0
S0>r
in order to compensate for risk.
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Risk Neutral Valuation: Summary
The risk-neutral probabilityQ thus defined, is amathematicalconstruct, a Change of Measure , which is useful to pricederivatives.
Note that in this risk-neutral world i.e. under Q all securitieshave the same expected return (equal to the risk-free rate):
EQ[RS] = r
EQ[RC] = r
underQ, any portfolio/trading strategy (a, b) has the sameexpected return (equal to the risk-free rate) :
EQ[V1] = EQ[aS1+b(1+r)] = a(1+r)S0+b(1+r)
= (1+r)V0
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Equivalent Martingale Measure
We have seen that
S0 = S0B0
= 1
1+rEQ[S1] = E
Q[ S11+r
] = EQ[S1B1
]
The probabilityQ = (q= qu= (1+r)S0Sd
SuSd, 1 q) has a nice
interpretationas the unique probability making the discounted stockprice S= { S0
B0, S1B1} move in a fair way: the expectation (underQ ) is
constant.
The probability measureQ = ( q, 1 q) = (qu,qd= 1 qu) is calledanequivalent martingale measure or equivalent risk neutral measure .
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Market Completeness
In this case, the One-Period Binomial Model is also complete(i.e. anycontingent claim is replicable): Any risk-neutral measure P mustsatisfy
S0
(1+r) = E[S1
] = pSu
+ (1 p)Sd,
and this condition uniquely determines the parameterp =P(up) as
P(up) =( 1+r)S0 Sd
Su Sd]0, 1[
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First Fundamental Theorem of Asset Pricing
Theorem (First Fundamental Theorem of Asset Pricing)There do not exist arbitrage opportunitiesif and only if there exists aprobabilityQ , called anequivalent martingale measure , such that
Q P i.e. they are equivalent
and S0B0
=E Q S1
B1
= E Q
S11+r
=
1
1+rEQ
S1
In other words, S= SB
= { S0B0, S1B1} is amartingaleunder the measureQ .
DefinitionOn ={up,down}, P is equivalent to P, written as PP, means thatP(up) = p (0, 1) and P(down) = 1 p.
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Second Fundamental Theorem of Asset Pricing
Definition (Complete Market)A financial market iscompleteif and only ifevery contingent claim isreplicable (or attainable).
Theorem (Second Fundamental Theorem of Asset Pricing)Assuming absence of arbitrage, there exists aunique EquivalentMartingale Measure (EMM)if and only ifthe market iscomplete.
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