3rd Day Lecture AIMS 2012 4x Printing

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Introduction to Mathematical Finance:Part I: Discrete-Time Models

AIMS and Stellenbosch UniversityApril-May 2012

1 / 25 R. Ghomrasni Last updated: 2-5-2012

Valuation Problem: Fair Price via Arbitrage Argument


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.



Question: What is a fair price for theses assets?



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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.


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


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).



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


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


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.


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.


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

]


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.


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


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


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


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


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 .


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[


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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3rd Day Lecture AIMS 2012 4x Printing