2nd 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 / 36 R. Ghomrasni Last updated: 1-5-2012

No Arbitrage = Law of One price

Remember

An arbitrage opportunityis a trading strategy that nevercosts you anything today (t= 0) or in the future (t= 1),in any contingency (or in any states of economy: up or

down), but has a strictly positive probability of having astrictly positive cash flow att= 1.

The law of one pricesays that assets promising the samefuture cash flows have the same price today.


What Does Arbitrage Free Mean?

Definition (Arbitrage)An arbitrage opportunity is defined as:

1. You do not need any money upfront: this is a zero-cost portfolio.The initial value of the portfolio is zero, that is the ability tomake zero net investment (i.e. some assets are held in positiveamounts, some in negative amounts and, perhaps, some in zeroamounts),

2. Have no probability of loss (The portfolio value at time 1 isnonnegative for all states),

3. Have a positive probability of gain (The portfolio value at time 1 isstrictly positive for some states).


Definition (Mathematical Definition of Arbitrage)An arbitrage portfolio is a portfolio which satisfies:

V0 = 0 costs nothing

(i) P(V1 0) = 1 never lose money

(ii) P(V1 > 0)> 0 sometimes win money

V0() = 0 costs nothing

(i) V1() 0 no chance of losing money

(ii) V1()> 0 some chance of making money

RemarkIf an arbitrage portfolio exists, there will exist infinitely many: forany arbitrage portfolio, scale all the asset holdings up or down by anarbitrary positive proportion; the result is also an arbitrage portfolio. make unbounded profits!



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Definition (Arbitrage)An arbitrage portfolio is a portfolio which satisfies:



(ii) P(V1 > 0)> 0 sometimes win money

The preceding conditions are also equivalent to:



(ii) E[V1]> 0 strictly positive payoff is expected.

E[V1] = pV1(up) + (1 p)V1(down)


Arbitrage Portfolio

Suppose we are able to set-up the following portfolios, which one (if any)are arbitrage portfolio?

V0 = 0

V1(up) = 1

V1(down) = 0

V0 = 0

V1(up) = 1

V1(down) = 1


Arbitrage Portfolio

another example, suppose we can construct the following portfolio,isthis an arbitrage portfolio?, is there an arbitrage opportunity in themarket?

V0 = 1

V1(up) = 1

V1(down) = 2

Answer: It depends on the interest rater!You can not tell whether a portfolio is an arbitrage portfolio by lookingonly on the payoffs (here 1 in upstate and 2 in downstate.)


Arbitrage PortfolioIf

V0 = 1

V1(up) = 1

V1(down) = 2

supposer= 0, B0 = 1

B1(up) = 1

B1(down) = 1

Long the portfolio and short the bond

0

0

1

arbitrage opportunity



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Arbitrage PortfolioIf

V0 = 1

V1(up) = 1

V1(down) = 2

withr= 1/2, B0 = 1

B1(up) = 1.5

B1(d) = 1.5

Long the portfolio and short the bond

0

0.5

+0.5

no arbitrage opportunity


Arbitrage Portfolio

Conclusion: Interest rate is extremely important, with differentinterest rateswe have the same portfolio either leading toarbitrage or not.

Question: Under which condition(s) is our simple one-periodBinomial model arbitrage free?


Conditions for Arbitrage-free


Necessary and Sufficient Conditions for Arbitrage-free inBinomial Market

Theorem (Necessary and Sufficient Conditions forArbitrage-free)In the one-period binomial market model consisted of a risky asset S anda risk-free asset B, we have

Themarket is arbitrage-free if and only if Sd< S0(1+r)< Su.

Proof.= Suppose that Sd< S0(1+r)< Su is false, e.g. S0(1+r)Su.Consider the following portfolio:

V =S+S0B0

B.



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Necessary and Sufficient Conditions for Arbitrage-free

The values at time t= 0 and at t= 1 are: V0 = S0+

S0B0

B0 = 0 ,

V1 = S1+ S0B0

B1 .

Therefore,

V1

(up) = S1

(up) + S0B0

B1

= S0

(1+r) Su 0 ,

V1(down) = S1(down) + S0B0

B1 = S0(1+r) Sd> 0 .

Thus, V satisfies,V0 = 0, V1 0 and

P(V1 > 0) P(S1 = Sd)> 0.

i.e. we have an arbitrage opportunity. Similarly, it can be shown (HMW)that S0(1+r) Sd would also contradict the arbitrage-free assumption.


Hint for HMW

ExerciseIn fact, if, for instance, one had S0(1+r) Sd, then one could makeunbounded riskless profits by initially borrowing an arbitrary amount ofmoney and buying an arbitrary number of shares in the stock at price S0

at time t= 0, followed by selling the stock at time t= 1 at a higherreturn level than r. In other words,

V =(S0)B+ S with > 0

Write down the details...


Remark (Interpretation of the Conditions for Arbitrage-free)

Sd< S0(1+r) Sd.

We have that the One-Period Binomial Model (OPBM) does not admitarbitrageif and only if

S0(1+r)]Sd, Su[={pSu+ (1 p)Sd| p]0, 1[}= {E[S1] | P P}

i.e. S0(1+r) is a convex combination ofSd and Su.



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Contingent Claim

DefinitionOn ={up, down}, Pis equivalent to P, written as P P, means thatP(up) = p (0, 1) and P(down) = 1 p.

Definition (Contingent Claim)A Contingent Claim is a contract which will pay something in the event

of something, in other words, a contract that pay a particular amountcontingent that something happen. For example, a function of the riskyassetS.

In the sequel we will work on the assumption that arbitrage doesnot occur and we will rely on this hypothesis for the pricing ofcontingent claims. i.e. we have

Sd< (1+r)S0 < Su


Valuation Problem: Fair Price


Fair Price(s) Problem

Before solving this problem, it is a priori not clear that the fair

price is necessarily unique!.


Sellers Objective=Writers Objective

1. From the claimCwriters perspective. At timet= 1 he has aliability/debtof(C1(up), C1(down)) 0, depending on which ofthe two states occurs.

2. At time t= 0, the writer is willing to accept any amountx 0which enables him to make good on his commitment at timet= 1:i.e. to have enough capital at timet= 1 to cover his obligation (tohedge the contingent claimCat time t= 1)

3. The sellers objective is, starting with the amountx 0 that S/hereceives at timet= 0, to find a portfolio strategy s.t. V0() = xand

V1() C()0 for all

4. In order to be competitive and win the deal, the seller will sellC forthe amount (the smallest one)

hup:= inf{x 0 | (a, b) R2 s.t V0 = a S0+b= x, V1() C() }



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Sellers Objective

IfCu= C(up) and Cd= C(down), then

V1() C()

aSu+b(1+r) Cu and aSd+b(1+r) Cd

aCu b(1+r)

Suand a

Cd b(1+r)

Sd

a max

Cu b(1+r)Su

,Cd b(1+r)

Sd

Hence, we will show if V1() C() we have

V0 = aS0+b Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su Sd

Proof next slides...


Sellers Objective

For all (a, b) R 2 :V(a,b)

1 C

V0 =aS0+b

b+S0maxCu b(1+r)

Su,Cd b(1+r)

Sd

=max b+S0Cu b(1+r)

Su

, b+S0Cd b(1+r)

Sd

=maxSu S0(1+r)

Sub+

CuS0Su

,Sd S0(1+r)

Sdb+

CdS0Sd

Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su Sd

Conclusion

hup Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su Sd


Geometrical Proof

positive slop SuS0(1+r)Su

b+ and negative slop SdS0(1+r)Sd

b+


Sellers Objective: Ask Price

By considering the following portfolio

a=Cu CdSu Sd

and b= CdSu CuSd(Su Sd)(1+r)

we can show that actually

hup= Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su SdAsk price

RemarkThe upper hedging price is the value of the least costly (self-financingportfolio) strategy composed of market instruments whose pay-off is atleast as large as the contingent claim pay-off.



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Buyers Objective

1. Let us determine the Buyers price. Let us consider a buyer whowants to buy a contingent claim with payoffC() 0 at maturityt= 1. Suppose the buyer choose to pay the pricex 0 for C attime 0.

2. The buyer does not want to run any risk of losing money. Thebuyer starts with the debt xand tries to find a portfolio so thatthe paymentCwhich (s)he receives at timet= 1 makes it possible

to cover the debt from timet= 0 by purchasing the c.c.Vx1 () +C() 0

3. the largest amount the buyer is willing to pay at time t= 0 is givenby

hlow:= sup{x 0 | (a, b) R2 s.t V0 = a S0+b= x,

Vx1 () +C() 0 }

hlow:= sup{x 0 | (, ) R2 s.t V0 = S0+ = x,

V1()C() }


Buyers Objective

V1() C()

aSu+b(1+r) Cu and aSd+b(1+r) Cd

a Cu b(1+r)Su

and a Cd b(1+r)Sd

a minCu b(1+r)

Su,Cd b(1+r)

Sd


Buyers Objective

For all (a, b) R 2 :V(a,b)

1 C

V0 =aS0+b

b+S0minCu b(1+r)

Su,Cd b(1+r)

Sd

=min b+S0 Cu b(1+r)

Su, b+S0

Cd b(1+r)

Sd

=minSu S0(1+r)

Sub+

CuS0Su

,Sd S0(1+r)

Sdb+

CdS0Sd

Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su Sd

Conclusion

hlow Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su Sd


Geometrical Proof



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Buyers Objective: Bid Price

By considering the following portfolio (same as the seller)

a=Cu CdSu Sd

= and b= CdSu CuSd(Su Sd)(1+r)

we can show that actually

hlow= Cu1+r

Su S0(1+r)

Su Sd+

Cd1+r

S0(1+r) Sd

Su SdBid price

Conclusion?


Seller and Buyers Objective: Conclusion

The Buyer and Seller should agree for the fair price: hup= hlow

hup= hlow= Cu

1+r

Su S0(1+r)

Su Sd+

Cd

1+r

S0(1+r) Sd

Su Sd

=The above analysis shows that the buyer and seller of options haveopposing interests that balance at one price only .


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 twofictitious assets 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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Question: What is a fair pricefor theses assets?



Question: What is a fair pricefor theses assets?

Idea: Make a p ortfolio 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


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2nd Day Lecture AIMS 2012 4x Printing