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Lecture 6

Arbitrage in multiperiod models

In multiperiod models arbitrage is defined similarly.

Definition

1) An arbitrage opportunity is some trading strategy H such that:

(a) V0= 0

(b) VT 0

(c) E VT >0

(d) H is self-financing

2) A self financing strategy His an arbitrage opportunity iff:

(a) V0 = 0 or (a) G

T 0

(b) VT 0 or (b) E G

T >0

(c) E VT >0 or (c) V

0 = 0

But the risk neutral measure the definition is a bit different:

Definition

A risk neutral measure (martingale measure) is a probability measure Q such that:

1) Q()> 0 for all

2) Sn is a martingale under Q for every n= 1, 2,...,N

EQ[S

n(t + s)|Ft] =S

n(t), t, s 0

orEQ[S

n(t + s) S

n(t)|Ft] = 0

orEQ[Sn(t + s)/B(t + s)|Ft] =Sn(t)/B(t)

= EQ[BtSn(t + s)/B(t + s)|Ft].

First fundamental theorem of finance:

There are no arbitrage opportunities if and only if there exists a martingale measure Q.

Proposition

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If the multiperiod model does not have any arbitrage opportunitity, then none of the underlyingsingle period models has any arbitrage opportunities in the single period sense.

Example

Consider a 2-period problem with = {1, 2,...,5} and one risky security:

S0() S1() S2()

1 6 5 3

2 6 5 4

3 6 5 8

4 6 7 6

5 6 2 8

Find: Vt,V

t ,G

t ,Gt for all strategies (H0, H1) and check for the existance of martingale measuresor linear pricing measure. Find all of them. Keep in mind Bt = (

10

9)t.

Is the strategy: H0(1)() = 2 for alls a self financing startegy?

H1(1)() = 3 for all s

H0(2)() =

1, f or = 1, 2, 3

2, f or = 4, 5

H1(2)() =

29/9, f or = 1, 2, 3

4, f or = 4, 5

The binomial model

Is a partocular case of the multiperiod model. Each period there are 2 possibilities: the securityprice either goes up by the factor u (u >1) orgoes down by a factord (0 < d

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Letk = # of up moves. in general, if# up steps= n

total # steps= t t n

then the # of path to reach state m = t

m

P(St = Sundtn) =

t

m

pn(1 p)tn n= 0, 1,...,t

The self financing equation now is

H0(t)(1 + R) + H1(t)St= H0(t + 1) + H1(t + 1)St

What equation does the risk neutral probability verify here?

S0= 1

1 + r EQ

[S1|F0], in general St = 1

1 + r EQ

[St+1|Ft]

q[u 1 r

1 + r ] + (1 q)[

d 1 r

1 + r ] = 0 q=

1 + r d

u d

In generalqis the conditional probability the next move is an up move given the information Ftat time t. Soq= 1+rd

ud for all Ft and t.

Also remark that in order for Q to exist we need 0< q

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3 S0= 5 S1= 4 S2 = 6

4 S0= 5 S1= 4 S2 = 3

then X=max{S2 K, 0}, forK= 5

X() =

4, = 11, = 2, 3

0, = 4

Example of a contingent claim (derivative security) that is not simple:

X=max{S0+ S1+ S2

3 5 , 0}

This is called an Asian or averaging option.

X() =

7/3, = 1

4/3, = 2

0, = 3, 4

Question:

The question of pricing is similer to the one-period model. Find(t, x) the price of the contingentclaim at time t= 0, 1,...,T.

Definition

A contingent claim is said to be marketable or attainable if there exists a self-financing strategysuch that VT() =X() for all . The corresponding portfolio or trading startegy H is saidto be replicate or generate X.

Risk neutral valuation principle

The time t value of a marketable contingent claim X is equal to Vt, the time t of the portfoliowhich replicates X. Moreover:

Vt =Vt/Bt = EQ[X/BT|Ft], t= 0, 1,...,T

for all risk neutral probability measures Q.

Binomial model review

Recall that at time T, each possibility forST is parameterized by K= # of ups from t= 0 to T.

So the value of the option at time T, if # ups = kVT(k) = (S0u

kdTk)

We saw that

V0= 1

(1 + r)TEQ[X] =

1

(1 + r)T

Tn=0

Tn

qn(1 q)Tnmax{0, S0u

ndTnk}

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wherek is the strike price.

But what is the option price at time 0 t T?

For any t T 1, there are at most t ups.

Vt(k) = 11 + r EQ[Vt+1|(#ups up to time t) =k]

= 1

1 + r(quVt+1(k+ 1) + (1 qu)Vt+1(k))

So, we have proved the recursion relation:

VT(k) = (S0ukdTk), k= 0, 1,...,T

t T 1, k t:

Vt(k) =

1

1 + R (quVt+1(k+ 1) + (1 qu)Vt+1(k))

The only question we might need to answer is what are the hedging strategies.

For hedging we need to start at t = 0 and work forward by using the 1-period model hedgingstrategy to obtain H= (x, y) which hedges the option given by:

V1(1), with probability qu

V1(0), with probability qd

Specifically,

x= 1

1 + r(uV1(0) dV1(1))

1

u d

y= 1

S0(V1(1) V1(0))

1

u d

This is obtain by solving the system:

(1 + r)x + S0uy= V1(1)

(1 + r)x + S0dy= V1(0)

In general: at time t T 1, for a fixed k {0, 1,...,t}

x= 11 + r (uVt+1(k) dVt+1(k+ 1)) 1u d

y= 1

S0(Vt+1(k+ 1) Vt+1(k))

1

u d

Call options on a stock index

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Options, so far, were defined in term of one underlying security. But in general one can defineoptions one two or more underlying securities. For example, given a function g : RN R+ onecan take X to be X = g(S1(T),...,SN(T)). Then if you know the joint probability distributionof the random variables S1(T),...,SN(T) under the martingale measure, it is easy to compute thetime 0 value of this contingent claim. In particular, with:

g(S1,...,SN) = (a1S1+ ... + aNSN e)+

for positive scalars a1,...,aN, you could have a call option on a stock index.

Or withg(S1,...,SN) =max{S1,...,SN, e}

you could have a contingent claim delivering the best ofNsecurities and the cash amount e.

Example

SupposeK= 9,N= 2,T = 2,r = 0 and the price process and information are as bellow. Also this

model (one can show with a few computations) has an unique probability measure Q, computed asbelow.

Consider a call option with exercise price 13 on the time T= 2 value of the stock indexS1(t)+S2(t)

What is the time 0 value of such a call option? What is the time 0 value of a contingent claim onthe 2 stocks and 8?

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