3: ELEMENTARY MARKET MODEL [.03in]M.Rutkowski (USydney) Slides 3: Elementary Market Model 3/36....

3: ELEMENTARY MARKET MODEL

Marek Rutkowski

School of Mathematics and Statistics

University of Sydney

Semester 2, 2016

M. Rutkowski (USydney) Slides 3: Elementary Market Model 1 / 36

Single Period Market Model

Only one period is considered.

The beginning of the period is usually set as t = 0.The end of the period is usually set as t = 1 (or T = 1).

At t = 0, the prices of all assets are known and the investor canchoose the investment.

At t = 1, the prices of all assets are observed and the investor obtainsa payoff corresponding to the current portfolio value.

A single period market model is not realistic, but it allows us toillustrate important economic principles without suffering from (orenjoying) sophisticated mathematics.

A full understanding of the features (also drawbacks) of a singleperiod market model is thus necessary for further developments.


Elementary Market Model M = (B, S)

The investor has initial wealth x at t = 0 and is allowed to invest inthe risk-free asset B (bank account) and the risky asset S (stock).

He purchases φ shares of the stock and invests the remaining fundsin his bank account (or borrows cash from the bank).

Notation:

Sample space: Ω = ω1, ω2.Probability measure: P(ω1) = p > 0 and P(ω2) = 1− p > 0.A deterministic interest rate r > −1.The price of a risky asset at time t is denoted by St.

We assume that S0 > 0 and we set u = S1(ω1)S0

and d = S1(ω2)S0

.

We suppose that 0 < d < u, so that there are two distinct values ofthe future stock price: S1(ω1) = uS0 > S1(ω2) = dS0.

We do not assume that d < 1 or u > 1 (although d stands for ‘down’and u stands for ‘up’).


Elementary Market Model

1 + r

S1( ω

1) = uS

0

S1( ω

2) = dS

0

p

1 − p

S0

t = 0 t = 1

1


Why the Elementary Market Model?

In real life, the stock price mouvements are more complicated thanindicated by the elementary market model.

Hence we do not pretend that the elementary model furnishes arealistic picture of the stock price fluctuations.

Nevertheless, even in this simplistic framework the random characterof the stock price is already visible and thus the problem of optionspricing is non-trivial.

There are two important reasons why we consider this model:

First, the concept of arbitrage-free pricing of derivative securities canbe clearly explained.Second, the binomial asset pricing model can be seen as aconcatenation of elementary market models.


Outline

We will examine the following issues:

1 Trading Strategies and Arbitrage-Free Models

2 Replication of Contingent Claims

3 Risk-Neutral Probability Measure

4 Put-Call Parity Relationship

5 Summary of the Elementary Market Model

6 Generalisation of the Elementary Market Model


PART 1

TRADING STRATEGIES AND ARBITRAGE-FREE MODELS


Trading Strategy and Wealth Process

For arbitrary real numbers x and φ, where x is the initial wealth andφ is the number of shares of stock purchased or sold at time 0, thepair (x, φ) is called the trading strategy.

The initial wealth (or value) equals

V0(x, φ) := x = (x− φS0)B0 + φS0.

For any trading strategy (x, φ), the investor obtains at time 1 therandom payoff V1(x, φ), which satisfies for i = 1, 2

V1(x, φ)(ωi) = (x− φS0) (1 + r) + φS1(ωi)

Definition (Value Process)

The wealth process (or value process) of the trading strategy (x, φ)is given by (V0(x, φ), V1(x, φ)) where V0(x, φ) = x and V1(x, φ) isthe random variable V1(x, φ) = (x− φS0) (1 + r) + φS1.


Arbitrage

An essential feature of an efficient market is that for any tradingstrategy can turn nothing into something, the investor who adoptsit must also face the risk of loss.

The following definition is thus a crucial step in the arbitrage pricingmethodology.

Definition (Arbitrage)

A trading strategy (x, φ) in the single-period market model is called anarbitrage opportunity if:

A.1. x = 0, that is, no initial investment is required,

A.2. V1(x, φ) ≥ 0, that is, no risk of losing money,

A.3. There exists an ωi such that V1(x, φ)(ωi) > 0.


Arbitrage-Free Model

Assume that A.2 holds. Then condition A.3 is equivalent to

A.3′. EP V1(x, φ) > 0, that is, a strictly positive expected payoff.

Definition (Arbitrage-Free Model)

A single-period market model is said to be arbitrage-free if no arbitrageopportunity exists in the model.

Real markets sometimes exhibit arbitrage, but the forces of supply anddemand take actions to remove it as soon as someone discovers it.

Market model that admit arbitrage are not suitable for our purposes.

Proposition (3.1)

The model M = (B,S) is arbitrage-free if and only if d < 1 + r < u.


Proof of Proposition 3.1

Proof of Proposition 3.1 (⇒).

To prove the ‘only if’ part, we argue by contradiction:

Assume first that d ≥ 1 + r:

At t = 0, the investor borrows the amount S0 of cash on the moneymarket and buys one share of the stock.At t = 1, the investor sells the stock and thus receives either uS0 ordS0. He also pays back (1 + r)S0.

Let us now assume that u ≤ 1 + r:

At t = 0, the investor borrows one share of the stock from a broker andsells it immediately (short-selling). He then invests S0 in the moneymarket.At t = 1, the investor obtains (1 + r)S0 from the bank account. Hespends either uS0 or dS0 to buy one share of the stock and returns itto the broker (hence to the original owner).

In both cases, the investor’s strategy is an arbitrage opportunity.Hence the proof of the ‘only if’ part is complete.



Proof of Proposition 3.1 (⇐).

The ‘if’ part is also easy to establish:

We start by noting that for x = 0 the equality

V1(x, φ) = (x− φS0) (1 + r) + φS1

becomesV1(0, φ) = φ(S1 − (1 + r)S0).

More explicitly,

V1(0, φ)(ω1) = φ(S1(ω1)− (1 + r)S0) = φS0(u− (1 + r))

V1(0, φ)(ω2) = φ(S1(ω2)− (1 + r)S0) = φS0(d− (1 + r))

It is thus clear that if d < 1 + r < u, then an arbitrage opportunitydoes not exist.


PART 2

REPLICATION OF CONTINGENT CLAIMS


European Options

Definition (European Call and Put Options)

A European call (put) option is a contract which gives the buyer theright to buy (sell) an asset at a future date T for a predetermined price K.

The underlying asset (for instance, a stock or a stock index), thematurity date T and the strike price K are given in the contract.

Payoff at T of a European call option:

If ST > K then the holder gets the net payoff ST −K > 0 byexercising the option, buying the stock at K and selling it at ST .If ST ≤ K then the holder does not exercise the option and this leadsto the null payoff at T .

Hence the payoff of the call option at time T is

CT = h(ST ) = max (ST −K, 0) = (ST −K)+.


European Options

Payoff of a European put option:If ST > K then the holder does not exercise the option and thus thepayoff equals 0.If ST ≤ K then the holder exercises the option and obtains the netpayoff K − ST > 0 by buying the stock at ST and selling it at K.

Hence the payoff of the put option at time T equals

PT = h(SY ) = max (K − ST , 0) = (K − ST )+.

European calls and puts are examples of contingent claims. Theirpayoffs CT and PT at the expiration date T are random, but theyonly depend on the stock price ST and strike K.

We will now address the following general question:

How to select an initial investment x and a trading strategy (x, φ) inorder to obtain the same wealth V1(x, φ) at time 1 as the payoffof a given contingent claim X = h(S1)?


Replication of a Contingent Claim

Definition (Replicating Strategy)

A replicating strategy (or a hedge) (x, φ) for the contingent claimX = h(S1) in the elementary market model M = (B,S) is a tradingstrategy which satisfies V1(x, φ) = h(S1), that is,

(x− φS0) (1 + r) + φS1(ω1) = h(S1(ω1)), (1)

(x− φS0) (1 + r) + φS1(ω2) = h(S1(ω2)). (2)

Definition (Arbitrage Price)

Assume that the elementary market model M = (B,S) is arbitrage-free.If (x, φ) is a replicating strategy of a contingent claim then x is called thearbitrage price (or price) for the claim at t = 0. We write x = π0(X).


Hedging of a Contingent Claim

Computation of the hedge and (unique) arbitrage price x of a contingentclaim X = h(S1):

By subtracting (2) from (1), we find the hedge ratio

φ =h(S1(ω1))− h(S1(ω2))

S1(ω1)− S1(ω2)=

h(uS0)− h(dS0)

(u− d)S0. (3)

Equality (3) is called the delta hedging formula.

One can substitute (3) into (1) or (2) in order to compute x.

To derive a convenient representation for x, we introduce the notation

p :=1 + r − d

u− d∈ (0, 1) (4)

and we define the probability P by: P(ω1) = p and P(ω2) = 1− p.


Pricing of a Contingent Claim

One can check that P satisfies

EP

(S1

1 + r

)=

1

1 + r[pS1(ω1) + (1− p)S1(ω2)] = S0. (5)

We rewrite (1) and (2) in the following way:

x+

[S1(ω1)

1 + r− S0

]φ =

1

1 + rh(S1(ω1)) (6)

x+

[S1(ω2)

1 + r− S0

]φ =

1

1 + rh(S1(ω2)) (7)

If we multiply (6) and (7) with p and 1− p, respectively, and add them,then we obtain

x+

1

1 + r[pS1(ω1) + (1− p)S1(ω2)]− S0

φ

=1

1 + r[ph(S1(ω1)) + (1− p) h(S1(ω2))] .

(8)


Model Completeness

In view of (5), the term with φ vanishes. Hence (8) yields thefollowing convenient representation for the price x

x =1

1 + r[ph(S1(ω1)) + (1− p)h(S1(ω2))] . (9)

It can be seen from (9) that the price x depends on p and 1− p, butit is independent of the real-world probabilities p and 1− p.

Note that equalities (3) and (9) hold for an arbitrary payoff functionh(S1). Hence for any contingent claim X we have a uniquereplicating strategy and a unique arbitrage price.

Definition (Completeness)

Since all contingent claims in the elementary market model havereplicating strategies, the model is called complete.


PART 3

RISK-NEUTRAL PROBABILITY MEASURE


Risk-Neutral Probability Measure

Definition (Risk-Neutral Probability Measure)

A probability measure Q on the sample space Ω = ω1, ω2 is called arisk-neutral probability measure (equivalent martingale measure) forthe market model M = (B,S) if Q is equivalent to P and the followingequality holds

EQ

(S1

1 + r

)= S0.

Proposition (3.2)

The risk-neutral probability measure for the model M = (B,S) isunique and it satisfies Q = P if and only if d < 1 + r < u.

If 1 + r ≤ d or u ≤ 1 + r, then no risk-neutral probability exists.



Proof of Proposition 3.2.

It suffices to observe that the equality

EQ

(S1

1 + r

)= S0

means that Q(ω1) ≥ 0, Q(ω2) ≥ 0, Q(ω1) +Q(ω2) = 1 and

Q(ω1)S1(ω1) +Q(ω2)S1(ω2) = (1 + r)S0.

The last equality above yields

Q(ω1) =1 + r − d

u− d= P(ω1). (10)

If d = 1 + r or u = 1 + r then Q given by (10) is well defined, but itis not equivalent to P since either Q = (1, 0) or Q = (0, 1).


Expected Rates of Return on Traded Assets

Assume that u < 1 + r < d. Then the risk-neutral probabilitymeasure Q = P exists and is unique.

The expected rate of return on the savings account B equals

EP(rB) = E

P

(B1 −B0

B0

)=

B1 −B0

B0= r.

The expected rate of return on the stock under P equals r. Thisfollows from the equality

EP(rS) = E

P

(S1 − S0

S0

)=

(1 + r)S0 − S0

S0= r.

The probability measure P is the unique probability measure Q underwhich the equality EQ(rB) = EQ(rS) holds.


Risk-Neutral Valuation Formula

Proposition (3.3)

For any claim X = h(S1), the arbitrage price of X at time 0 in thearbitrage-free elementary market model M = (B,S) satisfies

π0(X) =1

1 + rEP(h(S1)) = E

P

(X

1 + r

). (11)


We know (see (9)) that the price x satisfies

x =1

1 + r[ph(S1(ω1)) + (1− p)h(S1(ω2))] .

Formula (11) now follows immediately.

We will now give a shorter proof for (11).




It is assumed that u < 1 + r < d. Let (x, φ) be any trading strategy.From the equality

V1(x, φ)

1 + r= x+ φ

(S1

1 + r− S0

)

and formula (5), we deduce that investing is a ‘fair game’ under Pmeaning that for any strategy (x, φ) we have

EP

(V1(x, φ)

1 + r

)= x.

In particular, if (x, φ) replicates X then V1(x, φ) = X and thus weobtain the risk-neutral valuation formula (11) since x = π0(X).


PART 4

PUT-CALL PARITY RELATIONSHIP


Example: Call and Put Options

Example (3.1)

Consider the elementary market model M = (B,S) with parametersr = 1

3 , S0 = 1, u = 2, d = 12 , p = 3

5 and T = 1.

Recall that the risk-neutral probability measure P is given asP(ω1) = p and P(ω2) = 1− p where

p :=1 + r − d

u− d.

Hence the risk-neutral probability measure P equals

P(ω1) = p =1 + 1

3 −12

2− 12

=5

9, P(ω2) = 1− p =

4

9.



Example (3.1)

S1(ω1) = 2 C1(ω1) = 1 P1(ω1) = 0

S0 = 1

p=5/9

??

1−p=4/9

???

????

????

????

?K = 1 C1 = (S1 −K)+ P1 = (K − S1)

+

S1(ω2) = 0.5 C1(ω2) = 0 P1(ω2) = 0.5



Example (3.1 Continued)

The price of the European call with strike price K = 1 equals

C0 =1

1 + rEP(CT ) =

1

1 + r

(pCT (ω1) + (1− p)CT (ω2)

)

=1

1 + rp(uS0 −K) =

3

4×

5

9× 1 =

5

12.

The price of the European put with strike price K = 1 equals

P0 =1

1 + rEP(PT ) =

1

1 + r

(pPT (ω1) + (1− p)PT (ω2)

)

=1

1 + r(1− p)(K − dS0) =

3

4×

4

9×

1

2=

1

6.


Put-Call Parity

The arbitrage prices at time 0 computed in Example (3.1) satisfy

C0 − P0 =1

4= 1−

3

4= S0 −

1

1 + rK. (12)

Equality (12) is a special case of the put-call parity.

Proposition (3.4)

The put-call parity can be represented as follows, for t = 0, 1,

Ct − Pt = St −B(t, T )K (13)

where the zero-coupon bond equals B(t, T ) = (1 + r)−(T−t), so thatB(0, 1) = (1 + r)−1 and B(1, 1) = 1.

It is clear that CT − PT = (ST −K)+ − (K − ST )+ = ST −K.

Equality (13) is thus an easy consequence of Proposition 3.3.


PART 5

SUMMARY OF THE ELEMENTARY MARKET MODEL


Summary: Properties

Let us summarise the properties of the elementary market model:

1 The two-state single-period market model M = (B,S) isarbitrage-free if and only if d < 1 + r < u.

2 The arbitrage-free property of the model M = (B,S) does notdepend on the real-world probability measure P.

3 An arbitrary contingent claim X can be replicated by means of aunique trading strategy. Hence the model M = (B,S) is complete.

4 The initial value of a replicating strategy for X is called the arbitrageprice for X and it is denoted as π0(X).

5 The risk-neutral probability P exists and is unique if and only ifd < 1 + r < u (that is, whenever the model M is arbitrage-free). Bydefinition, P is equivalent to P.

6 The arbitrage price π0(X) of any claim X can be computed fromthe risk-neutral valuation formula.


Summary: Theorem

Theorem (3.1 Elementary Market Model)

The elementary market model M = (B,S) is arbitrage-freeif and only if d < 1 + r < u.

Any contingent claim X can be replicated so the model is complete.Formally, X = V1(x, φ) for some (x, φ) ∈ R2.

If d < 1 + r < u then any contingent claim X admits the uniquearbitrage price π0(X) := x where X = V1(x, φ).

The risk-neutral probability P for the model M exists andis unique if and only if d < 1 + r < u.

If 1 + r ≤ d or u ≤ 1 + r, then no risk-neutral probability Q exists.

If d < 1 + r < u, then the arbitrage price π0(X) of any contingentclaim X satisfies

π0(X) = EP

(X

1 + r

).


PART 6

GENERALISATION OF THE ELEMENTARY MARKET MODEL


Generalisation of the Elementary Market Model

We generalise the elementary market model by postulating that:

1 We still deal with two primary traded assets: B and S.2 The sample space Ω = ω1, ω2, . . . , ωk where k ≥ 3.3 Hence S1 = (S1(ω1), . . . , S1(ωk)) where we assume that

S1(ωk) < S1(ωk−1) < · · · < S1(ω2) < S1(ω1).

4 The model is arbitrage-free if and only if

S1(ωk) < S0(1 + r) < S1(ω1).

5 The risk-neutral probability measure Q exists if and only ifS1(ωk) < S0(1 + r) < S1(ω1), but it is not unique.

6 The model is incomplete: no replicating strategy exists for somecontingent claims X = (X(ω1), . . . ,X(ωk)).

7 We will not examine this model in detail, since it can be seenas a special case of a general single-period market model.


Generalisation of the Elementary Market Model

Theorem (3.2 Generalised Elementary Market Model)

The generalised elementary market model M = (B,S) with k ≥ 3is arbitrage-free if and only if S1(ωk) < S0(1 + r) < S1(ω1).

Only some (but not all) contingent claims X can be replicated,that is, are attainable. Hence M = (B,S) is incomplete.

If S1(ωk) < S0(1 + r) < S1(ω1), then any attainable claim X hasthe unique arbitrage price π0(X).

The risk-neutral probability Q for the model M exists if and only ifS1(ωk) < S0(1 + r) < S1(ω1). It is not unique.

If S1(ωk) < S0(1 + r) < S1(ω1), then the arbitrage price π0(X) ofany attainable claim X satisfies for any risk-neutral probability Q

π0(X) = EQ

(X

1 + r

).


3: ELEMENTARY MARKET MODEL [.03in]M.Rutkowski (USydney) Slides 3: Elementary Market Model 3/36....

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