fe 8507 lecture 2 more on dynamic security markets students (1).pptx

8/11/2019 FE 8507 Lecture 2 More on dynamic security markets students (1).pptx

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FE 8507 Stochastic Modelling

Asset PricingLecture 2SDF, Arb itrage, Complete Markets and Port

Choice

DR MANDY THAM

[email protected]

NANYANG BUSINESS S CHOOL

AY 2014-2015 MINI-TERM 2

FE8507/DR MANDY THAM/2014_2015


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Required


Read Chapter 8, BackRead Lecture 2 notes

Revise Chapter 1 and 4, Back

Read The limits of arbitrage JF article.

Read the Forbes article

Tutorial 2


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2.1 More on SDF

Recall ()

.(1)

If a risk-free security is not traded, then in (1) is the shadow risthe zero-beta rate.

(1) is not particularly intuitive until we have replaced the unobservaof some observables.

One way to do so is to choose the explicit form for the utility functio

Lets start with power utility (CRRA)

..(2)



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2.1 More on SDF

Recall that + ( ) ..(3)

Now, show that

ln()= (

)

(

)



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2.1 More on SDF: Lognormal distribu

A standard statistical result is that if X is lognormal, then

ln ~(, )

(+

)......(5)

where k = con

e.g. ln ~(0.1, 0.25)

2 (.+

.)



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2.1 Proofln ~(,

)

(+

)

ln 1

2

Recall +,+ 1

Let X = +,+and conditional on info at time t, also

ln +,+ ln(+,+)

ln(+,+)

(ln(+) (ln(,+) 1

2 ln(+)ln(,+)

(ln(+) (ln(,+)

ln(+

(ln(,+)+ ln +



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2.1 ProofNow, let

,+

(money market fund)

ln ln

ln 0

ln + , ln 0

Using (5)

(ln(+) (ln(,+)

ln(+

(ln(,+)+ ln +

We have,

(ln(+)ln() 1

2 ln(+) 0

Rearrange,

ln() (ln(+)

ln(+) .(7)



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2.1 ProofNext, take LN of (3)

ln + (+

)

(ln + ) ( +

)

(ln + (

+

)

Finally, sub into (7)

ln() (ln(+) 1

2 ln(+)

(

)

(

) ..(8) Q.E



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2.1 Economic IntuitionLet us analyse (8)

ln() ( +

) 1

2(

+

)

measures consumption growth, is interest rate, measures patience and is c

discount factor.

1) Interest rate is high when expected consumption growth is high.High interest rate is required to induce saving today in order to increase consump

2) Interest rate is high whenis low.When people are impatient (is low), a higher interest rate is required to induce



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ln() ( +

) 1

2(

+

)

3) When consumption growth is expected to be volatile, interest rate i You lose more utility as consumption falls than you gain utility from an equal r

consumption. Naturally, you would want to hedge the greater fall in utility by sis a form of precautionary savings.

4) When there is no risk aversion (=0), interest rate depends only on the sufactor (i.e. patience).



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ln() ( +

) 1

2(

+

)

5) As risk aversion () increases, interest rate becomes more responsivconsumption growth. Less willing to deviate from a smooth consumption across time

Eqn (8) indicates correlations and not causality between the LHS and RHS Conversely, we can interpret our correlations in opposite directions.



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2.1 Economic intuitionLet us analyse (8)

ln() ( +

) 1

2(

+

)

Example:

> 1(more risk-averse than a log-utility with 1), a higher interest rate raiseconsumption more than one-to-one. Substitution effects dominate income effec

Substitution effects => to switch from current consumption to savings.

Income effects=> higher income as real interest rate increases.

0 < < 1(less risk-averse than a log-utility with 1), a higher interest rate rperiod consumption less than one-to-one. Income effects dominate substitution

1(log-utility), a higher interest rate raises second period consumption one-tSubstitution effects equal income effect.



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2.1 Further remarks

from (8) ln() (

)

(

)

Cross-refer to Week 1- Reading Pennacchi Pg 3

ln() (1 )

(Pennacchi)

Pennacchi define

.

To reconcile our predictions (1)-(5) based on (8) with Pennacchi, let

1 Pennacchi also assumes that +is non-stochastic, therefore

0and

We should use (8) which is the generic case assuming +is stochastic.



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2.1 Further remarks


In previous analysis, we assume returns and prices are given are in realterms.

+,+ , where,+is the real payoff of asset i at time t

Suppose they are given in nominal terms, we must deflate by the consumer price time t.

+,

,

+

,

, 1

+,+ 1...(9)

Where +=

+


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2.2 Hansen-Jagannathan Bounds


Using +,+ 1

+, ,+ +)(,+ 1

Divide by + , +, ,+

+ ,+

1

+

,+ +, ,+ (+)((,+)

,

((,) = +, ,+ ( + ..(10a)Since 1 +, ,+ 1,

| ,

(,)|

= + .(10b) where =

(10b) gives us the Hansen-Jagannathan bounds


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Sharpe ratio =

,

(,)

Empirical test: The Sharpe ratio of any portfolio/assets should not exceed

Using the same power utility previously, + (

)

((

))

((

))

(

) ( (

) )

(

)

(


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Assume ln +/ ~(, ),

(

)

+

ln / ~(, (

)

),

(

) +

(

) +

(

) 1

1

1

1 1 .(11) (Hint: use Taylors expans


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2.3 CAPMKey assumption of CAPM:

(+) ,,+where k>0

(+), ,+ = k ,+, ,+ = ,+

(+), ,+ = ,+, ,+

Using +,+ 1 (note: This gives the expected return of an asset in eq

+) (,+ (+ , ,+) 1

,+

,,



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2.3 CAPM

Since +

()

() , ,+

(),,

()=

,

() (a)

,+ (),,

()=

,,,,

() (b)

(b)/(a) and rearrange,

,+ ,+, ,+

,+[ ,+ ]

,+ [ ,+ ] .(12)

,,,

,and ,+ >0



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2.3 CAPMNote:

1

(12) is CAPM and is also known as the Security Market line (SML).

Alpha is actual return minus expected return estimated from an APCAPM.

CAPM is an equilibriummodel because it is derived from +,

which is an equilibrium pricing model.Equilibrium => no mispricing => zero alpha

=> ,

,( )

,

, ,+



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2.3 Testing CAPM:

,+

[

,+

]

Test (1): ,+ >0

Test (2): alpha =0

Empirically, CAPM fits the data badly.

Rethink how to specify our SDF ?

Rolls critique

CAPM is untestable.

Joint test:

(1) Empirical market proxy is correct.

(2) CAPM is correct.



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2.3 CAPM and Hansen-JagannathanBounds

,

(,) = + , ,+ + ..(10a)

Under CAPM, + , ,+ 1

,

(,)=( +

From (10a), we conclude | ,

(,)| +

,

(,)

CAPM specifies an upper limit, namely that any other assets cannotabsolute Sharpe ratio larger than that of the market portfolio.



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2.4 Arbitrage and the law of one price

An arbitrage opportunity for a finite T horizon is a self-finaprocess such that either

(i)< 0and 0with probability 1 or

(i) 0and 0with probability 1 and > 0witprobability

Cross-reference Chapter 4.



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(i)

< 0and

0with probability 1

A third party sponsored your gamble, and your gamble paid off wsure.

(ii) 0and 0with probability 1 and > 0with

probabilityZero initial investment in the portfolio but the portfolio sometim

positive profit but never with losses.

A special case of arbitrage is when this zero-net investment portfrisklessreturn.


xamp e: r rage


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xamp e: r rageOpportunity

Assume no short-sales constraint.


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If there are no arbitrage opportunities for the finitethen there is a strictly positive SDF process , ,

If there are no arbitrage opportunities for each finitthen there is an infinite horizon strictly positive SDF , , .

Absence of arbitrage => SDF exists => Law of one p

State prices exist for each state => Market is complete.



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Example: Covered interest parityCovered interest parity condition links spot and forward fo

exchange markets to foreign and domestic money markets uof one price.

:current date 0 forward price for exchanging one unit o

currency T periods in the future. At t=T, you pay dollars of one unit of the foreign currency.

:Spot price of foreign exchange. At t=0, you pay dollaof one unit of the foreign currency.



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Example: Covered interest parityAt t = 0:

Construct a zero initial investment portfolio by selling forward oneexchange at price . i.e. We are obligated to deliver one unit of fat t=T and receive dollars.

Concurrently, buy

unit of foreign currency and invest

in a foreign

period return .

This will costs us

dollars.

We then borrow

dollars at per-period interest rate .

Net initial investment = 0



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Example: Covered interest parityAt t = T:

unit of foreign currency earnings per-period returns over T

unit of foreign currency

We receive dollars for delivery of one unit of foreign currency.

Our debt compounded over T periods:

(

)

Net proceeds: -

()

By law of one price, we must have

.(13

(3) is the covered interest parity condition.



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2.5 State prices and SDF

If we are willing to assume that there is a finitenumber offuture states of the world, we can express the pricing kerneuseful ways.

Suppose there are kpossible states of the world and nass

Denote the payoff of asset i in state j as.

Let security prices be given by the N-vector P.

Denoting portfolio weights by the N-vector .Cost of the portfolio = P

Payoff of the portfolio =



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2.5 State prices and SDFWithin this framework, a state-price vector is defined to be a vectorq=(, )satisfying

(, ) =

.

..(14)

Or simply,

A solution for q exists if X is full-rank(complete market).

=> law of one price holds

=> SDF exists



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2.5 State prices and SDFRemember that all uncertaintyin this model is about which state w

the next period.

Let probability of state s = , then from 1 E MR

1 ,= .(15)

Now, we have = ,=

1 ,= (16)



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2.5 State prices and SDFCompare (15) and (16),

(15): 1 ,=

(16): 1 ,=

We have

.(17)

( 8)

Thus, the SDF can also be represented by the state-price normalizedprobability of the state occurring.

If exists and is strictly positive, then exist and is strictly positiv



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Concept checkUsing (16): 1 ,

=

Derive the present value of an asset that pays out regardless of the realized statein the next period.

What do you call this asset?



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Concept checkGiven that states of nature occur randomly. Would you exp

price of a high dividend-yield stock to be higherduring ecrecessionsor during booms?



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2.5 State prices and the SDFIn sum,

When state prices exist, the SDF exists.

When the SDF exists, we have the pricing equation 1 Ethe law of one price.

A dynamic security market is said to satisfy the law of on

whenever W and W* are self-financing processequal at any date t.



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2.6 Limits to Arbitrage

One typically can arbitrage when there are no marNo short-sales constraints

Negligible transaction costs

Depth in the market

However there are limits to arbitrage in the real world.

Read The Limits of Arbitrage

by Andrei Shleifer and Robert W. Vishny, Journal of Finance, Vo(Mar., 1997), pp. 35-55.



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2.6 Limits to Arbitrage

Read Forbes article: TProof That Tech Is In A

Top chart: Total U.S. scap/GDP.

Bottom chart: S&P50

Suppose you thfair market cap/should be one, warbitrage in yea



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2.7 Risk-Neutral Probability/PricingWe can perform the same trick that produced a pricing ke

sort of probability" that we care to consider. Many results sometimes called risk-neutralprobabilities.

Using ,=

, ,,

=

, ,

= (20)

where ,

has the characteristics of probability.

1. 0

2. = ,

= ,

, 1



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2.7 Risk-Neutral Probability/PricingWe can then write

,

,= =

,

(,)

Since we have

,

Under risk-neutral pricing, we again have

, , =

, (21)

We have changed the probability measure from the statistprobabilityto the risk-neutral probability .



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2.7 Risk-Neutral Probability/PricingFormally,

Suppose a money market account exists. Index this as ass

free return from date t to t+1 is,

,

A risk-neutral probability for < is defined to be a promeasure Q where

1. For any event A, (A)=0 if and only if Q(A)=0.2. For any asset i=2,n,

is a martingale on the time horizon

relative to Q.

If 1 and 2 hold, and Q are called Equivalent Martingale Me



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2.7 Risk-Neutral Probability/PricingThus,

,

,

,

,

(,)

,

,

,= , (22)

Under risk-neutral probability Q, the expected return on each asset freereturn.

Sub

,

,

,+,

, into (22)

We have

(,+,)

, ,(23)

To compute the price at t, compute the expectation of its value at t+by risk-free return.



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2.7 Risk-Neutral Probability/Pricing

The mapping from -measure to Q-measure is called the

Nikodymderivative of Q with respect to , denoted by

Think of Radon-Nikodymderivative as the ratio of probabunder Q relative to .Now you should understand why Q and must be equivalent mea

Valuation via a risk-neutral probability is equivalent toan SDF process.



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2.7 Risk-Neutral Probability/PricingWe have

,

, ..(24)

Proof:

Recall , and ,= ,

Therefore,

,

, =

,= ,,

=

,

= =



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2.7 Risk-Neutral Probability/PricingWe have

,

,1 ..(25)

Proof:

Recall ,Since ,= , =

,, ( ,,

= 1) ,,1

where 1is an indicator function that takes on unity only when s=



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2.7 Risk-Neutral Probability/PricingWe also have

,

,

..(26)

Proof:

Recall ,

Since ,=,

,

,

,

,

,,



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2.8 Complete Markets1) A dynamic securities market is said to be completeif, for eve

random variable x that depends only on date-t information isdate-t payoff.

A random variable x is marketed if it is the payoff of some port

Recall

(, ) =

.

(1) means that []is linearly spanned by a set of marketed portfoliwords, [] is full rank and can be invested to find q.



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2.8 Complete Markets

If the market is completeand law of one price holds, thencan be at most one SDF processbecause the condition

implies that =, since both

depend on date-t information.

If the market is completeand there is no arbitrageopportunities, then there is a uniqueSDF process, and it is positive.



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2.9 Portfolio Choice in Complete MarkeSuppose an investor can choose any consumption process C satisfying

[ ]= ..(27)

Intertemporal budget constraint with no endowment is given by

+ ( )+

Using CRRA utility

()

Recall (

(

=>+

..(28)

Iterating on (28), we have

..(29)



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2.9 Portfolio Choice in Complete MarkeFrom (27): [ ]

=

Sub (29), we have

[

]= [

]=

Which can be solved as

+[

]

.(30)

Sub (30) into (29),

+[

]

(3



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Next week

Tutorial 2 presentation

Dynamic Programming, Chap 9, Back.


fe 8507 lecture 2 more on dynamic security markets students (1).pptx

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