Lecture 9: Course overview

Economics of Finance

School of Economics, UNSW

Copyright By cscodehelp代写 加微信 cscodehelp

Arbitrage-free Pricing

• financing company: stocks and bonds

• atomic or Arrow-Debreu securities

• bond valuation (duration)

• options: European,

• CAPM and APT

Optimal decisions/allocation

• exploring arbitrage

• minimum cost hedging

in incomplete markets

• optimal trade in Arrow-Debreu economy

• optimal portfolio allocations

A Road Map

The Law of One Price (LOP)

Definition: (LOP) In an arbitrage-free economy with no transactions costs, any given time-state claim will sell for the same price, no matter how obtained. This holds for any ’package’ of time-state claims.

• In the ’real world’ transactions costs are usually present;

• The lack of arbitrage opportunities only insures that prices for a given set of time-state claims will fall within a band narrow enough to preclude generating a positive profit net of transactions costs out of trading.

Definition: Valuation is the process of determining the present value of a security or productive investment.

Example: How much is a tree worth today (at time 0)?

Good weather

Bad weather

Present Value of a tree: PV = 0.285 · 63 + 0.665 · 48 = 49.875

Financing Methods

Say you’d like to set up an apple firm which consists of an apple tree, i.e., you need 49.875 apples to purchase the tree.

There are two ways to finance this investment, issue bonds or issue stocks. Assume your firm issues a bond:

The Apple Tree Firm promises to pay the holder 20

apples at the end of the year, no matter what the

weather has been.

This way the holder does not bear any face value risk (though other types of risk, e.g., default risk or interest rate risk, etc., remain).

If your firm issues a stock, instead:

The Apple Tree Firm promises to pay the holder all

the apples left over after the bondholder has been

This way the holder bear the risk of the apple production net the issued bond payment, BUT is entitled a voting right.

The bond represents the ownership of the money, i.e., prior claim; the stock represents the ownership of the firm, i.e., residual claim.

Principle of value additivity

pa × qfirm = pa × (qbond + qstock).

Pricing future desired payment c

With the payment matrix Q {states × securities}, having at

least as many securities (with linearly independent payoffs) as states

and given pS {1×securities} security prices

we find unique price p for any desired payments c {states×1} via replication portfolio n = Q−1c (from accounting Qn = c)

and even simpler using atomic prices

p = pSn = pSQ−1c = patomc = dffatomc = dfE∗(c) = E(m1c),

where m1 is the stochastic discount factor, and E and E ̃ are expectations taken over physical and risk-neutral measures.

If any security (derivative, option) deviates from this pricing, we have arbitrage, can construct profitable strategy using replicating portfolio.

Pricing future desired payment c

With the payment matrix Q {states × securities}, having at

least as many securities (with linearly independent payoffs) as states

and given pS {1×securities} security prices

we find unique price p for any desired payments c {states×1} via replication portfolio n = Q−1c (from accounting Qn = c)

and even simpler using atomic prices

p = pSn = pSQ−1c = patomc = dffatomc = dfE∗(c) = E(m1c),

where m1 is the stochastic discount factor, and E and E ̃ are expectations taken over physical and risk-neutral measures.

If any security (derivative, option) deviates from this pricing, we have arbitrage, can construct profitable strategy using replicating portfolio.

Reality check 1 How do we get Q?

Reality check 2 Infinite number of states (always incomplete)? 7 / 33

How much payment can you get with a dollar? Consider Bond and Stock:

Value relative

Opportunity Set

Make linear combination of value relatives for Bond and Stock:

GW security 3

BW security 1 1.5038 2

Arbitrage Opportunity

When value relative is above/below the opportunity set, there are arbitrage opportunities:

Z is preferred to ZZ. Example of realising arbitrage: short-sell replicated ZZ to buy Z with proceeds. Note ZZ period 1 payments are fully covered by Z plus there is risk-free profit.

Hedging with Minimum Cost

Hedging is a technology to construct a portfolio and offset risks in all future states.

• Essentially same technology of payment replication as arbitrage;

• Arbitrage: replicate an asset that is over/under valued in the market and generate profit;

• Hedging: given an asset position, offset all risks and liability by replicating this position.

In an incomplete market, hedging entails a constrained optimization problem:

min pS · n subject to Q · n ≥ c. n

• The multiperiod discount factor df

• The multiperiod certain cash flow cf

• Bond’svaluePV =df·cf,manybonds: p=df·Q

• Multi-period interest rate, yield curve i(t) = 1 1t − 1 df ( t )

• Yield-to-maturity is a constant interest rate, such that the present value of all bond’s payments equals its price.

• Duration

• The average waiting period for a bond to be paid back.

D = t cft

t (1+y)tpb

• Frequently used is the modified duration – negative relative change in bond’s value to per unit of interest rate change

md= D (1+y)

Multi-periods: Bond

• Two-period zero-coupon bond (no coupon payments)

• Its initial value is $1.00.

• Its price increases 5% of its prior value in every period.

gg 1 1.1025

1 1.1025 H b

PPq 1.1025

Multi-periods: Stock

• Its initial value is $1.00. It pays no dividends.

• Its price increases 26% of its prior value in good times. • Its price falls to 96% of its prior value in bad times.

gg 1 1.5876

1 1.2096 H b

PqP 0.9216

Planned Acquisitions

We write down the payment of these acquisitions in a matrix: B0 S0 Bb Sb

1.05 1.26 −1 −1 0 0 g

1.05 0.96 0 0 −1 −1 b

0 1.05 1.26 0 0 gg 0 1.05 0.96 0 0 gb

0 0 0 0 1.05 0.96 bb

Price Vector:

B0 S0 BgSgBbSb

pS = 1.00 1.00 0.0 0.0 0.0 0.0

To price a unit of payment at each state use patom = pS · Q−1 g b gg gb bg bb

patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444

1.05 1.26 bg

• Call option vs. Put option:

• Call option: entitles the right to buy an underlying asset

(say shares, foreign currency or commodity) at a specified

strike price, or, exercise price(X).

• Put option: entitles the right to sell the underlying asset at

a specified strike price X.

• European option vs. American Option

• European put or call option: can be exercised only on expiration date.

• American put or call option: can be exercised exercised on any date up to and including its expiration date.

• European: use atomic security prices to price all net

payoffs at the end of its life

• American: must consider a possibility of an earlier exercise

European-style option: Cash and Call

European-style option: Underlier and Put

Put-Call Parity

pCall +PV(X)=pPut +punderlier

• Irrespective of the value of the underlier at expiration, both

portfolios will have the same payoffs;

• Law of One Price – same payoffs should have the same price • Only applies to European options.

Arrow-Debreu equilibrium:

• Individual expected utility maximisation problem max U=u(c0)+β πs1 ·u(cs1)

c0,cs1,as1 ∀s1∈S1

• Given budget constraints under all states;

c0 + qs1 · as1 = e0, s1 ∈S1

cs1 =as1 +es1,∀s1 ∈S1 • Prices taken as given

• Set up Lagrangian and take first order conditions;

•Deriveprices:q =λs1=βπu′(cs1),∀s∈S s1 λ0 s1 u′(c0) 1

• Combine with market clearing:

kc0 =ke0;kcs1 =kes1,∀s1 ∈S1

• Characterise equilibrium c, a, qs.

Summary of Gains from Trade and Pareto improvement

• Heterogeneous consumers benefit from trade relatively to autarky

• Consumption smoothing

• Risk sharing due to difference in endowments • Risk sharing due to difference in risk aversions

• Competitive equilibrium (prices are taken as given) leads to Pareto improvement – making at least one consumer better off without making anyone worse off .

Risk neutral probabilities and Stochastic discount Factor

Forward price and stochastic discount factor

• Forward prices – risk-neutral probabilities

qs1 qs1 u′(cs1) u′(cs1)

fs1 =df(1)= q =πs1 u′(c) πs1 u′(c),∀s1∈S1

s1 0s1∈S1 0

ms1 = βu′ (cs1) u′ (c0)

such that df(1) = E(m1)

s1 ∈S1 • Stochastic df

Expected mean-variance utility

Eu = e − (s2/t),

• increasing w.r.t. e (monotonicity); • decreasing w.r.t. s (risk aversion); • t risk tolerance;

Market opportunities

Market opportunities are presented by portfolios of available securities

• the stochastically dominant section is called efficient frontier

• a single stock market: two symmetric line sectors from the risk free rate

• two general stocks: a rightward parabola in e − v space, or a hyperbola in e − s space

Efficient frontier

Sharpe ratio

Sharpe ratio:

measures marginal return for risk.

SM = eM −rf is the slope to the tangent line and therefore the sM

best Sharpe ratio available on the market

S = e − rf , s

Capital allocation line and separation theorem

All investors invest in the combination of the risky-free asset and the same market portfolio. The share of the market portfolio and risk-free asset is determined by their risk tolerance (risk-aversion).

Systematic vs Idiosyncratic risk

Capital Asset Pricing Model

Capital asset pricing model (CAPM) is a model used to determine an appropriate expected return of any asset

• only systematic risk is valued

• replicate any desired expected asset return ej using the market portfolio (fraction βj) and the risk-free asset (fraction 1–βj)

ej =rf +βj(eM −rf)

• βj: sensitivity of asset j to market movements

Security market line

Factor models

CAPM provides good benchmark, but reality is more complicated: market risk is just one factor, but there are others

Rj =rf +βj,1f1+,…,+βj,KfK +εj,

• Rj is the expected return of the asset (or portfolio) j

• εj idiosyncratic, unexplained part of return E(εj) = 0, E(Rj) = ej

• rf is the risk-free rate

• fk is the factor risk premium

• βj,k is the sensitivity of portfolio j to factor k

• K is the number of factors.

This is not pure arbitrage, but statistical arbitrage

• Monday, 15 August, 13:00-17:00 time frame • Two hours + 10 minute reading time

• Comprehensive – covers all course material • Open book, but individual exam

• Multiple-choice, compute questions

• Exam counts for 45% of the overall mark

Final exam

myExperience

• The best way to thank me is to fill in myExperience survey

• We take this very seriously so please make your comments clear and constructive

程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: kyit630461@163.com