Asset Pricing Chapter X. Arrow-Debreu pricing II: The Arbitrage - - PowerPoint PPT Presentation

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Asset Pricing Chapter X. Arrow-Debreu pricing II: The Arbitrage - - PowerPoint PPT Presentation

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10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting

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SLIDE 1

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Asset Pricing

Chapter X. Arrow-Debreu pricing II: The Arbitrage Perspective June 22, 2006

Asset Pricing

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SLIDE 2

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

10.1 Market Completeness and Complex Security

Completeness: financial markets are said to be complete if, for each state of nature θ, there exists a θ, i.e., for a claim promising delivery of one until of the consumption good (or, more generally, the numeraire) if state θ is realized and nothing otherwise. Complex security: a complex security is one that pays off in more than one state of nature. (5, 2, 0, 6) = 5(1, 0, 0, 0)+2(0, 1, 0, 0)+0(0, 0, 1, 0)+6(0, 0, 0, 1), pS = 5q1 + 2q2 + 6q4.

Asset Pricing

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SLIDE 3

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Proposition 10.1 If markets are complete, any complex security

  • r any cash flow stream can be replicated as a

portfolio of Arrow-S Proposition 10.2 If M=N and all the M complex securities are linearly independent, then (i) it is possible to infer the prices of the A-D state-contingent claims form the complex securities’ prices and (ii) markets are effectively complete Linearly independent = no complex security can be replicated as a portfolio of some of the other complex securities.

Asset Pricing

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SLIDE 4

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

(3, 2, 0) (1, 1, 1) (2, 0, 2) (1, 0, 0) = w1(3, 2, 0) + w2(1, 1, 1) + w3(2, 0, 2) Thus, 1 = 3w1 + w2 + 2w3 = 2w1 + w2 = w2 + 2w3   3 1 2 2 1 0 0 1 2     w1

1 w2 1 w3 1

w1

2 w2 2 w3 2

w1

3 w2 3 w3 3

  =   1 0 0 0 1 0 0 0 1  

Asset Pricing

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SLIDE 5

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

t = 1 2 3 ... T −I0 ˜ CF 1 ˜ CF 2 ˜ CF 3 ... ˜ CF T NPV = −I0 +

T

  • t=1

N

  • θ=1

qt,θCFt,θ. (1)

Asset Pricing

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SLIDE 6

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure

Table 10.2: Risk-Free Discount Bonds As Arrow-Debreu Securities

Current Bond Price Future Cash Flows t = 0 1 2 3 4 ... T −q1 $1, 000 −q2 $1, 000 ... −qT $1, 000 where the cash flow of a “j-period discount bond” is just t = 0 1 ... j j + 1 ... T −qj $1, 000 Asset Pricing

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SLIDE 7

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

(i) 7 7

8% bond priced at 10925 32, or $1097.8125/$1, 000 of face

value (ii) 5 5

8% bond priced at 100 9 32, or $1002.8125/$1, 000 of face

value The coupons of these bonds are respectively, .07875 ∗ $1, 000 = $78.75 / year .05625 ∗ $1, 000 = $56.25/year

Asset Pricing

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SLIDE 8

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Table 10.3: Present And Future Cash Flows For Two Coupon Bonds Bond Type Cash Flow at Time t t = 0 1 2 3 4 5 77/8 bond: −1, 097.8125 78.75 78.75 78.75 78.75 1, 078.75 55/8 bond: −1, 002.8125 56.25 56.25 56.25 56.25 1, 056.25 Table 10.4 : Eliminating Intermediate Payments Bond Cash Flow at Time t t = 0 1 2 3 4 5 −1x 77/8 bond: +1, 097.8125 −78.75 −78.75 −78.75 −78.75 −1, 078.75 +1.4x 55/8 bond: −1, 403.9375 78.75 78.75 78.75 78.75 1, 478.75 Difference: −306.125 400.00 Price of £1 in 5 years = £0.765 Asset Pricing

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SLIDE 9

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Table 10.5: Date Claim Prices vs. Discount Bond Prices Price of a N year claim Analogous Discount Bond Price ($1,000 Denomina- tion) N = 1 q1 = $1/1.06 = $.94339 $ 943.39 N = 2 q2 = $1/(1.065113)2 = $.88147 $ 881.47 N = 3 q3 = $1/(1.072644)3 = $.81027 $ 810.27 N = 4 q4 = $1/1.09935)4 =$ .68463 $ 684.63 Table 10.6: Discount Bonds as Arrow-Debreu Claims Bond Price (t = 0) CF Pattern t = 1 2 3 4 1-yr discount

  • $943.39

$1,000 2-yr discount

  • $881.47

$1,000 3-yr discount

  • $810.27

$1,000 4-yr discount

  • $684.63

$1,000 Asset Pricing

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SLIDE 10

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Replicating 80 80 80 1080

Table 10.7: Replicating the Discount Bond Cash Flow Bond Price (t = 0) CF Pattern t = 1 2 3 4 08 1-yr discount (.08)(−943.39) = −$75.47 $80 (80 state 1 A-D claims) 08 2-yr discount (.08)(−881.47) = −$70.52 $80 (80 state 2 A-D claims) 08 3-yr discount (.08)(−810.27) = −$64.82 $80 1.08 4-yr discount (1.08)(−684.63) = −$739.40 $1,080 Asset Pricing

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SLIDE 11

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Evaluating a CF: 60 25 150 300

p = ($60 at t=1) „ $.94339 at t=0 $1 at t=1 « + ($25 at t=2) „ $.88147 at t=0 $1 at t=2 « + ... = ($60) 1.00 1 + r1 + ($25) 1.00 (1 + r2)2 + ... = ($60) 1.00 1.06 + ($25) 1.00 (1.065113)2 + ... Evaluating a risk-free project as a portfolio of A-D securities=discounting at the term structure. Asset Pricing

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SLIDE 12

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Appendix 10.1 Forward Prices and Forward Rates

(1 + r1)(1 + 1f1) = (1 + r2)2 (1 + r1)(1 + 1f2)2 = (1 + r3)3 (1 + r2)2(1 + 2f1) = (1 + r3)3, etc.

Asset Pricing

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SLIDE 13

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Table 10.10: Locking in a Forward Rate t = 1 2 Buy a 2-yr bond

  • 1,000

65 1,065 Sell short a 1-yr bond + 1,000

  • 1,060
  • 995

1,065 Table 10.11: Creating a $1,000 Payoff t= 1 2 Buy .939 x 2-yr bonds

  • 939

61.0 1,000 Sell short .939 x 1-yr bonds + 939

  • 995.34
  • 934.34

1,000

Asset Pricing

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SLIDE 14

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Diversifiable risk is not priced

˜ zc = A˜ za + B˜ zb, for some constant coefficients A and B (2) prices of the three assets: pc = Apa + Bpb. pi =

  • s

qszsi, i = a, b (3)

pc = X

s

qszsc = X

s

qs(Azsa + Bzsb) = X

s

(Aqszsa + Bqszsb) = Apa + Bpb Asset Pricing

slide-15
SLIDE 15

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Diversifiable risk is not priced

Suppose a and b are negatively correlated. c is less risky, yet pc must be «in line» with pa and Pb Suppose a and b are perfectly negatively correlated. Can be combined to form d, risk free pd must be such that holding d earns the riskless rate How can the risk of a and b be remunerated

Asset Pricing

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SLIDE 16

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Proposition 10.3 A necessary as well as sufficient condition for the creation of a complete set of A-D securities is that there exists a single portfolio with the property that options can be written on it and such that its payoff pattern distinguishes among all states of nature. Proposition 10.4 If it is possible to create, using options, a complete set of traded securities, simple put and call options written on the underlying assets are sufficient to accomplish this goal

Asset Pricing

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SLIDE 17

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

It is assumed that ST discriminates across all states of nature so that Proposition 8.1 applies; without loss of generality, we may assume that ST takes the following set

  • f values:

S1 < S2 < ... < Sθ < ... < SN, where Sθ is the price of this complex security if state θ is realized at date T. Assume also that call options are written on this asset with all possible exercised prices, and that these

  • ptions are traded. Let us also assume that Sθ = Sθ−1 + δ for

every state θ.

Asset Pricing

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SLIDE 18

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

Consider, for any state ˆ θ, the following portfolio P: Buy one call with K = Sˆ

θ−1

Sell two calls with K = Sˆ

θ

Buy one call with K = Sˆ

θ+1

At any point in time, the value of this portfolio, VP, is VP = C

  • S, K = Sˆ

θ−1

  • − 2C
  • S, K = Sˆ

θ

  • + C
  • S, K = Sˆ

θ+1

  • .

Asset Pricing

slide-19
SLIDE 19

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization The payoff from such a portfolio thus equals: Payoff to P = 8 < : if ST < S ˆ

θ

δ if ST = S ˆ

θ

if ST > S ˆ

θ

q ˆ

θ =

1 δ h C “ S, K = S ˆ

θ−1

” + C “ S, K = S ˆ

θ+1

” − 2C “ S, K = S ˆ

θ

”i . Payoff Diagram for All Options in the Portfolio P

Payoff d −2CT(ST, K = Sq) ST CT(ST, K = Sq_1) Sq_1 CT(ST , K = Sq+1) Sq Sq+1 d

Asset Pricing

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SLIDE 20

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

10.7 Recovering Arrow-Debreu Prices form Option Prices: A Generalization

(i) Suppose that ST, the price of the underlying portfolio (we may thin of it as a proxy for M), assumes a "continuum" of possible values. (ii) Let us construct the following portfolio: for some small positive number ε >0 Buy one call with K = ˆ ST − δ

2 − ε

Sell one call with K = ˆ ST − δ

2

Sell one call with K = ˆ ST + δ

2

Buy one call with K = ˆ ST + δ

2 + ε.

Asset Pricing

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SLIDE 21

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

Payoff Diagram: Portfolio of Options

ˆ Payoff e CT(ST, K = ST– 2) ST CT(ST , K = ST– 2 – e) ST Value of the portfolio at expiration CT(ST, K = ST+ 2) ST – 2–e ST – 2 ST+ 2 ST+ 2+e ˆ CT(ST, K = ST+ 2+e) ˆ ˆ ˆ

d – d – d – d – d – d – d – d –

ˆ ˆ ˆ ˆ ˆ Asset Pricing

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SLIDE 22

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

A Generalization

Let us thus consider buying 1/ε units of the portfolio. The total payment, when ˆ ST − δ

2 ≤ ST ≤ ˆ

ST + δ

2, is ε · 1 ε ≡ 1, for any

choice of ε. We want to let ε → 0, so as to eliminate payments in the ranges ST ∈

  • ˆ

ST − δ

2 − ε, ˆ

ST − δ

2

  • and

ST ∈

  • ˆ

ST + δ

2, ˆ

ST + δ

2 + ε

  • . The value of 1/ε units of this

portfolio is: 1 ε

  • C(S, K = ˆ

ST − δ 2 − ε) − C(S, K = ˆ ST − δ 2) −

  • C(S, K = ˆ

ST + δ 2) − C(S, K = ˆ ST + δ 2 + ε)

  • ,

Asset Pricing

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SLIDE 23

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization lim

ε→0

1 ε n C(S, K = ˆ ST − δ 2 − ε) − C(S, K = ˆ ST − δ 2 ) − ˆ C(S, K = ˆ ST + δ 2 ) − C(S, K = ˆ ST + δ 2 + ε) ˜o = − lim

ε→0

8 > > > > > < > > > > > : C “ S, K = ˆ ST − δ

2 − ε

” − C “ S, K = ˆ ST − δ

2

” −ε | {z }

≤0

9 > > > > > = > > > > > ; + lim

ε→0

8 > > > > > < > > > > > : C “ S, K = ˆ ST + δ

2 + ε

” − C “ S, K = ˆ ST + δ

2

” ε | {z }

≤0

9 > > > > > = > > > > > ; = C2 „ S, K = ˆ ST + δ 2 « − C2 „ S, K = ˆ ST − δ 2 « . Asset Pricing

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SLIDE 24

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

Suppose, for example, we have an uncertain payment with the following payoff at time T: CFT =

  • if ST /

∈ [ˆ ST − δ

2, ˆ

ST + δ

2]

50000 if ST ∈ [ˆ ST − δ

2, ˆ

ST + δ

2]

  • .

The value today of this cash flow is: 50, 000 ·

  • C2
  • S, K = ˆ

ST + δ 2

  • − C2
  • S, K = ˆ

ST − δ 2

  • .

q

  • S1

T, S2 T

  • = C2
  • S, K = S2

T

  • − C2
  • S, K = S1

T

  • .

(4)

Asset Pricing

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SLIDE 25

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

Payoff Diagram for the Limiting Portfolio

Payoff ST ST ST_ _

2

ST+ _

2

1

d d

Asset Pricing

slide-26
SLIDE 26

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization A Generalization

Table 10.8: Pricing an Arrow-Debreu State Claim

Cost of Payoff if ST = K C(S, K) Position 7 8 9 10 11 12 13 ∆C ∆(∆C) = qθ 7 3.354

  • 0.895

8 2.459 0.106

  • 0.789

9 1.670 +1.670 1 2 3 4 0.164

  • 0.625

10 1.045

  • 2.090
  • 2
  • 4
  • 6

0.184

  • 0.441

11 0.604 +0.604 1 2 0.162

  • 0.279

12 0.325 0.118

  • 0.161

13 0.164 0.184 1 Asset Pricing