Assignment 8

Arbitrage A-D Examples Bonds

Definitions I Arbitrage is a riskless profit opportunity that any individual who prefers more wealth to less will choose to exploit.

I Riskless profit opportunity I The opportunity never losses money today or at any subsequent date.

I The opportunity sometimes makes money today or a some point in the future.

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Arbitrage A-D Examples Bonds

Theory I Financial theory assumes that there are no arbitrage opportunities because any opportunities that might arise are eliminated so quickly that we never observe them.

I no arbitrage ) relate the prices of some assets to the prices of other assets.

I Model of time and uncertainty I states of the world I di§erent time periods

I [Event Tree: with two dates, two states, and two probabilities]

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Arbitrage A-D Examples Bonds

Event tree I pi is the probability of state i occurring tomorrow I we assume that pi ≥ 0 for any i and Â

i pi = 1

I Xi,j is the payo§ tomorrow in state i on asset j

Asset 1 Asset 2 State 1 X1,1 X1,2 State 2 X2,1 X2,2

I Pj is the price today of asset j:

Asset 1 Asset 2 Price P1 P2

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Redundancy I A redundant asset has payo§s that can be obtained by combining other assets

I Formally, the set of payo§s for one asset is a linear combination of the payo§s of another group of assets.

I We may always have redundant assets. We must have redundant assets when the number of assets exceeds the number of distinct states of the world.

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Arbitrage A-D Examples Bonds

Redundancy example I model of asset market

Asset 1 rain coat shares

Asset 2 umbrella shares

State 1 1 2 State 2 0 0 Price 1/2 unknown

I states: State 1 - rain and State 2 - no rain I Is asset k asset redundant?

 j 6=k

qjXi,j = Xi,k

I What is its price? Pk = Â

j qjPj

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Arbitrage A-D Examples Bonds

I solve for q1 with two equations and one unknown variable I State 1:

1

 j=1 q1X1,1 = X1,2

q1(1) = 2

q1 = 2

I State 2:

1

 j=1 q1X2,1 = X2,2

q1(0) = 0

0 = 0 true for any q1

I for q1 = 2 (a portfolio of 2 shares of asset 1) the portfolio has the same payo§s as asset 2. Asset 2 is redundant.

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Arbitrage A-D Examples Bonds

No arbitrage (1) I Principle 1: Redundant assets are priced as linear combinations of non-redundant assets.

I in the example P2 = 2P1 because Xi,2 = 2Xi,1 for every i.

I State 1: X1,2 = 2 = 2(1) = 2X1,1 I State 2: X2,2 = 0 = 2(0) = 2X2,1

I price of 2 shares of asset 1 = price of 1 share of asset 2 (no arbitrage)

P2 = 2P1 P2 = 2(1/2) P2 = 1

I Now, we assume that all redundant assets have been dropped from the system of assets. We are left with a number of assets that is less than or equal to the number of states.

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Arbitrage A-D Examples Bonds

Another example I model of asset market

Asset 1 rain coat shares

Asset 2 umbrella shares

State 1 5 15 State 2 2 6 Price unknown 6

I states: State 1 - rain and State 2 - no rain I Is the kth asset redundant? (k = 1)

 j 6=k

qjXi,j = Xi,k

I What is its price? Pk = Â

j qjPj

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Arbitrage A-D Examples Bonds

I solve for q2 with two equations and one unknown variable I State 1:

2

 j=2 q2X1,2 = X1,1

q2(15) = 5

q2 = 1/3

I State 2:

2

 j=2 q2X2,2 = X2,1

q2(6) = 2

q2 = 1/3

I for q2 = 1/3 (a portfolio of 1/3 shares of asset 2) the portfolio has the same payo§s as asset 1.

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Arbitrage A-D Examples Bonds

Price of Asset 1 I initially unknown but related to price of asset 2 I price of 1 share of asset 1 = price of 1/3 of a share of asset 2 (no arbitrage)

P1 = (1/3)P2 P1 = (1/3)6 P1 = 2

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Arbitrage A-D Examples Bonds

Find arbitrage opportunity I Practice question: What is the source of the arbitrage opportunity and how you would trade to exploit it?

Asset 1 corn co shares

Asset 2 soy co shares

State 1 4 10 State 2 2 5 Price 3 7

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Arbitrage A-D Examples Bonds

Complete markets I Complete markets: For each state i an asset exists, or can be constructed from other assets, to pay $1 in state i and $0 in all other states.

I Arrow-Debreu securities I Arrow-Debreu prices or state prices I state price for state i is Si

I Example 1: The original assets are Arrow-Debreu securities

Asset 1 rain coat shares

Asset 2 ice cream shares

State 1 1 0 State 2 0 1 Price P1 P2

I Note that S1 = P1 and S2 = P2, and therefore, these two securities are the Arrow-Debreu securities.

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Arbitrage A-D Examples Bonds

No arbitrage (2) I Principle 2: State prices are positive I Why? If there is an Arrow-Debreu security with a zero or negative price, buy it.

I What happens? I pay nothing or even get money today I next period you do not lose anything in all states and in one state you hit the lottery.

I This is clearly an arbitrage opportunity.

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Arbitrage A-D Examples Bonds

No arbitrage (3) I Principle 3: There exist state prices, all of which are positive, that satisfy Pj = Â

i SiXi,j, and thus, are consistent with

observed asset prices. I Portfolio 1: 1 unit of asset j w/ price Pj and payo§s Xi,j. The payo§ in state i is Xi,j for asset j

I Portfolio 2: Xi,j units of A-D security i for each state i. The payo§ in state i is Xi,j ∗1 for the portfolio of A-D securities

I If Pj > Â i SiXi,j buy one unit of portfolio 2 and sell one unit

of portfolio 1) arbitrage opportunity I If Pj < Â

i SiXi,j buy one unit of portfolio 1 and sell one unit

of portfolio 2) arbitrage opportunity

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Arbitrage A-D Examples Bonds

Complete markets example I Complete markets: For each state i an special asset exists, or can be constructed from other assets, to pay $1 in state i and $0 in all other states.

I Arrow-Debreu securities (A-D securities) I state price for state i is Si

Asset 1 rain coat shares

Asset 2 ice cream shares

State 1 1 −1/4 State 2 −1/2 1 Price 1/4 3/4

I How do we find the A-D securities? What is the state price for each state? What is the price of the asset that pays 3 in state 1 and 2 in state 2?

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Arbitrage A-D Examples Bonds

A-D security for state 1 I two equations and two unknown variables I State 1 equation:

2

 j=1 qjX1j = 1

q1(1)+q2(−1/4) = 1 q1 = 1+(1/4)q2

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Arbitrage A-D Examples Bonds

I State 2 equation:

2

 j=1 qjX2,j = 0

q1(−1/2)+q2(1) = 0 (1+(1/4)q2)(−1/2)+q2(1) = 0

−1/2− (1/8)q2 +q2 = 0 (7/8)q2 = 1/2

q2 = 4/7

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Arbitrage A-D Examples Bonds

I What is q1?

q1 = 1+(1/4)q2 = 1+(1/4)(4/7) = 8/7

I portfolio for state 1 A-D security is q1 = 8/7 and q2 = 4/7 I What is S1 (no arbitrage)?

S1 = Â j qjPj

S1 = q1P1 +q2P2 S1 = (8/7)(1/4)+(4/7)(3/4) S1 = 2/7+3/7 S1 = 5/7

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Arbitrage A-D Examples Bonds

A-D security for state 2 I two equations and two unknown variables again I State 1 equation:

2

 j=1 qjX1,j = 0

q1(1)+q2(−1/4) = 0 q1 = (1/4)q2

I State 2 equation:

2

 j=1 qjX2,j = 1

q1(−1/2)+q2(1) = 1 (1/4)q2(−1/2)+q2(1) = 1

q2 = 8/7

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Arbitrage A-D Examples Bonds

I What is q1?

q1 = (1/4)q2 = (1/4)(8/7) = 2/7

I portfolio for state 2 A-D security is q1 = 2/7 and q2 = 8/7 I What is S2 (no arbitrage)?

S2 = Â j qjPj

S2 = q1P1 +q2P2 S2 = (2/7)(1/4)+(8/7)(3/4) S2 = 1/14+6/7 S2 = 13/14

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Arbitrage A-D Examples Bonds

Construction of other asset I trade to build asset 3

Asset 3 movie shares

State 1 3 State 2 2 Price unknown

I replicate from A-D securities: 3 units of A-D for state 1 and 2 units for A-D for state 2

Pj = Â i SiXi,j

P3 = 3S1 +2S2 P3 = 3(5/7)+2(13/14) P3 = 28/7 = 4

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Arbitrage A-D Examples Bonds

Riskless asset I Payo§ is equal to one in all states

riskless State 1 1 State 2 1 Price unknown

I Replicate from A-D securities: buy 1 unit of A-D for state 1 and buy 1 unit for A-D for state 2

Pf = Â i (1)Si

Pf = 5/7+13/14 Pf = 23/14

Rf = (1/Pf )−1 = 14/23−1 = −9/23

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Arbitrage A-D Examples Bonds

Arbitrage example 1 I Redundant assets: Assets whose payo§s can be obtained by combining other assets

I Formally, the set of payo§s for one asset is a linear combination of the payo§s of another group of assets.

Asset 1 corn co shares

Asset 2 soy co shares

State 1 1.5 9 State 2 1 6 Price 3 16

I What is the arbitrage opportunity?

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Arbitrage A-D Examples Bonds

Arbitrage example 2 I An A-D security or a portfolio of other securities that replicates the payo§s of an A-D security has a zero or negative price.

Asset 1 corn co shares

Asset 2 soy co shares

State 1 2 1 State 2 5 1.5 Price 3 1.5

I What is the arbitrage opportunity?

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Arbitrage A-D Examples Bonds

Arbitrage example 3 I The price of any asset must match the value of the portfolio of building block with the same payo§s.

Asset 1 Asset 2 Asset 3 Asset 4 State 1 0 0 1 15 State 2 1 0 0 7 State 3 0 1 0 4 Price 2/7 1/4 3/5 11

I What is the arbitrage opportunity?

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Arbitrage A-D Examples Bonds

No arbitrage principles I Principle 1: Redundant assets are priced as linear combinations of non-redundant assets.

I Principle 2: State prices are positive I If there is an Arrow-Debreu security with a zero or negative price, it is time to buy. You pay nothing or even get money today and next period you do not lose anything in all states and in one state you hit the lottery. This is clearly an arbitrage opportunity

I Principle 3: There exist state prices, all of which are positive, that satisfy Pj = Â

i SiXi,j, and thus, are consistent with

observed asset prices. This set of state prices is unique if markets are complete.

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Arbitrage A-D Examples Bonds

Stochastic discount factor I consider asset j where

Pj = S1X1,j +S2X2,j

I rewrite

Pj = p1 S1 p1 X1,j + p2

S2 p2 X2,j

Pj = p1M1X1,j + p2M2X2,j Pj = E[MXj]

I M such that Mi = Si pi is called the stochastic discount factor

(sdf) I Pj is probability weighted average of the sdf in state i and payo§ in same state

I Pj is expectation of the sdf times payo§ I M is the same random variable for all assets (riskless asset?)

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Arbitrage A-D Examples Bonds

Alternative interpretation

Pj = S1X1,j +S2X2,j

Pj = (S1 +S2) !

S1 S1 +S2

X1,j + S2

S1 +S2 X2,j

"

Pj = 1

1+Rf [p∗1X1,j + p

∗ 2X2,j]

Pj = 1

1+Rf E∗ [Xj]

I S1 S1+S2

and S2S1+S2 are positive and must sum to 1 I {p∗i } is set of “pseudo-probabilities” or “risk-adjusted probabilities” (not necessarily equal to actual probabilities)

I Pj is pseudo-probability weighted average of Xi,j

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Arbitrage A-D Examples Bonds

Fixed income securities I Fixed-income securities (bills, notes, and bonds) make fixed payments in either nominal or real terms.

I Discount bonds make a single payment at the maturity date that can be normalized to $1.

I Coupon bonds pay $C each period up to and including the maturity date (C is the coupon rate if the principal payment is normalized to $1). At the maturity date these bonds also make principal payment.

I Essentially, coupon bonds are a package of zero-coupon bonds of di§erent maturities.

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Arbitrage A-D Examples Bonds

Definitions I The yield to maturity for a bond is the per-period average return if the bond is held to maturity (with per-period compounding); equivalently, it is the discount rate that equates the present value of the bond’s stream of payments to the bond’s price.

I The term structure of interest rates is the set of yields to maturity, at a given time, for bonds of di§erent maturities.

I The holding period return on a bond is the return over a specific holding period less than the bond’s maturity. Usually, we work with one-period holding periods.

I Bonds of di§erent maturities can be combined to guarantee an interest rate on a fixed-income investment to be made in the future. This guaranteed interest rate is called a forward rate.

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Arbitrage A-D Examples Bonds

Yields, Prices, and Returns I Imagine that there is a m-period discount (zero coupon) bond with a price of Pm,t.

I What is the return on the bond if it is held to maturity? What is the yield to maturity for the bond?

I Pay Pm,t today and get $1 at maturity

1+Rm,t,t+m = 1 Pm,t

= (1+Ym,t) m

I Using log returns

rm,t,t+m = −pm,t = mym,t

ym,t = ln(1+Ym,t)

pm,t = ln(Pm,t)

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Arbitrage A-D Examples Bonds

One-period HPR I This is the gross return on a m-period bond held for one period (bought at time t with maturity m and sold at time t +1 with maturity m−1)

1+Rm,t+1 = Pm−1,t+1 Pm,t

= (1+Ym,t)m

(1+Ym−1,t+1)m−1

I In logs

rm,t+1 = pm−1,t+1 −pm,t rm,t+1 = mym,t − (m−1)ym−1,t+1 rm,t+1 = ym,t − (m−1)(ym−1,t+1 −ym,t)

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