# Arbitrage and Pricing

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```					Chapter 5

5 Arbitrage and Pricing
In a standard consumption-investment problem, investors have two types of controls available. In the 1st period, they invest wealth in investment assets after consumption. In the 2nd period they sell these assets to buy the various consumption goods. Therefore, investment consists of a series of transfers. 1. Notation: Let the number of outcome states and assets be finite. States are indexed by s (s=1, … S) and assets by i (i = 1, …, N). The one period investment problem is characterized by the table of end-of-period prices on the N assets in the S states Y’ = (y1, …, yS) The investor chooses a portfolio represented by the vector n where ni is the number of units (shares) held of asset i. The initial values of the assets are v, and the budget constraint is n’v = W0. The realized return (1 + rate of return) on the ith asset in the state s is Zsi = Ysi / vi. The state space table of return is Z =YDv-1 Where Dv is a diagonal matrix with ith element vi. We usually assume that vi  0. The vector z.i is the ith column of Z and represents the S possible returns on asset i. The vector zj.’, the jth row of Z, represents the returns on the N assets in state s=j.

z is a random vector with realization zs.; z i is a scalar random variable which is the
~

~

~

ith element of z . In return space, investments are characterized by a vector  i = ni vi. If the investment satisfies W0 = where

ni vi = constant > 0, then we normalize  i to wi = ni vi / W0. In this case, the investment is a portfolio and w is the vector of portfolio weights. w and η contain the same information as vi is nonzero. If wi < 0, the asset has been sold short. Question: Find the definition of short sale and some application of such activity. Example: Consider the economy with two assets and three states:  1 3   Z =  2 1 . 3 2   T An investment commitment of = (1,3) would have state-by-state outcomes of -1-



Chapter 5 Z = (10,5,9)T and an original commitment of 4. The corresponding portfolio would be (10/24, 5/24, 9/24)T. Definition: An arbitrage portfolio is defined as a vector of commitments summing up to 0. Arbitrage portfolios will be denoted by ω . We usually assume that ω  0. This gives an insight into how such a portfolio is financed. Definitions: A riskless portfolio is a portfolio with the same return in every state: Zw = R1, 1T w =1 R is the reiskless return and R-1 is the risk-free rate. In general, this is not true, so we define

R = Max [ Min (Zw)]


w
_

S

R = Min [ Max (Zw)] w S These are usually linear programming problems.
Remark: The concept of arbitrage in economics is, in reality, there is never an opportunity to make a risk-free profit that gives a greater return than that provided by the interest from a bank deposit. In real life, arbitrageurs make profit by borrowing bank money to invest, they take on risks to get better return. Exercise: For each of the following, state whether or not there is an arbitrage opportunity: 1) An individual need change as offering you a 10 pound note in return for 9 pounds change. (/) 2) England football team is playing France tonight, a friend says that he will pay you 20 pounds if England loses and if you support France during the game. (/) 3) Mr A offers to purchase a car for 12 000 pounds, Mr B offers to buy the same car for 15 000 pounds. (/) 4) You and your friend are betting on the weather tomorrow. If it does not rain, your friend will pay you 2 pounds, if it rains, you pay your friend 2 pounds. (no, gamble ≠ arbitrage) 2. Redundant Assets Let w1 denote a specific portfolio. This portfolio is duplicable if there exists w2  w1 such that Zw2 = Zw1. If w2 and w1 are duplicable portfolios, then any w is duplicable as w + w2 - w1 matches it. Definition: A redundant (primitive) asset is one of the assets (a column of Z) that contributes to the duplicability. Or equivalently, one or more of the primitive assets in Z is redundant if there exists a nontrivial arbitrage portfolio such that:

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Chapter 5 Z ω = 0, 1T ω = 0, ω  0. Example: Are any of the assets in the economy 0 0 1 Z=  1 2 0    redundant? Solution: Solve Z ω = 0, 1T ω = 0, the only possible solution is ω = 0. Hence none of the assets in Z is redundant.  Z Question for thought: What can be said of the matrix  T  if one of the assets in Z is 1    redundant? 3. Contingent Claims and Derivative Assets Definition: A Contingent Claim or Derivative Asset’s end of term payoff is exactly determined by the payoffs on one or more of the other assets. Common types of derivative assets are put and call options, warrants, rights offerings, futures and forward contracts, and convertible or more exotic types of bonds. Some examples of these. Definitions: 1) A (one period) forward contract = the obligation to purchase, at the end of the period, a set number of units (shares) of a particular primitive asset at a price specified at the beginning of the period. 2) A short position in a (one period) forward contract = the obligation to sell, at the end of the period, a set number of units (shares) of a particular primitive asset at a price specified at the beginning of the period. 3) A call option = the right, but not the obligation, to purchase, at the end of the period, a set number of units (shares) of a particular primitive asset at a price specified at the beginning of the period. - the price is known as striking or exercise prices. 4) A put option = the right, but not the obligation, to purchase, at the end of the period, a set number of units (shares) of a particular primitive asset at a price specified at the beginning of the period. 5) An insurable portfolio pays off only in one particular state. Hence, state s is insurable if there exists a solution to Z = es where es is the sth column of the identity matrix. 1T is the cost per dollar of the insurance policy against the occurrence of state s. Theorem: A state is insurable iff the asset returns in this state are linearly independent of the returns on the assets in the other states.

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Chapter 5 Proof: If zs. is linearly dependent of the other state return vectors, then zs. =  λσ zσ
 s

Thus for all

, zs.T =

 

λσ zσ T

s

Choose the particular

in the definition, we have

 

λσ zσ T =0 because Z

= es

s

. But on the lhs, same condition implies zs.T

On the other hand, if zs. is not linearly dependent of the other state return vectors, then Z = es has nonzero solution - by algebraic construction.

3 2     2 Example: Consider the economy Z=  1 2  . The portfolio   has return pattern  1    2 4    4   st nd rd  0  . So insurance against 1 state is possible at cost ¼. The returns in the 2 and 3 0   states are collinear, so insurance is not available for either.
4. Dominance and arbitrage One portfolio w1 is said to be dominant of w2 if Z w1  Z w2 . (*) Here the  is strict for at least one component. Since the dominated portfolio never outperforms the dominating one, investor should prefer to hold the dominating one. Remark: The existence of a dominated portfolio here demonstrates that there is no bounded optimal portfolio for any investor To show this, let w1 be a portfolio that dominates w2 and define x = Z(w1 – w2). From (*), x≥0. Now any portfolio w is dominated by w + γ(w1 – w2) for any γ>0. This is because 1T (w + γ(w1 – w2)) = 1+γ(1-1) = 1 Z (w + γ(w1 – w2)) = Zw + γx ≥ Zw hence a portfolio

and the dominance is verified. Since γ>0 is arbitrary, it can be as large as possible, hence the dominance is unbounded. If a solution to 1T η ≤ 0, Zη ≥ 0. (**) exists, we call that solution an arbitrage opportunity of the first type. Note that at least one of the components of Zη is strictly non-negative. When the first equation in (**) is an equality, η is an arbitrage portfolio creating a zero investment arbitrage opportunity.

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Chapter 5 A negative investment arbitrage or an arbitrage opportunity of the second type exists is there is a solution to 1T η < 0, Zη ≥ or = 0. (***) Remark: The principle distinction is that arbitrage of the first kind guarantees positive returns with non-positive commitments (think of borrowing and lending at different rates) whereas the second kind guarantees non-negative returns with negative commitments (a "free lunch"). Examples: 1) An economy in which there are arbitrage opportunities only of the second type is

2   5 Z=    5  2 .    For any η, we clearly have Z η = ((5 η1 + 2 η2), -(5 η1 + 2 η2)) which can not satisfy (**) unless η is identically = 0, however, η’ =(2, -5) satisfy (**): 1T η = -3 < 0, Zη = 0.
2) An economy with arbitrage opportunities only of the first type is 0 1   Z =   1  1 1 1   The pay off for any investment is then Z η = (η1 , - η1 - η2, η1 + η2), this vector is non-negative only if η1 + η2= 0, creating an arbitrage of the first type. It can be shown that there is no arbitrage for second type. Exercise: Show that if there exists a positive investment portfolio with semi-positive return, that is, Zw ≥0, then an arbitrage opportunity of the second type guarantee the existence of an arbitrage opportunity of the first type. 5. Pricing in the absence of arbitrage Investment equilibrium seems can only be achieved without the presence of arbitrage. We first look at some relevant discussions. Definition: A vector p is said to be a pricing vector supporting the economy Z if Z’p = 1. Theorem: There exists a nonnegative pricing vector p which supports the return tableau of an economy if and only if there are no arbitrage opportunities of the second type. Proof: Consider the linear programming problem Minimize 1’η subject to Zη ≥ or = 0 (*) This minimum is obviously ≤ 0 since η=0 is admissible. -5-

Chapter 5

If there are no AO2 (arbitrage opportunities of the second type), by (***), the minimum is 0. Now from the theory of duality, a finite objective minimum for a primal linear programming problem guarantees that the dual is also feasible. The dual of (*) is Maximize 0’ p Subject to Z’p = 1, p ≥ 0.

Since Minimize 1’η = Maximize 0’ p = 0, we have the desired result. Conversely, if there is nonnegative pricing support vector, then the dual problem in (*) is feasible. By necessity its objective is 0. By theorem of duality, the primal problem in (*) has a minimal objective of zero. Thus, no arbitrage opportunities exist. QED Theorem: There exists a positive pricing vector p which supports the returns tableau of an economy if and only if there are no arbitrage opportunities of 1st or 2nd types. Proof (necessity): Assume η is an arbitrage opportunity of the 1st type. Then Zη ≥ 0 and p > 0 imply P’Zη > 0 Since p is a pricing support vector, so p’Z = 1’ and we get 1’η>0 which implies that arbitrage of 1st type does not exist. (Sufficiency omitted) Example: This theorem does not prohibit the existence of pricing vectors with some negative prices. For example, for the economy in Section 1, Z =  1 3 ,  
 2 1 3 2  

The pricing support equations are p1 + 2 p2 + 3 p3 = 1 3p1 + p2 + 2 p3 = 1 with solutions p1 = 0.2(1 – p3 ), p2 = 0.2 – 1.4 p3. This problem has a positive solution p’ = (0.18, 0.26, 0.1), hence no arbitrage opportunities of 2nd type available. Nevertheless, the pricing vector (0.1, -0.3, 0.5) also supports the economy. Example: Consider the economy in Section 3, Z= The pricing support equations are 3p1 + p2 + 2 p3 = 1 2p1 + 2 p2 + 4 p3 = 1 with solutions p1 = 0.25, p2 = 0.25 - 2 p3. Thus there is no arbitrage opportunities.
3 2   , 1 2   2 4  

where state 1 is insurable.

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Chapter 5
5 Example: For the economy in Section 4, Z =    2   ., the price support equations    5  2

are 5p1 - 5p2 = 1 2p1 - 2p2 = 1 This has no solution. Hence arbitrage of 2nd type possible.

5 2   Example: For the economy Z =  3 1  . The solutions to the pricing support 2 1   equations are p1 = 2 – p3 , p2 = -3 + p3, it is obvious that p1 and p2 have opposite signs, hence arbitrage of 2nd type possible.
Example: For the pricing support equations p1 - p2 + p3 = 1 - p2 + p3 = 1. The solution must have p1 =0. This economy has an arbitrage of 1st type. Remark: The positivity of the pricing vector assures the absence of arbitrage and the linearity guarantees the absence of any monopoly power in the financial market.

6. Riskless Issues
Riskless = there exists a portfolio w1 with 1’w1 = 1, Zw1=R1. Proposition: If there is a rsikless asset or portfolio with return R, then in the absence of arbitrage, the sum of the state prices of any valid supporting pricing vector is equal to 1/R Proof: By assumption, there exists a portfolio w1 with 1’w1 = 1 Zw1=RI. Then for all valid supporting price vectors (p’Z = 1’), we have p’(Zw1) = (p’Z)w1 = p’1R = 1’w1 = 1, ∑ps = 1/R.

Proposition: If there is no way to construct a riskless portfolio, then the sum of the
_  _ 

state prices is bounded below by 1/ R and 1/ R where R and R are defined by

R = Max [ Min (Zw)]


w
_

S

R = Min [ Max (Zw)] w S
Proof: By definition of R , there exists a portfolio w1 such that Zw1 ≤ R 1. Since p’Z = 1’ for any valid pricing vector,
_ _

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Chapter 5
_ _

p’ (1 R ) ≥ p’(Zw1) = 1’w1 = 1,

∑ps ≥ 1/ R .
_

Similarly, by the definition of R , there exists a portfolio w2 such that Zw2 ≥ R I.


Again since p’Z = 1’ for any valid pricing vector, p’ (1 R ) < p’(Zw2) = 1’w2 = 1,
_

∑ps < 1/ R .

_

Example (The above bounds are tight): Consider the economy in Section 1,  1 3   Z =  2 1  , the two pricing constraint eqns are 3 2   p1 + 2p2 + 3p3 = 1, 3p1 + p2 + 2p3 = 1. Solve these equations to get p1 = 1/7+ (1/7)p2 p3 = 2/7- (5/7)p2 The values of p2 between 0 and 2/5 gives valid pricing vector. The sum of the state prices takes on all values between 3/5 and 3/7 which are the
_

reciprocals of R and R .


The single price law of markets: Two investments with the same payoff in every state must have the same current value. That is if n1 and n2 are two vectors of asset holdings and Y n1 = Y n2, then v’ n1 = v’ n2 or equivalently, Ym = 0 → v’m = 0.

Using portfolio returns, let η=Dv m and Dv is a diagonal matrix with Dii = vi, we have (Y Dv –1 )(Dv m) = Zη = 0 → (v’ Dv –1 )(Dv m) = 1’η = 0 (*) Any violation of (*) implies the existence of an arbitrage opportunity of the second type.

7. Applications
Example 1: Consider a security market consisting of two securities, A and B. At time t=0, they have prices of PA(0) and PB(0). At t=1, we look at two possibilities: Security Time 0 price P(0) £ 5 9 Market up at t=1 P(1) £ 6 12 Market down at t=1 P(1) £ 4 8

A B

There is an arbitrage opportunity here.

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Chapter 5 I can buy 1 unit of B and sell 2 units of A at time t=0, this way, I have 1 pounds at t=0. At time t=1, I have to do the reverse, sell 1 unit of B and buy 2 units of A,  When market is up, this brings me 0 pounds.  When market is down, this brings me also 0 pounds.  Overall, I made 1 pound without any risk of lose. However, if we want to do the same thing but with Security Time 0 price P(0) £ 5 10 Market up at t=1 P(1) £ 6 12 Market down at t=1 P(1) £ 4 8

A B

Then we get no profit. The opportunity for arbitrage has been eliminated.- the prices are now consistent. Example 2: Security Time 0 price P(0) £ 7 7 Market up at t=1 P(1) £ 14 14 Market down at t=1 P(1) £ 3 2

A B

In this case, an arbitrage opportunity exists. Buy 1 unit of A and sell 1 unit of B will achieve no risk profit.. As the tendency of buying A and selling B increases, arbitrage opportunities will be eliminated. In this case, obviously, PA > PB. Example 3: Forward contracts At time t=0, say, an agreement is made between two parties A and B that A will buy from B a specified amount of an asset S and a specified price on a specified future date. The person selling is said to hold a short forward position. The person buying is said to hold a long forward position. Let S(r) be the price of the underlying asset price at time r – this is usually unknown a priori. K be the price agreed at t=0 for asset S. It is usually assumed that there is a `risk-free` force of interest δ over the term of the contract.

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Chapter 5 Ignore other factors such as inflation and interest rate, we use the following example to explain the concept of forward contract: Investor I agrees to sell 1000 BP shares in 6 months’ time to Investor II at a price of £ 7 per share. The current price is £ 6.50 per share. £ 7 is the forward price. 6 months later, the share price is £6.90. I has to sell 1000 shares at £7 a piece. He sold it for £7000, but the shares really worth 6900. Hence I made a profit of £100. II , by contrast, made a lose of £100. Example 4: Forward price of a security without income or cost. Consider the following two portfolios: A: At t=0, enter a forward contract to buy one unit of an asset S, with forward price K maturing at time T. Simultaneously invest an amount Ke-δT in the risk-free investment – the price of the portfolio is Ke-δT. B: At t=0, buy one unit of the asset, at the current price S(0) – the price of the portfolio is S(0). At time t = T: The price of the portfolio A is: receiving K from the risk-free investment, pay K to buy one unit of S – the payout of the portfolio is one unit of S share. The price of the portfolio B is one unit of S share. If we follow the principle of no arbitrage, the two portfolio has the same payout, hence the initial price is the same: Ke-δT = S(0) this implies K = S(0)eδT.

This gives us a guide on the price of the value of K. Exercise: A three-year forward contract exists in a zero-coupon corporate bond with a current price per £100 nominal of £50. The yield available on three-year government securities is 6% pa effective. Calculate the forward price. In the above problem, what if the three-year forward contract is replaced by six-month forward contract ? (note: a zero-coupon bond is a security that people purchase at time t=0 for a specified lump sum at some specified future date.)

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Chapter 5 Example 5: Forward price of a security with fixed cash income. Consider the following problem: at a time t1 where 0< t1 < T, the security underlying the forward contract provides a fixed amount c to the holder. Portfolio A: Enter a forward contract to buy one unit of S, with forward price K, maturing at T, simultaneously invest an amount Ke-δT + ce-δ(t1) in the risk-free investment ---- the price at t=0 is hence Ke-δT + ce-δ(t1) ---- the payout at t=T is hence K + ce-δ(T - t1) – K + one unit of asset with price S(T) = ce-δ(T - t1) + one unit of asset with price S(T).

Portfolio B: Buy one unit of the asset, at the current price S(0). At time t1, invest c in the risk-free investment ---- the price at t=0 is hence S(0) ---- the payout at t=T is hence ce-δ(T - t1) + one unit of asset with price S(T). Since yield from A and B are the same, no arbitrage assumption gives Ke-δT + ce-δ(t1) = S(0) → K = S(0)eδT - ce-δ (T - t1) Example 6: The forward price for a security with known dividend yield The dividend yield for an equity is defined to be: Dividend yield = Dividend per share / Price per share Let D be the known dividend yield per annum. Assume that dividends are received continuously, and are immediately reinvested in the security of S. (Note: The actual amount of dividend is varying with the price). Start with one unit of S at time t = 0, the accumulated holding at time T would be eDT units of the security. Now consider the following two portfolios: Portfolio A: Enter a forward contract to buy one unit of S, with forward price K, maturing at time T; simultaneously invest an amount Ke-δT in the risk-free investment - At t = 0, the price of Portfolio A is Ke-δT. Portfolio B: Buy e-DT units of S, at the current price S(0). Reinvest dividend income in S immediately after it is received – At t=0, the price of Portfolio B is S(0)e-DT. At t = T, the net portfolio A is: 1 unit of S. At t = T, the net portfolio B is: 1 unit of S. Using the no arbitrage assumption the prices must also be the same - that is: Ke-δT = S(0)e-DT => K = S(0) e(δ-D)T

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Chapter 5 The following is a concrete numerical case: Assume that the dividend yield of a portfolio of shares with current price of £1,000 is 4% pa. Calculate the forward price of a one-year forward contract, based on the portfolio. If we assume dividends are received continuously and the risk-free rate of interest is 4.6028% pa effective. 4.6028% effective is equivalent to a force of interest of 4.5%. Therefore K = 1000 e0.045-0.04 = £ 1005.012521 Exercise: The current share price of a stock is £100. Dividends are paid continuously and the current dividend is £2 pa. Calculate the forward price of a five-year contract on the asset if the risk-free force of interest is 5% pa and the dividend yield remains constant. Exercise: Deduce a formula for the forward price, K, for an equity forward contract in T years’ time (T is an integer). Assume a constant dividend yield D, and that dividends are received in the middle of each year and are immediately reinvested. Exercise: In the above example, is it possible to calculate K if the dividend is fixed amount of cash provided 1) The dividend is reinvested in shares of S. 2) The dividend is reinvested in risk-free investment. Hedging is a general term which describes the use of financial instruments (including stocks, bonds, forward contracts and more complex financial contracts such as options) to reduce or eliminate a future risk of loss. Example 7: An investor agrees to sell an asset with no income in 2 years' time. The current price of the asset is 100 and the risk-free force of interest is 10%. We assume that the investor borrows and invests money at the risk-free rate. The forward price must be 100 e2x0.1 = 122.1402758. This is the amount that the investor will receive at time 2 for selling the asset. In order the sell the asset, the investor first needs to buy the asset. Suppose the investor chooses to buy the asset now and so the accumulated profit at time 2 is 0. If the price of the asset at time 2 turns out to be 110, then the investor would have made a profit if she had waited and purchased the asset at time 2. The accumulated profit would have been 122.1402758 -110. If the price of the asset at time 2 turns out to be 130, then the investor would have made a lose if she had waited and purchased the asset at time 2. The accumulated lose would have been 130 - 122.1402758. This is called a "static hedge" since the hedge portfolio, which consists of the asset to be sold plus the borrowed risk-free investment, does not change over the term of the

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Chapter 5 contract. For more complex financial instruments, such as options, the hedge portfolio is more complex, and requires (in principle) continuous rebalancing to maintain. This is called a "dynamic hedge". Exercise: Assume that there is no interest or dividend income Consider a forward contract agreed at time t = 0, with a forward price K, for buying one unit of S. The maturity date of the contract is time T. Find the value of the contract at an arbitrary intermediate time t. Construct a numerical example to illustrate the point. Hint: These values can be found using the "no arbitrage" assumption and similar techniques to those used to find the forward price.

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