the green line is a question.

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Consider a standard binomial model of a risky asset where there are two states of nature, the UP state and the DOWN state. The former is denoted by U and the latter, by D. The state space is 2 = {U, D}. A generic element of this space is denoted by w when a statement or formula holds in the up and down states separately. This risky asset's price at time zero is So. Its payoff at time one is unknown at time zero. However, it is known to pay Si (U) when U occurs, and pay Si (D) whenever D occurs at time 1. The important point is that for risky assets, we necessarily have S. (U) # Si (D). In many applications where the asset in question is a limited liability stock, or equity, we have nonnegative payoffs in each state and in many cases model S (U) = (1+u) So in state U and pay S.(D) = (1 + d) So, where So > 0. In these situations we typically make the following assumption: Assumption: ud and u >0. We allow for the possibility that d 0. Additional restrictions are imposed indirectly by the absence of arbitrage when there is also a safe asset, or bond available (see Theorem 1 below). There are other kinds of risky assets including derivative securities written on an underlying stock. Derivatives include put and call options as well as forward contracts. We will cover these examples in a class meeting. There is also a single safe asset, or bond. Its price at time zero is Bo > 0. The rate of interest is r > 0. At time one this bond pays B1 = (1+r) Bo whether state U or state D occurs it pays the same in either state. A binomial asset model (or, simply binomial model) has one or more risky assets and typically a single safe asset. However, these ingredients can be mixed and matched in order to focus on various asset pricing problems, e.g. derivatives pricing. We suppose for now that there is just one risky asset and a safe asset. A portfolio is a pair of numbers (x, y) corresponding to the quantity of the risky and safe assets held by the investor. That portfolio costs at time zero are: To (, y) = So + yBo. The signs of the x and y are not restricted (that is, long and short positions are permissible). The portfolio (r, y) pays 71 (W) = Si (w) + y B1, where w = U, D respectively. The portfolio (x, y) is an arbitrage portfolio provided So + y Bo 0 (zero net outlay); Si (U) + y (1 + r) Bo > 0; xSi (D) + y (1 + r) Bo > 0, and at least one of the payoffs at time one offers a strictly positive payoff. An arbitrage porfolio is self- financing since the investor does not have to employ his or her own money to finance the purchase and sale of the underlying safe and risky assets. "Other people's money" (OPM) is all that's needed to form an arbitrage portfolio." The main behavioral hypothesis is that more money is always preferred to less even a bit more cash in one state compared to the status quo, and no less cash in any other state, represents a more preferred alternative. ALL arbitrage theories make a nonsatiation assumption of this form! The main economic priniciple for a financial market equilibrium is the NO ARBITRAGE Principle: there are no arbitrage portfolios. Abbreviate this principle as (NAP). Theorem 1 (NAP) implies u>r>d. OPM gets its name from a movie (and play) of the same title about a corporate raider (Danny DeVito in the film version). Proof. Begin by noticing that if r>u>d, then the safe asset's payoff per unit invested always exceeds the payoff of the risky asset. This implies that the safe asset dominates the risky asset no investor who prefers more cash (in each state) to less will EVER choose to invest in this risky asset. Likewise, if u >d>r>0, the risky asset (per unit) dominates the safe asset. Hence, we can confine attention to showing u >r > d must hold when the (NAP) obtains. Suppose the condition (NAP) is true, but the conclusion of the theorem is false. This means (NAP) holds, d> u and r> 0 hold, but dr. In this case, there will be an arbitrage portfolio. This contradiction implies the theorem is true. Thus, the main argument is to construct an arbitrage portfolio assuming u > dr. The first step is to find the possible zero net outlay portfolios. These are combinations of u and y such that So+yBo = 0. This restriction implies there are two possible cases: Case I: x > 0 and y = -rSo/Bo. This is a long position in the stock and a short position in the bond (borrowing money). Case II: y > 0 and r = -yBo/S. This is a long position in the bond (lending money) and a short position in the stock. The proof for Case I is shown below. The proof for Case II is a homework problem. Let x > 0 and consider the zero net outlay portfolio with y=-So/Bo. This portfolio's payoff in the U state is: 71 (U) Si (U)+(1+r) Bo = 1(1+u) So+y (1+r) Bo S. S. (1+u) - (1+r) B. = xS. (u-r) (after simplifing) > S. ( dr) (by u > d) > 0 (as dr). Therefore, this portfolio delivers a positive payoff in state U. A similar string of inequalities shows the portfolio pays a nonnegative amount in the Down state. To see this: 1 (D) Si (D) +y (1 + r) Bo = (1 + d) So + y(1+r) Bo xS = So (1+d) - (1 + r) Bo Bo = S, ( dr) (after simplifing) > 0 (as dr). This last inequality shows that the payoff in the Down state is nonnegative. Therefore, the portfolio (x, y) so constructed is an arbitrage portfolio. This is impossible if the condition (NAP) holds! The proof is completed by verifying there is an arbitrage portfolio in Case II. You are to work this out as a homework assignment. Hence, the condition (NAP) implies u>>d. I The next result is known as the Law of One Price. Basically it says that assets with the same payoff in every state must have the same asset price at time zero, or there is an arbitrage opportunity. This result underlies many applications in derivatives pricing. It is a valid theorem for any finite state space. However, only the binomial version is treated here. The binomial model underlying our statement and proof the of the Law of One Price takes as given two risky assets and one safe asset. The two risky assets will have the same payoffs in each state of nature. The safe asset will correspond to the payoff from lending (or, borrowing) money realized at time one. Both risky assets are defined on {12, 22, P}. Asset A has initial price SA and it pays SA (U) in the UP state, and it pays SA (D) in the down state with probability p and 1 p, respectively. Assume SA (U) > S4 (D). Asset B is similar. It has an initial price S and it pays $ (U) in the UP state, and it pays SP (D) in the DOWN state with probability p and 1 P, respectively. Assume that S (U) > SP (D). The third asset is safe it pays the same in each state of nature. It is basically a bank deposit account paying an interest rate r > 0. In the formal theorem given below this interest rate is assumed to be zero. That is, the "zero lower bound" has been achieved (a somewhat fanciful representation of today's short term money market). However, the formal theorem and proof do NOT depend on this. That is, the theorem and its proof can be easily be reworked to admit a positive rate of interest. Theorem 2 (LAW of ONE PRICE)' Assume the market for these three assets satisfies the No Arbitrage Principle. IF SA (W) = S(w) for each wes, then SA = S. The logic of this proof is fairly simple when viewed from an arbitrageur's perspective. If the asset's prices are not equal, then there is a self-financing trade that makes a sure profit whichever state of nature occurs. This is contrary to the (NAP) condition. The self-financing trade comes down to shorting asset the asset with the higher price and simultaneously buying the cheaper asset (buy low, sell high). This can be done and even leave some cash in your hands that can be lent to the bank and earn the risk free rate of interest! The result is that when you realize the payoffs from your position once Nature reveals the true state, you will have made some money after covering any debts incurred in the first place! You have an arbitrage opportunity. This is the source of the contradiction that proves the theorem as you assume NO ARBITRAGE is available in the market in the first place! The first theorem implies that the return on the safe asset is intermediate between the UP and DOWN rates of return for the risky assets. More on the notion of rate-of-return in each state in the lead-up to Theorem 3. The formal proof works out all the details. The notion of self-financing means that you start with $0 cash and find a way to go long on one asset and short the other using none of your funds whatsoever! This is a recurring idea in arbitrage pricing theories. Proof. (Law of One Price). Suppose, without loss of generality, that SA > Sp. The opposite inequality argument will be similar. The basic approach is to show by construction of an arbitrage portfolio that the maintained (NAP) is violated, a contradiction. Put differently, the argument is to find an arbitrage opportunity starting from a $0 initial cash position and obtain a contradiction to (NAP). Borrow $S4 and simultaneously BUY one unit of asset B to pay $56. Henceforth, drop the $ signs as these values are understood to be measured in some monetary unit! The resulting cash position is To = -$++($A - SB) = 0. The first term, -S4 represents the liability you have undertaken by borrowing security A's price indeed, the security since you take possession of it - it is a short sale). Your liability is measured by a negative number. Your purchase of asset B with the borrowed funds is picked up in the second term as S, which is now your asset. Record the purchase as money coming to you, so it is a positive number. The net cost of this transaction is the amount (S4 - S) > 0 (by assumption) it is the net cash your action yields. NOTICE how all this cancels out to produce a zero net outlay! NEXT wait until Nature reveals her choice of a state, w = U or w = D. Whichever state actually occurs sell asset B back to the market at its then ruling price, SP w). This is the state-contingent revenue from this sale. Simultaneously cover the short or borrowed position by repurchasing asset A at the then ruling price, S1 (w). This registers as a negative number corresponding to a cash outflow to the market. BY ASSUMPTION: SA (w) = S(w) for each w 2. This implies that the net cost of reversing the position undertaken prior to Nature's state revelation is a ZERO cost transaction once Nature tells us the true state, w. So, the payoff, conditioned on the state of Nature, is 71 (w) -S1 (w) + S(w)+(SM-S) (SA - $) >0 as SA (w) = S(w). These last two lines use the zero interest rate clearly, a positive interest rate does NOT change the conclusion! Our arbitrageur has achieved a strictly positive payoff (for sure) in each of the possible states 2 The probalities only enter here to be emphasize that assets A and B are random variables. 3 You should do this just to make sure you understand all the proof's steps. This may be the earliest formal notion of a no arbitrage condition. It appears as early as the work of Huygens on probability theory in 1657, although he did not use the term Law of One Price to identify this. See Rubinstein (4] for details. of Nature! This means that if S > S, then there is an arbitrage opportunity, a contradiction to the maintained assumption (NAP). I Notice that the actual probabilities in the given probability space captured in the measure P play ab- solutely NO ROLE in either Theorem 1 or Theorem 2. This is not an accident. It is a recurring theme in derivatives pricing theories. Indeed, the implementation of derivatives pricing models works, in part, because the No Arbitrage pricing technique only draws on the observed market prices before Nature chooses the true state, and the payoff structures in each state. However, it turns out that the absence of an arbitrage opportunity contains information about a market probability measure that makes the asset prices and their payoffs look like fair gambles. These market probabilities are derived next and are sometimes known in the literature as the Martingale probability measure. Fair gambles, with these probabilities, will turn out to imply that the expected rate of return under the Martingale probabilities equals the safe asset's rate of return the rate of interest). A risk neutral arbitrageur is just indifferent between holding this risky asset and the safe asset in the absence of an arbitrage opportunity. For this reason, the market probability measue, or Martingale probability measure, is also known in economic jargon as a risk neutral probability measure. Some writers in the financial asset pricing literature refer to the switch from the measure P to the risk neutral probability measure as the Martingale change of measure. The idea of a rate-of-return is always interpreted as the ratio of the profit to the cost in any transaction. Consider a binomial asset model where the risky asset has a positive initial price So and state contingent payoffs Sw). We can define a new random variable on our probability space, the rate-of-return by the formula: S. (U)-So if w=U; S. (D-S if w=D So For purposes of our presentation assume that S (D) = (1 + u) So in state U and S(D) = (1+d) So in state D. Then u if w=U; r's (w)= d if w=D A risky asset might not have a well-defined rate-of-return this is the case with a forward contract in which the initial price is zero! We will discuss this case in class and see there is a workaround for this asset type! Theorem 1 tells us that for (NAP) that u >r>d must hold. Define a probability measure P* on {12,29} by letting p* denote the probability of the UP state and (1 - p*) the probability of the DOWN state. These probabilities are formally defined using the expressions: r's (w) = { ={ red p* u-d u-r 1-p* u-d By Theorem 1, we see that 0

0. Summarize this as the following result. Theorem 3 Existence of a Martingale (Risk Neutral) Probability Measure) The binomial model erhibiting the No Arbitrage condition implies there is a risk neutral probability measure, P*, such that E* [rs] = r. Moreover, this risk neutral measure can be used to value ANY risky (or safe) asset. COMMENT. The converse of Theorem 3 also holds and forms a major result in the general theory it is part of the Representation Theorem). The investor's knowledge of the state-contingent claims IS a strong assumption and it is critical for this theoretical model's results. Several examples of risk neutral pricing (using Theorem 3) as well as the Law of One Price will be given in class and homeworks. There are many dedicated texts on binomial models of asset pricing in the math finance literature as well as volumes that present substantial material on these models. Here is a small sample. Consider a standard binomial model of a risky asset where there are two states of nature, the UP state and the DOWN state. The former is denoted by U and the latter, by D. The state space is 2 = {U, D}. A generic element of this space is denoted by w when a statement or formula holds in the up and down states separately. This risky asset's price at time zero is So. Its payoff at time one is unknown at time zero. However, it is known to pay Si (U) when U occurs, and pay Si (D) whenever D occurs at time 1. The important point is that for risky assets, we necessarily have S. (U) # Si (D). In many applications where the asset in question is a limited liability stock, or equity, we have nonnegative payoffs in each state and in many cases model S (U) = (1+u) So in state U and pay S.(D) = (1 + d) So, where So > 0. In these situations we typically make the following assumption: Assumption: ud and u >0. We allow for the possibility that d 0. Additional restrictions are imposed indirectly by the absence of arbitrage when there is also a safe asset, or bond available (see Theorem 1 below). There are other kinds of risky assets including derivative securities written on an underlying stock. Derivatives include put and call options as well as forward contracts. We will cover these examples in a class meeting. There is also a single safe asset, or bond. Its price at time zero is Bo > 0. The rate of interest is r > 0. At time one this bond pays B1 = (1+r) Bo whether state U or state D occurs it pays the same in either state. A binomial asset model (or, simply binomial model) has one or more risky assets and typically a single safe asset. However, these ingredients can be mixed and matched in order to focus on various asset pricing problems, e.g. derivatives pricing. We suppose for now that there is just one risky asset and a safe asset. A portfolio is a pair of numbers (x, y) corresponding to the quantity of the risky and safe assets held by the investor. That portfolio costs at time zero are: To (, y) = So + yBo. The signs of the x and y are not restricted (that is, long and short positions are permissible). The portfolio (r, y) pays 71 (W) = Si (w) + y B1, where w = U, D respectively. The portfolio (x, y) is an arbitrage portfolio provided So + y Bo 0 (zero net outlay); Si (U) + y (1 + r) Bo > 0; xSi (D) + y (1 + r) Bo > 0, and at least one of the payoffs at time one offers a strictly positive payoff. An arbitrage porfolio is self- financing since the investor does not have to employ his or her own money to finance the purchase and sale of the underlying safe and risky assets. "Other people's money" (OPM) is all that's needed to form an arbitrage portfolio." The main behavioral hypothesis is that more money is always preferred to less even a bit more cash in one state compared to the status quo, and no less cash in any other state, represents a more preferred alternative. ALL arbitrage theories make a nonsatiation assumption of this form! The main economic priniciple for a financial market equilibrium is the NO ARBITRAGE Principle: there are no arbitrage portfolios. Abbreviate this principle as (NAP). Theorem 1 (NAP) implies u>r>d. OPM gets its name from a movie (and play) of the same title about a corporate raider (Danny DeVito in the film version). Proof. Begin by noticing that if r>u>d, then the safe asset's payoff per unit invested always exceeds the payoff of the risky asset. This implies that the safe asset dominates the risky asset no investor who prefers more cash (in each state) to less will EVER choose to invest in this risky asset. Likewise, if u >d>r>0, the risky asset (per unit) dominates the safe asset. Hence, we can confine attention to showing u >r > d must hold when the (NAP) obtains. Suppose the condition (NAP) is true, but the conclusion of the theorem is false. This means (NAP) holds, d> u and r> 0 hold, but dr. In this case, there will be an arbitrage portfolio. This contradiction implies the theorem is true. Thus, the main argument is to construct an arbitrage portfolio assuming u > dr. The first step is to find the possible zero net outlay portfolios. These are combinations of u and y such that So+yBo = 0. This restriction implies there are two possible cases: Case I: x > 0 and y = -rSo/Bo. This is a long position in the stock and a short position in the bond (borrowing money). Case II: y > 0 and r = -yBo/S. This is a long position in the bond (lending money) and a short position in the stock. The proof for Case I is shown below. The proof for Case II is a homework problem. Let x > 0 and consider the zero net outlay portfolio with y=-So/Bo. This portfolio's payoff in the U state is: 71 (U) Si (U)+(1+r) Bo = 1(1+u) So+y (1+r) Bo S. S. (1+u) - (1+r) B. = xS. (u-r) (after simplifing) > S. ( dr) (by u > d) > 0 (as dr). Therefore, this portfolio delivers a positive payoff in state U. A similar string of inequalities shows the portfolio pays a nonnegative amount in the Down state. To see this: 1 (D) Si (D) +y (1 + r) Bo = (1 + d) So + y(1+r) Bo xS = So (1+d) - (1 + r) Bo Bo = S, ( dr) (after simplifing) > 0 (as dr). This last inequality shows that the payoff in the Down state is nonnegative. Therefore, the portfolio (x, y) so constructed is an arbitrage portfolio. This is impossible if the condition (NAP) holds! The proof is completed by verifying there is an arbitrage portfolio in Case II. You are to work this out as a homework assignment. Hence, the condition (NAP) implies u>>d. I The next result is known as the Law of One Price. Basically it says that assets with the same payoff in every state must have the same asset price at time zero, or there is an arbitrage opportunity. This result underlies many applications in derivatives pricing. It is a valid theorem for any finite state space. However, only the binomial version is treated here. The binomial model underlying our statement and proof the of the Law of One Price takes as given two risky assets and one safe asset. The two risky assets will have the same payoffs in each state of nature. The safe asset will correspond to the payoff from lending (or, borrowing) money realized at time one. Both risky assets are defined on {12, 22, P}. Asset A has initial price SA and it pays SA (U) in the UP state, and it pays SA (D) in the down state with probability p and 1 p, respectively. Assume SA (U) > S4 (D). Asset B is similar. It has an initial price S and it pays $ (U) in the UP state, and it pays SP (D) in the DOWN state with probability p and 1 P, respectively. Assume that S (U) > SP (D). The third asset is safe it pays the same in each state of nature. It is basically a bank deposit account paying an interest rate r > 0. In the formal theorem given below this interest rate is assumed to be zero. That is, the "zero lower bound" has been achieved (a somewhat fanciful representation of today's short term money market). However, the formal theorem and proof do NOT depend on this. That is, the theorem and its proof can be easily be reworked to admit a positive rate of interest. Theorem 2 (LAW of ONE PRICE)' Assume the market for these three assets satisfies the No Arbitrage Principle. IF SA (W) = S(w) for each wes, then SA = S. The logic of this proof is fairly simple when viewed from an arbitrageur's perspective. If the asset's prices are not equal, then there is a self-financing trade that makes a sure profit whichever state of nature occurs. This is contrary to the (NAP) condition. The self-financing trade comes down to shorting asset the asset with the higher price and simultaneously buying the cheaper asset (buy low, sell high). This can be done and even leave some cash in your hands that can be lent to the bank and earn the risk free rate of interest! The result is that when you realize the payoffs from your position once Nature reveals the true state, you will have made some money after covering any debts incurred in the first place! You have an arbitrage opportunity. This is the source of the contradiction that proves the theorem as you assume NO ARBITRAGE is available in the market in the first place! The first theorem implies that the return on the safe asset is intermediate between the UP and DOWN rates of return for the risky assets. More on the notion of rate-of-return in each state in the lead-up to Theorem 3. The formal proof works out all the details. The notion of self-financing means that you start with $0 cash and find a way to go long on one asset and short the other using none of your funds whatsoever! This is a recurring idea in arbitrage pricing theories. Proof. (Law of One Price). Suppose, without loss of generality, that SA > Sp. The opposite inequality argument will be similar. The basic approach is to show by construction of an arbitrage portfolio that the maintained (NAP) is violated, a contradiction. Put differently, the argument is to find an arbitrage opportunity starting from a $0 initial cash position and obtain a contradiction to (NAP). Borrow $S4 and simultaneously BUY one unit of asset B to pay $56. Henceforth, drop the $ signs as these values are understood to be measured in some monetary unit! The resulting cash position is To = -$++($A - SB) = 0. The first term, -S4 represents the liability you have undertaken by borrowing security A's price indeed, the security since you take possession of it - it is a short sale). Your liability is measured by a negative number. Your purchase of asset B with the borrowed funds is picked up in the second term as S, which is now your asset. Record the purchase as money coming to you, so it is a positive number. The net cost of this transaction is the amount (S4 - S) > 0 (by assumption) it is the net cash your action yields. NOTICE how all this cancels out to produce a zero net outlay! NEXT wait until Nature reveals her choice of a state, w = U or w = D. Whichever state actually occurs sell asset B back to the market at its then ruling price, SP w). This is the state-contingent revenue from this sale. Simultaneously cover the short or borrowed position by repurchasing asset A at the then ruling price, S1 (w). This registers as a negative number corresponding to a cash outflow to the market. BY ASSUMPTION: SA (w) = S(w) for each w 2. This implies that the net cost of reversing the position undertaken prior to Nature's state revelation is a ZERO cost transaction once Nature tells us the true state, w. So, the payoff, conditioned on the state of Nature, is 71 (w) -S1 (w) + S(w)+(SM-S) (SA - $) >0 as SA (w) = S(w). These last two lines use the zero interest rate clearly, a positive interest rate does NOT change the conclusion! Our arbitrageur has achieved a strictly positive payoff (for sure) in each of the possible states 2 The probalities only enter here to be emphasize that assets A and B are random variables. 3 You should do this just to make sure you understand all the proof's steps. This may be the earliest formal notion of a no arbitrage condition. It appears as early as the work of Huygens on probability theory in 1657, although he did not use the term Law of One Price to identify this. See Rubinstein (4] for details. of Nature! This means that if S > S, then there is an arbitrage opportunity, a contradiction to the maintained assumption (NAP). I Notice that the actual probabilities in the given probability space captured in the measure P play ab- solutely NO ROLE in either Theorem 1 or Theorem 2. This is not an accident. It is a recurring theme in derivatives pricing theories. Indeed, the implementation of derivatives pricing models works, in part, because the No Arbitrage pricing technique only draws on the observed market prices before Nature chooses the true state, and the payoff structures in each state. However, it turns out that the absence of an arbitrage opportunity contains information about a market probability measure that makes the asset prices and their payoffs look like fair gambles. These market probabilities are derived next and are sometimes known in the literature as the Martingale probability measure. Fair gambles, with these probabilities, will turn out to imply that the expected rate of return under the Martingale probabilities equals the safe asset's rate of return the rate of interest). A risk neutral arbitrageur is just indifferent between holding this risky asset and the safe asset in the absence of an arbitrage opportunity. For this reason, the market probability measue, or Martingale probability measure, is also known in economic jargon as a risk neutral probability measure. Some writers in the financial asset pricing literature refer to the switch from the measure P to the risk neutral probability measure as the Martingale change of measure. The idea of a rate-of-return is always interpreted as the ratio of the profit to the cost in any transaction. Consider a binomial asset model where the risky asset has a positive initial price So and state contingent payoffs Sw). We can define a new random variable on our probability space, the rate-of-return by the formula: S. (U)-So if w=U; S. (D-S if w=D So For purposes of our presentation assume that S (D) = (1 + u) So in state U and S(D) = (1+d) So in state D. Then u if w=U; r's (w)= d if w=D A risky asset might not have a well-defined rate-of-return this is the case with a forward contract in which the initial price is zero! We will discuss this case in class and see there is a workaround for this asset type! Theorem 1 tells us that for (NAP) that u >r>d must hold. Define a probability measure P* on {12,29} by letting p* denote the probability of the UP state and (1 - p*) the probability of the DOWN state. These probabilities are formally defined using the expressions: r's (w) = { ={ red p* u-d u-r 1-p* u-d By Theorem 1, we see that 0

0. Summarize this as the following result. Theorem 3 Existence of a Martingale (Risk Neutral) Probability Measure) The binomial model erhibiting the No Arbitrage condition implies there is a risk neutral probability measure, P*, such that E* [rs] = r. Moreover, this risk neutral measure can be used to value ANY risky (or safe) asset. COMMENT. The converse of Theorem 3 also holds and forms a major result in the general theory it is part of the Representation Theorem). The investor's knowledge of the state-contingent claims IS a strong assumption and it is critical for this theoretical model's results. Several examples of risk neutral pricing (using Theorem 3) as well as the Law of One Price will be given in class and homeworks. There are many dedicated texts on binomial models of asset pricing in the math finance literature as well as volumes that present substantial material on these models. Here is a small sample