Advanced Financial Models. Michael R. Tehranchi

Transcription

1 Advanced Financial Models Michael R. Tehranchi

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3 Contents 1. Standing assumptions: complications we ignore 5 2. Prerequisite knowledge 8 Chapter 1. One-period models 9 1. One-period models 9 2. Arbitrage, martingale deflators and 1FTP 1 3. Equivalent martingale measures Contingent claim pricing Markets with an infinite number of assets Call prices from moment generating functions Forward contracts and forward prices 29 Chapter 2. Discrete-time models Investment and consumption A motivating utility maximisation problem and arbitrage Arbitrage and the first fundamental theorem Elements of the proof of the harder direction of 1FTAP Numéraires and equivalent martingale measures Contingent claims Super-replication of American claims 55 Chapter 3. Brownian motion and stochastic calculus Brownian motion 6 2. Itô stochastic integration 6 3. Itô s processes and quadratic variation Itô s formula Girsanov s theorem A martingale representation theorem 69 Chapter 4. Continuous-time models The set-up Admissible strategies Arbitrage and local martingale deflators The structure of local martingale deflators Replication and super-replication The Black Scholes model and formula Markovian markets and the Black Scholes PDE PDEs and local martingales Risk-neutral pricing and PDEs 86 3

4 1. Examples Local volatility models Black Scholes volatility 94 Chapter 5. Crashcourse on probability theory Measures Random variables Expectations and variances Special distributions 1 5. Conditional probability and expectation, independence Probability inequalities Characteristic functions Fundamental probability results 12 4

5 Financial mathematics as a subject is young (as compared to, say, number theory), but it is mature enough now that there has emerged some consensus on the notation, vocabulary and important results. These notes are an attempt to present many of the main ingredients of this theory, mainly concerning the pricing and hedging of derivative securities. But before launching into the story, we will begin by acknowledging some of the real-world complications that will not be discussed at length hereafter. 1. Standing assumptions: complications we ignore Unfortunately, actual financial markets are very complicated. Of course, in order to develop a systematic financial theory, it is prudent to concentrate on the essential features of these markets and ignore the less essential complications. Therefore, the theory that will be presented in these notes is concerned with the analysis of market models that have plenty of simplifying assumptions. That is not to say that these complications are not important. Indeed, there is active ongoing research attempting to remove these simplifying assumptions from the canonical theory. Below is a list of these assumptions Dividends. The total stock of a publicly traded firm is divided into a fixed number N of shares. The owner of each share is then entitled to the fraction 1/N of the total profit of the firm. 1 A portion of the firm s profit is usually reinvested by management, for instance by building new factories, but the rest of the profit is paid out to the shareholders. In particular, the owner of each share of stock will receive periodically a dividend payment. However, in this course, we will assume that there are no dividend payments. Actually, this assumption is not as terrible as it sounds. Example sheet 1 will show how to adapt the theory developed for assets that pay no dividends to incorporate assets that have non-zero dividend payments Tick size. Financial markets usually have a smallest increment of price, the tick. (The tick refers back to the days when prices were quoted on ticker tape.) Indeed, the tick size can vary from market to market, and even for assets traded in the same market. However, in this course, we will assume that the tick size is zero. This is a convenient assumption for those who prefer continuous mathematics to discrete. It is usually a harmless assumption, unless the prices of interest are very close to zero. 1 Actually, things are even more complicated. For instance, stocks can be classified as either common or preferred, with implications on dividends, voting rights and claims on the firm s assets in case of bankruptcy. Also, the number N of shares outstanding is not necessarily fixed. 5

6 1.3. Transactions costs. Financial transactions are processed by a string of middle men, each of whom charge a fee for their services. Usually the fee is nearly proportional to the size of the transaction. However, in this course, we will assume that there are no transactions costs. This assumption is justified by by the fact that transactions costs are often very small relative to the size of typical transactions. But one must always remember that in some applications, it might not be wise to neglect these costs Short-selling constraints. In the real world, it is actually possible for someone to sell an asset that he does not own. The essential mechanism is to borrow a share of that asset from a broker, and then immediately to sell it to the market. This procedure is called short selling. Brokers, however, place contraints on this behaviour. Indeed, they usually require collateral and charge a fee for their service. Furthemore, if the market price of the asset increases, or if the price of the collateral decreases, the broker may ask the short seller to put up even more collateral. However, in this course, we will assume that there are no short-selling constraints. Indeed, the theory of discrete-time trading is cleaner without additional assumptions on the sizes of trades. But we will see that to overcome some technical problems in the theory of continuous-time trading, it will be natural to restrict trading to what are called admissible strategies Divisibility of assets. There is another real-world trading constraint of a rather technical nature. The smallest unit of stock is the share. A share cannot be further divided it is generally impossible to buy half a share of a particular stock. However, in this course, we will assume that assets are infinitely divisible Bid-ask spread. Real-world trading is asymmetrical since the price to buy a share is usually higher than the price to sell it. The reason is that are two different ways to buy or sell an asset listed on an exchange: the limit order and the market order. A limit buy order is an offer to buy a certain number of shares of the asset at a certain price. A limit sell order is defined similarly. The collection of unfilled limit orders is called the limit order book. At any time, there is the highest price for which there is an order to buy the asset. This is called the bid price. The lowest price for which there is an order to sell is called the ask price. The bid/ask spread is the difference. Figure 1 illustrates the evolution of a hypothetical limit order book as various orders arrive and are filled. A market order are instructions to execute a transaction at the best available price. In particular, if the market order is to buy, then the lowest limit sell order is filled first. Therefore, for small market buy orders, the per share price paid is the ask price. Similarly, 6

7 if a market sell order arrives, then the highest limit buy order is filled first, and hence the per share price received is the bid price. However, in this course, we will assume that there are no bid-ask spreads. This assumption is justified by the observation that in many markets, the spread is very small. However, in times of crisis, this assumption is not usually applicable, and hence the theory breaks down dramatically. Figure 1. Top left. The bid price is 8 and the ask is 11. Top right. A limit sell order for three shares at 11 arrives. Bottom left. A limit buy order for two shares at 8 is cancelled. Bottom right. A market order to buy five shares arrives. Note that four shares are sold at 11 and one at 12. After the transaction, the ask price is Market depth. As described above, there are only a finite number of limit orders on the book at one time. If a large market buy order arrives, for instance, then the lowest limit sell order is filled first. But if the market order is bigger than the total shares available to buy at the ask price, then the limit orders at the next-to-lowest price are filled, and progresses up the book until the market order is finally filled. In this way, the ask price increases. 7

8 The market depth is the number of shares available to buy or sell at the ask or bid price respectively. Equivalently, the depth of a market is a measure of the size of a market order necessary to move quoted prices. However, in this course, we will assume that there is infinite market depth. Equivalently, we will assume that investors are small relative to the limit order book, so they are price takers, not price makers. However, the most recent financial crisis shows that this assumption does not always approximate reality just ask the traders at Lehman Brothers! 2. Prerequisite knowledge The emphasis of this course is on some of the mathematical aspects of financial market models. Very little is assumed of the reader s knowledge of the workings of financial markets. However, some mathematical background is needed. Our starting point is the famous observation (sometimes attributed to Niels Bohr) that it is difficult to make predictions, especially about the future. Indeed, anyone with even a passing acquaintance with finance knows that most of us cannot predict with absolute certainty how the the price of an asset will fluctuate otherwise we would be much richer! Therefore, the proper language to formulate the models that we will study is the language of probability theory. An attempt is made to keep this course self-contained, but you should be familiar with the basics of the theory, including knowing the definition and key properties of the following concepts: random variable, expected value, variance, conditional probability/expectation, independence, Gaussian (normal) distribution, etc. Familarity with measure theoretical probability is helpful, though a crashcourse on probability theory is given in an appendix. Please send all comments and corrections (including small typos and major blunders) to me at m.tehranchi@statslab.cam.ac.uk. 8

9 CHAPTER 1 One-period models 1. One-period models We consider a market with n assets. The identity of the assets is not important as long as the standing assumptions (zero dividends, zero tick size, zero transaction costs, no short-selling constraints, infinite divisibility, zero bid-ask spread, infinite market depth) are fulfilled. We usually think of the assets as being stocks and bonds, but they also can be more exotic things like pork belly futures. The models we will encounter in this course will be of form (Pt 1,..., Pt n ) t T where Pt i will model the price of asset labelled i at time t T. In this course, the index set T will be one of the three sets {, 1} for single period models, Z + = {, 1, 2,...} when time is discrete, and R + = [, ) when time is continuous. As simple as it seems, much of the financial aspects of this course already appear in oneperiod models where T = {, 1}. It is, therefore, appropriate to devote a significant portion of the course to this important special case. We now describe our first model. As we shall see later, this set-up captures most of the essential features of the general discrete-time model. We think of time as the present, where we have full information. Time 1 is the future, so outcomes are uncertain. We model this as follows P R n is not random and P 1 is a R n -valued random vector How can we tell if a model is good? Note that the non-random vector P is already observed, so the only thing left to model is the distribution of the random vector P 1. A statistical criterion would be to say a model is good if it fits data well. However, we are in a situation where we have only one realisation of P 1, which will happen in the future. In particular, at time we have no observations of P 1 and hence conventional statistics is impossible! Therefore, we think about the question economically. We consider an investor in the market. Suppose his initial wealth is the non-random amount x R. He chooses a nonrandom portfolio H R n, where the real number H i denotes the number of shares held in asset i. (If H i > then the position is said to be long, and if H i < then the position is said to be short.) The investor consumes the remaining wealth x H P, where we are using the notation n a b = a i b i 9 i=1

10 for the usual Euclidean inner (or dot) product in R n. At time 1, the agent then liquidates the position, receiving the random amount H P 1 which is consumed. We are lead to the following problem: Problem. Given x R and the function U : R 2 R { } which is strictly increasing in both arguments. Find H R n to (*) maximise E[U(c, c 1 )] subject to c = x H P, c 1 = H P Arbitrage, martingale deflators and 1FTP When does the above optimal investment problem (*) have a solution? To answer the question, we introduce a crucial definition: Definition. An arbitrage is a portfolio H R n such that H P H P 1 almost surely, and P(H P = = H P 1 ) < 1. Example. Consider a market with two assets with prices given by (P 1, P 2 ) (4, 7) 1/2 (3, 4) 1/2 (2, 3) (The above diagram should be read P 1 = 3, P(P1 2 = 7) = 1/2, etc.) Consider the portfolio H = ( 4, 3). It costs H P = to buy at time, but the time-1 wealth H P 1 is strictly positive in all states of the world: 5 1/2 1/2 At this point, you should check that the portfolio H = ( 3, 2) is also an arbitrage in the example above. Proposition. If the market has an arbitrage, then the optimal investment problem (*) has no maximiser. Conversely (and more importantly for us) if the problem (*) has a maximiser then the market has no arbitrage. [Technical point. One should be careful about integrability. Recall for that expected value E(Z) is always defined whenever Z almost surely. But for non-negative random variables, the expected value may take the value +. On the other hand, the expected value E(Z) for a general real-random variable Z is defined only if either E(Z + ) < or E(Z ) < (or if both expectations are finite) in which case E(Z) = E(Z + ) E(Z ). This convention avoids 1 1

11 having to define. By the way, here we are using the standard notation z + = max{z, } and z = max{ z, } for real z. To ensure that the objective function of (*) is well-defined, we implicitly assume that the given initial wealth x, utility function U, the initial price vector P, and the terminal price random vector P 1 have the following property: either E[U(c, c 1 ) + ] < or E[U(c, c 1 ) ] < for all portfolios H, where c = x H P and c 1 = H P 1. Easy-to-check sufficient conditions which would imply this assumption are either to assume that either the random vector P 1 is bounded or to assume that U is bounded from either above or below. These points were omitted in lecture to avoid getting too bogged down on technicalities...] Proof. Fix an initial wealth x and for a portfolio H R n, let c (H) = x H P be the initial consumption and c 1 (H) = H P 1 the terminal consumption, and let F (H) = E[U(c (H), c 1 (H))] be the expected utility of consumption as a function of the portfolio. Suppose H is the optimal portfolio. That is, assume that F (H ) is finite (that is, neither or + ) and F (H) F (H ) for all H R n. Assume, for the sake of finding a contraction, that K is an arbitrage. Hence c (H +K) = c (H ) K P and c 1 (H + K) = c 1 (H ) + K P 1, and the event that at least one of the inequalities is strict has positive probability. Since U is strictly increasing in both arguments, we have F (H + K) > F (H ), contradicting the optimality of H. Markets with an arbitrage opportunity would be nice we all would be a lot richer. But for the sake of building realistic models, we usually assume that markets are free of arbitrages. Indeed, the existence of arbitrage is not very economical. In particular, note that if H is an arbitrage, so are the portfolios 2H, 3H,.... That is, if we spot an arbitrage in the market, we could scale it to larger and larger proportions and consume more and more. Eventually, the assumption that we are price takers (meaning we are so small relative to the market that we can trade with no price impact) becomes unrealistic. We now stop briefly to discuss other notions of arbitrage that make sense in this oneperiod framework: Definition. An arbitrage relative to a portfolio K R n is a portfolio H R n such that H P K P and K P 1 H P 1 almost surely, and P(H P = K P and K P 1 = H P 1 ) < 1. An arbitrage relative to the portfolio is sometimes called an absolute arbitrage, but this is just our ordinary definition of arbitrage introduced above. Aside. Note that if H is an absolute arbitrage, and K is any portfolio, then H + K is an arbitrage relative to K. Similarly, if H is an arbitrage relative to K then the portfolio H K is an absolute arbitrage. These comments indicate that in this one-period framework there is little motivation to introduce the notion of relative arbitrage. 11

12 But to see why one might want to distinguish between relative and absolute arbitrage, suppose we want to build a theory where we restrict portfolios to a set A R n. In the theory described in this chapter, we are letting A = R n. Since R n is a vector space, we have that H, K A implies that both H + K A and H K A. But one may want to develop a theory where, for instance, short-selling is prohibited, in which case we would take A = {H R n : H i for all i}. Note that if H, K A the vector H + K is in A but the vector H K need not be an element of A. Hence, for this version of the theory, an absolute arbitrage still gives rise to relative arbitrage, but one cannot necessarily construct an absolute arbitrage from a relative arbitrage. We will return to the notion of relative arbitrage again when we discuss continuous time models. But until then, the term arbitrage will mean absolute arbitrage. In many discussions of arbitrage theory, there is the assumption that at least one asset is a numéraire: Definition. An asset is a numéraire iff its price is strictly positive for all time, almost surely. More generally, a portfolio η R n is a numéraire portfolio is η P > and η P 1 > almost surely. Having a numéraire in the market simplifies the story in some ways. For instance, when we discuss arbitrage theory, we no longer have to allow for intermediate consumption. Definition. A terminal consumption arbitrage a portfolio K such that K P = K P 1 almost surely. P(K P 1 > ) >. Proposition. If H is an arbitrage and η is a numéraire portfolio then K = H H P η P η is a terminal consumption arbitrage. Proof. Note that K P = by construction. Also we have K P 1 = H P 1 η P 1 η P H P. Note that K P 1 almost surely. Also, since η P 1 η P > almost surely, we have P(K P 1 = ) = P(H P = = H P 1 ) < 1. Aside. There is a further reason why we traditionally look at markets with no arbitrage. To explain, we take a moment to ask where do prices come from? The terminal price P 1 is unknown at time but is revealed at time 1, so we model it as random. What determines the randomness? One could argue that all that matters is the beliefs of the market participants, not the underlying mechanism that causes the apparent randomness. So, we assume that there are J investors, and each investor j has a probability measure P j, where j = 1,..., J modelling the distribution of P 1. We also assume that each agent j comes to the market with initial capital x j. The market already has the n assets, with total supply of asset i 12

13 given by S i and S = (S 1,..., S n ). The agents trade with each other until each arrive at an optimal allocation Hj and collectively determine an initial price P. To formalise this, we have the following definition: Definition. Given initial wealths x j, utility functions U j and probability measures P j, for j = 1,... J which determine the distribution of the random vector P 1, let H j (p) = arg max{e j [U(c, c 1 )] : c = x j H p, c 1 = H P 1 } be the optimal portfolio for agent i assuming the initial price is P = p. An equilibrium price P is a solution to the equation J H j (P ) = S, where the notation E j denotes expectation with respect to P j. j=1 Note that the above condition says that for the equilibrium initial price P, the agents portfolios Hj solve their version of the optimal investment problem (*). A consequence of the previous proposition is the following motivating result: Proposition. If the market is in equilibrium then no agent can believe there is an arbitrage. We now return to our market model P, P 1. Our goal is to develop a structure theorem that characterises markets without arbitrages. In order to achieve this goal, we now explore heuristically the implication of having a maximiser to an optimal investment problem. Given the utility maximisation problem (*), it is very natural consider the dual problem. The following calculations are purely formal, with no claim to rigorous justification. Also, we assume for the moment some knowledge of Lagrangian duality, at the level of IB Optimisation, but nothing here is needed for the rest of the course. We define the Lagrangian L(c, c 1, H; Y, Y 1 ) = E[U(c, c 1 )] + Y (x c H P ) + E[Y 1 (H P 1 c 1 )] = E[U(c, c 1 ) Y c Y 1 c 1 ] + Y x + H [E(Y 1 P 1 ) Y P ] where the Lagrange multipliers are a real number Y and a real-valued random variable Y 1. To identify the dual feasibility constraint, we ask for which Y and Y 1 does the Lagrangian have an unconstrained maximiser over the triple c, c 1, H. Since H can take any value in R n, in order for the maximum of the Lagrangian over H to exist, we need E(Y 1 P 1 ) = Y P. Similarly, since the function U is strictly increasing in both variables, the existence of a maximiser implies that Y > and Y 1 > almost surely. Indeed, the maximisers are related by [ ] U E = Y and U = Y 1. c c 1 Given the above motivation, we now make a definition: 13

14 Definition. A martingale deflator for a market model P, P 1 consists of a non-random Y >, a random variable Y 1 > almost surely such that for all i = 1,..., n. E(Y 1 P i 1 ) <, and E(Y 1 P i 1) = Y P i Now we come to first theorem of the course, and one of the most important theorems in financial mathematics. It is no surprise that it is often called the first fundamental theorem of asset pricing. Theorem (First fundamental theorem of asset pricing). A market model has no arbitrage if and only if there exists a martingale deflator. The origin of the terminology martingale deflator is that the process (Y t P t ) t {,1} is a martingale, in the most trivial sense of the word. However, we will use this observation to define martingale deflators in both discrete and continuous time. Note that if Y = (Y t ) t {,1} is a martingale deflator, so is cy for any constant c >. In particular, in the one period setting, it is convenient to work with the normalised random variable ρ = Y 1 /Y. Definition. A pricing kernel (or stochastic discount factor or state price density) is a ρ be a positive random variable such P = E(ρP 1 ) Aside. This is a comment on the origin of the term state price density. Suppose the sample space Ω = {1,..., m} is finite. Then the random vector P 1 has the representation m P 1 = 1 {j} P 1 (j). j=1 That is to say, that each price P1 i is a linear combination of the indicator functions 1 {j}. Each indicator function can be interpreted as the value of a security which pays one unit of money if the outcome (or state) is ω = j and zero otherwise. These are called Arrow Debreu securities. The formula m P = E(ρP 1 ) = P{j}ρ(j)P 1 (j). j=1 shows that we can think of P{j}ρ(j) as the time- price of the Arrow Debreu security of state j. Hence, the quantity ρ(j) is the state-price density (with respect to the probability measure). Proof of the easy direction. First we prove that if there exists a martingale deflator, then there is no arbitrage. Letting X t = H P t for t =, 1, we have by the definition of martingale deflator and the linearity of expectations E(Y 1 X 1 ) = H E(Y 1 P 1 ) = H (Y P ) = Y X. 14

15 Now suppose X and X 1 almost surely. Since Y and Y 1 almost surely, we have Y X = E(Y 1 X 1 ) so that Y X = = E(Y 1 X 1 ). Since Y, we see that X =. Also, see that Y 1 X 1 = almost surely by the pigeonhole principle (Recall the pigeonhole principle: if Z a.s and E(Z) = then Z = a.s.) Again, since Y 1 almost surely, we conclude that X 1 = almost surely. In particular, the portfolio H is not an arbitrage. Proof of the hard direction. We now suppose that the market has no arbitrage, so that for any vector H R n that has the property that H P H P 1 almost surely, it must be the case that H P = = H P 1 almost surely. We will show that, given any positive random variable Z, there exists a martingale deflator Y, Y 1 such that the product Y 1 Z is bounded by a constant. This extra boundedness assumption is much stronger than what we need, but it comes for free from the proof and we will find it useful later in the course. Let ζ = e P 1 2 /2 1 + Z. Define a function F : R n R by F (H) = e H P + E[e H P 1 ζ]. The positive random variable ζ is introduced to ensure integrability. Indeed note that the integrand e H P 1 ζ e H 2 /2 is bounded for each choice of H. In particular, the function F is finite everywhere and (by the dominated convergence theorem) smooth. We will show that no investment-consumption arbitrage implies that the function F has a minimiser H. By the first order condition for a minimum, we have and hence we may take = F (H ) = e H P P E[e H P 1 ζp t ] Y = e H P and Y 1 = e H P 1 ζ. Note that Y 1 Z < C for some constant C > (which depends on H in general). So let (H k ) k be a sequence such that F (H k ) inf H F (H). If (H k ) k is bounded, we can pass to a convergent subsequence, by the Bolzano Weierstrass theorem, such that H k H. By the smoothness of F we have inf H F (H) = lim k F (H k) = F (lim k H k ) = F (H ) so H is our desired minimiser. It remains to show that no arbitrage implies that there exists a bounded minimising sequence (H k ) k. So for the sake of finding a contradiction, suppose every minimising sequence (H k ) k is unbounded. Now we arrive at a little technicality. Let and let U = {u R n : u P = = u P 1 a.s.} R n V = U. 15

16 Notice that if u U and v V then F (u + v) = F (v). Hence, by projecting a given minimising sequence onto the subspace V, we consider an unbounded minimising sequence (H k ) k taking values in V. We can pass to a subsequence such that H k. Now let Ĥ k = H k H k. Note that Ĥk = 1 and that Ĥk V. Since (Ĥk) k is bounded, we can again pass to a convergent subsequence such that Ĥk Ĥ. Notice once more that Ĥ = 1 and that Ĥ V. We know that the sequence F (H k ) is bounded (since it is convergent) but we also have F (H k ) = (eĥk P ) H k + E[(e Ĥk P 1 ) H k ζ] so we must conclude that Ĥ P Ĥ P 1 a.s. (since otherwise the right-hand side would blow up). By the assumption of no arbitrage we conclude that Ĥ P = = Ĥ P 1 a.s., which means Ĥ U. But we also know that Ĥ V. Since the subspaces are orthogonal, we have U V = {}, and in particular, we have H =. But this contradicts the fact that Ĥ = Equivalent martingale measures In this section, we introduce the notion of an equivalent martingale measure. We will see that an equivalent martingale measure is, essentially, the same as a pricing kernel. However, unlike a pricing kernel, the definition of equivalent martingale measure depends crucially on the existence of a numéraire. The primary purpose of this section is to reconcile concepts and terminology used by other authors to the theory developed so far. We will also find that equivalent martingale measures can be used to simplify some calculations later in the course. Our goal is to define an equivalent martingale measure. We begin with another definition: Definition. Let (Ω, F) be a measurable space and let P and Q be two probability measures on (Ω, F). The measures P and Q are equivalent, written P Q, iff iff P(A) = 1 Q(A) = 1 P(A) = Q(A) =. The above definition says that equivalent probability measures have the same almost sure events. Complementarily, equivalent probability measures have the same null sets. It turns out that equivalent measures can be characterised by the following theorem. When there are more than one probability measure floating around, we use the notation E P to denote expected value with respect to P, etc. Theorem (Radon Nikodym theorem). The probability measure Q is equivalent to the probability measure P if and only if there exists a P-a.s. positive random variable Z such that Q(A) = E P (Z1 A ) 16

17 for each A F. The random variable Z is called the density, or the Radon Nikodym derivative, of Q with respect to P, and is often denoted Z = dq dp. As a mnemonic device, note that the Radon Nikodym derivative satisfies the identity dq Q(A) = dp dp. Also, note that P has a density with respect to Q given by dp dq = 1 Z. We only need the easy direction of the theorem, that the existence of a positive density implies equivalence, for this course. Here is a proof. The proof of the harder direction is omitted since we do not need it. Proof. Suppose P(Z > ) = 1 and that E P (Z) = 1. Define a set function Q by A Q(A) = E P (Z1 A ). Note that Q is countably additive by the monotone convergence theorem. Also, Q(Ω) = E P (Z) = 1, so Q is a probability measure. If P(A) =, then the event {1 A = } is P-almost sure and hence Q(A) = E P (Z1 A ) =. Conversely, if Q(A) = we can conclude that {Z1 A = } is P-a.s. by the pigeon-hole principle since {Z1 A } is P-a.s. But since {Z > } is P-a.s., we must conclude that {1 A = } is P-a.s., i.e. P(A) =. Thus Q and P are equivalent. Example. Consider the sample space Ω = {1, 2, 3} with the set F of events all subsets of Ω. Consider probability measures P and Q defined by P{1} = 1, P{2} = 1, and P{3} = 2 2 Q{1} = 1 999, Q{2} =, and Q{3} =. 1 1 Then P and Q are equivalent. We may take their density Z = dq to be dp Z(1) = 1 999, Z(2) =, Z(3) =. 5 5 (Since both measures don t see the event {3}, we can let Z(3) be any value.) Definition. Let (P t ) t {,1} be a market model defined on a probability space (Ω, F, P). The measure P is called the objective (or historical or statistical) measure for the model. Suppose the market has a numéraire portfolio η. Let N t = η P t for t =, 1. An equivalent martingale measure relative to the numéraire is any probability measure Q equivalent to P such that E Q ( P1 /N i 1 ) < and ( ) P E Q i 1 = P i N 1 N for all i {1,..., n}. 17

18 Remark. The idea is that a numéraire can be used to count money. Hence, we can speak in terms of prices relative to (or discounted by) the numéraire. As a preview of what s to come, the term equivalent martingale measure is appropriate since the discounted price processes (Pt i /N t ) t {,1} are a martingale for Q with respect to the filtration (F t ) t {,1} where F = {, Ω} and F 1 = F. We will elaborate on this in the multi-period case. When the market has a numéraire, then the notion of a pricing kernel and that of an equivalent martingale measure are essentially the same. Proposition. Consider a market with numéraire portfolio η, and let N t = η P t for t =, 1. Let Q be an equivalent measure. Then Q is an equivalent martingale measure relative to the numéraire if and only if for a pricing kernel ρ. dq dp = ρn 1 N Proof. This is a matter of chasing the definitions. Putting these ingredients together yields the more common version of the first fundamental theorem: Theorem (First fundamental theorem of asset pricing). Suppose that the market has a numéraire. Then there is arbitrage if and only if there exists an equivalent martingale measure relative to the numéraire. Aside. Recall that if the market model admits a numéraire portfolio, then there is no arbitrage if and only if there is no terminal consumption arbitrage. Since most books assume from the beginning that a numéraire exists, it is rare to see this distinction addressed. Finally, we introduce some more definitions which are frequently used in financial modelling. They will crop repeatedly through these lectures. To avoid repeating the same definition multiple times, we give the following definitions in the context of a possibly multi-period market model. Definition. A (riskless) bond is an asset whose price at a fixed time T > is a positive constant. Unless otherwise indicated, we will adopt the convention that the time-t price of a bond is exactly one unit of money. Definition. An equivalent martingale measure relative to a bond is called a forward measure. Of course, in one period, the fixed time is T = 1. The next definition is also stated in terms of a possibly multi-period model: Definition. A riskless asset (or more generally a portfolio) in a discrete-time model is one whose time t price is known at time t 1, for all t 1. Definition. An equivalent martingale measure relative to a riskless numéraire is called a risk-neutral measure. 18

19 In one period, a riskless numéraire is a bond, and assuming no arbitrage, a bond is a riskless numéraire. Hence the notions of forward measure and risk-neutral measures coincide in this context. But be careful - in general multiperiod models, bonds need not be riskless assets and riskless numéraires need not be bonds. 4. Contingent claim pricing The setting of this section is as follows. We find ourselves in a market with prices (P t ) t {,1}. A new asset (a contingent claim) is introduced to the market with time 1 price ξ 1, a given random variable. The question is this: what is a good value for the initial price ξ of this new asset? Example (Call option). A European call option gives the owner of the option the right, but not the obligation, to buy a given stock at time 1 at some fixed price K, called the strike of the option. Let S 1 denote the price of the stock at the maturity date 1. There are two cases: If K S 1, then the option is worthless to the owner since there is no point paying a price above the market price for the underlying stock. On the other hand, if K < S 1, then the owner of the option can buy the stock for the price K from the counterparty and immediately sell the stock for the price S 1 to the market, realising a profit of S 1 K. Hence, the payout of the call option is ξ 1 = (S 1 K) +, where a + = max{a, } as usual. The hockey-stick graph of the function g(x) = (x K) + is below. We will assume that the original market (P t ) t {,1} has no arbitrage, since otherwise it is difficult to formulate a reasonable answer to the question. Now recall that given a utility function U, a consumption stream c = (c t ) t {,1} is preferred to consumption stream c iff E[U(c)] > E[U(c )], and say that the agent is indifferent between consumption streams c and c iff E[U(c)] = E[U(c )]. Thus, an agent seeks to maximise the expected utility of the consumption stream subject to the constraint that the initial wealth x is fixed. The budget constraint amounts to c = x H P and c 1 = H P 1, where H is the portfolio. Since the consumption stream is parametrised by H, it makes sense to talk of preferences over portfolios. Now add the claim to the market. For which initial price ξ would you be willing to hold one share of the claim? Using the reasoning above, the answer is iff there exists a portfolio H R n such that the augmented portfolio (H, 1) R n+1 (the portfolio of holding (H ) i 19

20 shares of asset i for i = 1,..., n and holding one share of the claim) is preferred to the augmented portfolio (H, ) for all H R n. That is to say, you would be willing to pay ξ for one share of the claim iff (1) max{e[u(c)] : c = x ξ H P, c 1 = ξ 1 + H P 1 )} This leads us to a definition: max{e[u(c)] : c = x H P, c 1 = H P 1 )}. Definition. Given the market (P t ) t {,1}, the payout of the claim ξ 1, the initial wealth x and the utility function U, the utility indifference price ξ is defined to any solution to (2) max{e[u(c)] : c = x ξ H P, c 1 = ξ 1 + H P 1 )} = max{e[u(c)] : c = x H P, c 1 = H P 1 )}. The idea is that if U is increasing in both arguments, so you prefer to consume more rather than less, then the indifference price ξ is an upper bound for the price you would be willing to pay for one share of the claim. Although the notion of the indifference price does provide an answer to our question, it is a bit unsatisfactory since it depends both the model of the future prices and our preferences. While there is plenty of price data available and hence there are many popular statistical models for prices, we do not directly observe our utility function. So in practice, it is hard to compute the utility indifference price. The following concept tackles the pricing problem in a preference-independent way. Definition. A portfolio H R n super-replicates the claim with payout ξ 1 iff H P 1 ξ 1 almost surely. The connection between super-replication and indifference pricing is this: Proposition. Suppose H P 1 ξ 1 almost surely. If U is strictly increasing in both arguments, then H P ξ. This says that if a portfolio super-replicates the claim, then the initial cost of this portfolio is an upper bound for your utility indifference price for the claim, and we have already argued that your utility indifference price is an upper bound for the price you would be willing to pay for the claim. Proof. For any portfolio K we have and using the assumption U(c, ) is increasing we have E[U(x H P K P, ξ 1 + K P 1 )] E[U(x (H + K) P, (H + K) P 1 )] max E[U(x J P, J P 1 )] J = max E[U(x ξ J P, ξ 1 + J P 1 )] J where the last line is just the definition of indifference price. Now by maximising over K and using the assumption that U(, c 1 ) is strictly increasing, we have H P ξ which is what we wanted to show. We finally come to a theorem which answers the question of when super-replication is possible. 2

21 Theorem (Characterisation of super-replication). Consider a market (P t ) t {,1} that is free of arbitrage and let ξ 1 be the time-1 payout of a contingent claim. There is a constant x such that E(ρξ 1 ) x for all pricing kernels ρ such that ρξ 1 is integrable if and only if there exists a portfolio H R n such that H P x and H P 1 ξ 1 a.s. Proof of the easy direction. Suppose that for some portfolio H we have H P x and H P 1 ξ 1 a.s. Then for any suitably integrable ρ we have E(ρξ 1 ) E(ρH P 1 ) = H P x. Proof of the harder direction. Suppose E(ρξ 1 ) x for every pricing kernel ρ such that ρξ 1 is integrable. We need to show that there exists a H R n such that H P x and H P 1 ξ 1 a.s. To that end, let F (H, γ) = e γ(x H P ) + E[e γ(h P 1 ξ 1 ) ζ] where the factor ζ = e P ξ 1 is introduced to ensure integrability. (This function is motivated by the utility maximisation problem introduced in the last chapter. The parameter γ > is the investor s risk aversion. We plan to send γ +, which corresponds the limit where the investor can tolerate no losses.) For each γ >, by the proof of the first fundamental theorem of asset pricing, there exists a unique H γ V such that where We now show that F (H γ, γ) = inf H F (H, γ), V = {u R n : u P = = u P 1 a.s.}. sup F (H γ, γ) <. γ 1 Firt note that by the first order condition for a minimum H F H=Hγ =, we see that by setting Y γ = e γ(hγ P x) and Y γ 1 = e γ(ξ 1 H γ P 1 ) ζ we have found a martingale deflator, and ρ = Y 1 /Y is a pricing kernel. Note that γ F H=H γ = Y γ (H γ P x) + E[Y γ 1 (ξ 1 H γ P 1 )] = H γ (Y γ P E[Y γ 1 P 1 ]) + E[Y γ 1 ξ 1 ] Y γ ξ by since ρ is a pricing kernel and the assumption that E(ρξ 1 ) x. 21

22 Next note that γ H γ is differentiable. (Indeed, recall that H γ is defined as the root of the function F (, γ) : V V, and D 2 (, γ) is a strictly positive definite operator on V, so the differentiability of H γ follows from the implicit function theorem.) Furthermore, since H γ is the minimiser of F γ and hence F (H γ, γ) F (H γ±ε, γ) g F (H g, γ) g=γ =. Putting this together implies γ F (H γ, γ) is nonincreasing, which proves the claim. Now we consider the sequence (H k ) k where the risk-aversion parameter takes the values γ = k N. If (H k ) k is bounded, then we can find a convergent subsequence such that H k H. Note that since the sequence F (H k, k) = (e H k P x ) k + E[(e ξ 1 H k P 1 ) k ζ] is bounded, we must have that x H P and ξ 1 H P 1 a.s. So it remains to rule out the case that the sequence (H k ) k is unbounded. Suppose that it was unbounded, for the sake of finding a contradiction. Then we can pass to a subsequence that H k. Like in our proof of the 1FTAP, let Ĥ k = H k H k and pass to a subsequence such that Ĥk Ĥ. Note that we have that Ĥ V and that Ĥ = 1. But by the formula x F (k, H k ) = H (eĥk P k ) k Hk + E[(e ξ 1 H k Ĥk P 1 ) k Hk ζ] we see that boundedness forces Ĥ P Ĥ P 1. By no arbitrage, we have Ĥ P = = Ĥ P 1 a.s. Since Ĥ V we conclude that Ĥ =, contradicting Ĥ = 1. The take-away message about super-replication is the following proposition: Proposition. Suppose that the market with prices (P t ) t {,1} has no arbitrage. Introduce a contingent claim with prices (ξ t ) t {,1}. If H super-replicates the time-1 payout ξ 1 of the claim and if augmented market (P, ξ) has no arbitrage, then ξ H P. Now we focus on claims that can be both super-replicated and sub-replicated by the same portfolio. Definition. A contingent claim with payout ξ 1 is replicable or attainable iff there exists a portfolio H such that H 1 P 1 = ξ 1 almost surely. Theorem (Characterisation of replicable claims). Suppose that the market model has no arbitrage, and let ξ 1 be the payout of a contingent claim. The claim is attainable if and only if there exists an x R such that E(ρξ 1 ) = x for all pricing kernels ρ such that ρξ 1 is integrable, if and only if there is a unique initial price ξ = x such that the augmented market (P, ξ) has no arbitrage. 22

23 Proof. Suppose E(ρξ 1 ) = x for all suitably integrable ρ. By the characterisation of super-replication (applied to ξ 1 ) there exists H such that H P x and H P 1 ξ 1 a.s. Similarly, the characterisation of super-replication (applied to ξ 1 ) yields the existence of K such that K P x and K P 1 ξ 1 a.s. Adding gives us (H + K) P (H + K) P 1 a.s. Since there is no arbitrage by assumption, we have (H + K) P = = (H + K) P 1 a.s. Therefore, ξ 1 H P 1 = K P 1 ξ 1 so both H and K replicate the claim. By the same argument, both have initial price H P = K P = x. Remark. For any contingent claim with payout ξ 1 there is an interval [ξ, ξ ] where ξ = inf{h P : H P 1 ξ 1 } is the cost of the cheapest super-replicating portfolio and ξ = inf{h P : H P 1 ξ 1 } is the cost of the most expensive sub-replicating portfolio. However, if the claim is replicable, then the interval collapses into a single price, which can be calculated by computing the expected value of ξ 1 ρ for any pricing kernel ρ. Since attainable claims have unique no-arbitrage prices, we single out the markets for which every claim is attainable: Definition. A market is complete if and only if every contingent claim is replicable. A market is incomplete otherwise. We can characterise complete markets: Theorem (Second Fundamental Theorem of Asset Pricing). An arbitrage-free market model is complete if and only if there exists a pricing kernel. Proof. Suppose that there is a unique pricing kernel. Let ξ 1 be any random variable. By the proof of the first fundamental theorem, we know that the random variable ξ 1 ρ is integrable. In particular, there is a number x such that x = E(ρξ 1 ) for all (the unique) pricing kernels. By the characterisation of attainable claims, there exists a portfolio such that H P 1 = ξ 1. Hence the market is complete. Conversely, suppose that the market is complete. Let ρ and ρ be pricing kernels. Fix for the moment claim with payout ξ 1 1 (The bound is just to ensure integrability). Since ρ+ρ ξ 1 is attainable, there exists x such that E(ρξ 1 ) = x = E(ρ ξ 1 ) E[ξ 1 (ρ ρ )] =. 23

24 Now letting ξ 1 = ρ ρ (ρ + ρ ) 2 Since we have E[(ρ ρ ) 2 Z] = where Z = (ρ+ρ ) 2 >, the pigeon-hole principle yields ρ = ρ almost surely as desired. This box summarises the fundamental theorems: 1FTAP: No arbitrage Existence of martingale deflator 2FTAP: Completeness + No arb Uniqueness of martingale deflator In complete markets have even more (arguably too much) structure: Theorem. If the market model P with n assets is complete, there there exists at most n events of positive probability. In particular, the n-dimensional random vector P 1 takes values in a set of at most n elements. Proof. Suppose A 1,..., A k are a collection of disjoint events with P(A i ) > for all i. Claim: the set {1 A1,..., 1 Ak } is linearly independent, and in particular, the dimension of the span of {1 A1,..., 1 Ak } is exactly k. To prove this claim, we must show that if a 1 1 A a k 1 Ak = a.s. for some constants a 1,..., a k, then a 1 = = a k =. To this end, note that if i j the sets A i and A j are disjoint and hence 1 Ai 1 Aj =. By multiplying both sides of the equation by 1 Ai we get a i 1 Ai =. But since P(A i ) > it must be the case that a i =. Now if the market is complete, each of the 1 Ai is replicable. Hence span{1 A1,..., 1 Ak } {H P 1 : H R n } = span{p 1 1,..., P n 1 } Looking at the dimensions of the spaces above, we must conclude k n. Example. (Put-call parity formula) Suppose we start with a market with three assets with prices (B, S, C). The first asset a bond, so its time-1 price B 1 = 1 is not-random. The next asset is a stock. The last asset is a call option on that stock with strike K and maturity T, so that C T = (S T K) +. Suppose that this market is free of arbitrage. Now we introduce another claim, called a put option. A put option gives the owner of the option the right, but not the obligation, to sell the stock for a fixed strike price at a fixed maturity date. If the strike is K and maturity date is T, then a similar argument as we used for the call option, the payout of a put option is P T = (K S T ) +. It turns out that the put option is replicable in the market (B, S, C). Indeed, we have the identity P 1 = (K S 1 ) + = K S 1 + (S 1 K) + = (K, 1, +1) (B 1, S 1, C 1 ). Hence H = (K, 1, +1) is a replicating portfolio. 24

25 Now, suppose we want to assign a price P to the put for t =. The cost of replication is the unique indifference price (for all increasing utility functions) and also the unique price such that the augmented market (B, S, C, P ) has no arbitrage. Hence we usually set P to satisfy This is the famous put-call parity formula. P C = B K S. Figure 1. A plot of P C versus K, where t = corresponds to 23 October 217 and and t = 1 is 17 November 217, and the underlying asset is the S&P 5 index with S = 2, The price of the calls and puts is taken to be the last traded price on the day (as opposed to the bid or ask price). All data is taken from 5. Markets with an infinite number of assets We now consider the seemingly unrealistic situation where the market is allowed to have an infinite number of assets. Rather than being an exercise for mathematicians to generalise needlessly, we will see shortly that this modelling framework does have practical applications. We now let I be an arbitrary index set, and we assume that for each i I there is an asset i with price Pt i at time t. For instance, if we let I = {1,..., n} we recover the case where there are n assets. For a finite subset J I we will let Pt J denote the J -dimensional vector (P j t ) j J of asset prices indexed by J. Consider a portfolio of holding H j shares of asset j for each j J. We let H J denotes the J -dimensional vector (H j ) j J and so the time t price of this portfolio is given by H J P J 1 = j J H j P j 1. 25

26 Definition. A sequence of finite-dimensional portfolios (H J ) J I asymptotically superreplicates a contingent claim with payout ξ 1 iff lim inf J H J P J 1 The sequence asymptotically replicates the claim iff lim J HJ P J 1 ξ 1 almost surely. = ξ 1 almost surely. Rather than give some abstract theory around these definitions, we move to a concrete example. We consider a market consisting of a risk-free bond with time-1 price B 1 = 1, a stock with time-1 price S 1, and a family of call prices indexed by the strike K with time-1 price C 1 (K) = (S 1 K) +. Theorem. Let ξ 1 = g(s 1 ). If g C 2 [, ) and S 1, then there exists a family of portfolios that asymptotically replicates the claim with payout ξ 1. If g C 2 (, ) and S 1 >, then the same conclusion holds. Proof. Before we begin, note that put options are replicable by put-call parity, so we can and will assume that the market also has put options of all strikes. Now the following formula holds identically g(s 1 ) = g(a) + g (a)(s 1 a) + a g (K)(K S 1 ) + dk + a g (K)(S 1 K) + dk for any a >. Note that the integrand of the first integral is zero unless min{s 1, a} K a. Similarly, the integrand of the second integral is zero unless a K max{s 1, a}. In particular, the ranges of both integrals are bounded intervals on which g is assumed continuous, so both integrals are ordinary Riemann integrals. (One way to prove the identity is to fix S 1 and let h(a) equal the right-hand side. By the standard rules of calculus, we have h (a) = and hence h(a) is a constant. To evaluate that constant, let a = S 1 and note that both integrals vanish since the ranges of integration have zero length.) To exhibit a family of portfolios approximating g(s 1 ), we fix an a > and consider a family of finite subsets of (, ) defined by K n = {K n 1,..., K n n} such that n i = K n i+1 K n i for 1 i < n, that max 1 i<n n i and K n n as n. By a Riemann integration theory the sum g(a) ag (a) + g (a)s 1 + i:k n i <a n i g (K n i )P 1 (K n i ) + i:k i a n i g (K n i )C 1 (K n i ) converges to g(s 1 ) as n. Hence the desired asymptotically replication is achieved by the family of portfolios consisting of holding g(a) ag (a) shares of the bond, holding g (a) shares of stock, and holding n i g (Ki n ) puts of strike Ki n < a and n i g (Ki n ) calls of strike Ki n a for i = 1,..., n. 26