Computational Finance The Martingale Measure and Pricing of Derivatives

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1 1 The Martingale Measure 1 Comutational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure The Martingale measure or the Risk Neutral robabilities are a fundamental concet in the no-arbitrage ricing of instruments which links rices to exectations. This will be a very useful later on, because as we will see, there are very good randomized algorithms (Monte Carlo) for estimating exectations. 1.1 Two Period Stock Bond Economy To begin we revisit the revious an examle economy we already studied, as illustrated in the following figure. t=0 t=t B: B(0) B(T) B(T) State 2 State State 2 P u S: C: 100 C(0) State 1 State 2 State 1 At t = 0, two instruments are available. Any amount (even fractional) of either instrument may be sold or uschased at the secified market rice - i.e., arbitrary short or long ositions are allowed. A risk free asset or bond, B, and a stock, S. At t = 0 (the first eriod), the bond is worth B(0), and, the stock is worth S(0) = 100. At t = T (the second eriod), the economy can be in one of two states. In both states, the bond is valued at B(T) and hence is risk free. In the first state the stock is valued at S(T) = 100 and in the second state, the stock is valued at S(T) = 150. Suose that P u is the robability that the market goes u. We have thus comletely secified the market dynamics for our simle economy. In this simlified economy, it is clear that one can guarantee an amount B(T) at t = T by investing B(0) in the bond at t = 0. The risk free discount factor is defined by D(T) = B(0) B(T) = e rt

2 1.2 Discrete Two Period Economies 2 where r > 0 is the risk free interest rate for continuous comounding. There is also a Euroean call otion on the stock with a strike rice of 100, exiring at time T. At t = T, the call otion will either be worth 50, if the market went u, or 0 if not. Previouslywehaveshownthatifthereisnoossibilityof makingmoneyoutofnothing, then the following two conditions must hold, 1. 2/3 < D(T) < 1 2. C(0) = 100(1 D(T)) indeendent of P u. The restriction on the discount factor revents one making guaranteed money from buying stock and shorting bond (or vice versa). The rice of the call otion was determined by constructing a ortfolio with a stock and short one otion, then the ortfolio value at time T is indeendent of the state and hence acts like a bond. The otion hedges against the risk in the stock. The imortant things to note are: 1. The exact nature of the instruments that constitute the market are not relevant, only that they follow the secified market dynamics. 2. To rice the third instrument (given D(T), or alternatively, the interest rate), it was notnecessarytoknowp u (therobabilitiesofofthevariousstates). Oneonlyneedsto know what states are ossible. Among other things, this imlies the counter intuitive fact that the rice of the otion would be the same whether P u = or P u = That the actual robability P u lays no role in the ricing of the instruments is not a quirk of the articular economy that we haen to have chosen. We will now develo the general framework (the Martingale Measure framework) which essentially shows that the real world robabilities have very little to say about ricing derivatives. We need to formalize the notion of the economy, and some of the notions described already, such as the hrase not being able to make money out of nothing. 1.2 Discrete Two Period Economies First, we need to set u the notation to describe the economy/market. Suose that there are N instruments, whose rices at time t are given by S i (t), i = 1,...,N which we will denote by the column vector S(t), that reresents the state of the economy at time t. For the moment, we focus on the two times t = 0 and t = T. Suose that at t = T, there are K ossible states of the world, in other words, the state S(T) can be one of the K ossible vectors S 1,...,S K. Let the robability of state j occurring at t = T be P j, j = 1...K, reresented by the column vector P. We define the ayoff matrix at t = T as the matrix Z(T) = [S 1,S 2,...,S K ]. The element Z ij reresents the value of one unit of instrument i in state j for i = 1,...,N and j = 1,...,K. A articular row of Z reresents the ossible

3 1.2 Discrete Two Period Economies 3 values that a articular instrument can take at time T. An instrument is risk free if its value in every state is a constant (i.e., if the row is a constant). A articular column reresents the values of all the instruments in a articular state. Having secified S(t), Z(T) and P, we have fully secified our economy, including its dynamics. For the examle in the revious section, we have B(0) B(T) B(T) ] S(0) = 100 C(0) Z(T) = [ 1 Pu P(T) = A ortfolio θ = [θ 1,...,θ N ] is a column vector of N comonents that indicates how many units of each instrument is held. The value of such a ortfolio at time t is given by θ T S(t). We use the notation ( ) T to denote the transose. Thus, the value of the ortfolio at t = 0 is θ T S(0), the ossible values that the ortfolio can take on at time T are given by the row vector θ T Z(T), and the exected value of the ortfolio at time T is given by θ T Z(T)P(T). Using this notation, let s re-examine the derivation of the result that the rice of the call otion in the very simle stock bond economy must be 100(1 D(T)). Suose that C(0) = 100(1 D(T))(1+ρ) where ρ > 0. Construct the ortfolio 100/B(T) θ = 1 1/(1+ρ) At t = 0, the value of this ortfolio is [ 100 B(T) ] B(0) 1 1 (1+ρ) 100 = 0 100(1 D(T))(1+ρ) P u At t = T, the ossible values of the ortfolio are 1 [ 100 B(T) 1 1 (1+ρ) ] B(T) B(T) = 0 50 [ 0 ] 50ρ 1+ρ 0 In every state of the world, the ortfolio is worth at least zero, and in at least one state of the world, the ortfolio is worth a strictly ositive amount. Certainly, any rational erson would want to consume an infinite amount of this ortfolio since there is no downside, and only otential uside. Thus the economy cannot be in equilibrium. Hence we rule out the 1 For vectors, we use the notation x 0 to signify a vector that has every comonent 0 and at least one comonent < 0. x 0 if and only if x 0. We use the notation x < =0 to signify a vector with every comonent 0, allowing the ossibility that every comonent = 0. x > = 0 if and only if x < =0.

4 1.3 The No-Arbitrage Axiom 4 existence of such ortfolios ortfolios because any sane individual would want to hold an unlimited amount of such ortfolios, leading to disequilibrium, and so we rule out the ossibility of ρ > 0. Exercise 1.1 Suose that ρ < 0. Construct a similar ortfolio and hence argue that if the market is in equilibrium, then we can rule out the orribility of ρ < 0. Thus the only remaining ossibility is that ρ = 0, which it turns out does not lead to any ossibility of such ortfolios. The roof of this in this secific case requires some insight since it requires to show that no such ortfolio exists. It will follow from a more general result we will rove later. The next ste is to formulate recisely what these tyes of ortfolios are that we wish to rule out. 1.3 The No-Arbitrage Axiom Informally, arbitrage is the ability to make money from nothing. In an earlier discussion we defined arbitrage as the ability to access a non-negative stream of cash flows at least one of which was ositive. This definition can be viewed as deterministic arbitrage. The examle of arbitrage resented in the discussion above is not of this form, because one cannot guarantee a at least one ositive cash flow. In one state of the world, all the cashflows were negative. In the other, they were all non-negative with at least one ositive. It turns out that one construct a ortfolio in the above examle which guarantees at least one ositive cash flow with all others being non-negative, and hence we have essentially deterministic arbitrage. Exercise 1.2 Construct a ortfolio in the above examle in which all states of the world lead to a stream of cash flows in which all non-negative and at least one of which is ositive. In general one cannot guarantee the existence of a ortfolio as in the revious exercise and have there are two different tyes of robabilistic arbitrage that we would like to define. Definition 1.1 (Tye I Arbitrage) An arbitrage oortunity of tye I exists if and only if there exists a ortfolio Θ such that θ T S(0) 0 and θ T Z(T) 0

5 1.3 The No-Arbitrage Axiom 5 Tye I arbitrage is robabilistic notion of arbitrage in the sense that one is not guaranteed free money. The cost of the ortfolio at time 0 is θ T S(0) 0. If this is 0, then it costs you nothing to own this ortfolio. On the other hand the value of the ortfolio in all states of the world is at least 0, with at least one state having value strictly ositive. Thus this tye of arbitrage gives the owner of the arbitrage ortfolio θ the ossibility of money at no or negative cost. The mere ossibility of free money alone should create disequilibrium, i.e. any rational individual should want to consume an infinite amount of such a ortfolio. Definition 1.2 (Tye II Arbitrage) An arbitrage oortunity of tye II exists if and only if there exists a ortfolio Θ such that Θ T S(0) < 0 and Θ T Z(T) > = 0 This kind of arbitrage ortfolio has strictly negative cost, i.e., to own this ortfolio, you get money (ositive cashflow). And in every future state of the world, your cash flow is non-negative. This tye of arbitrage ortfolio is the generalization of our revious notion of deterministic arbitrage and is the guarantee of free money. Certainly any rational individual would want to consume an unlimited amount of such a ortfolio. An economy in equilibrium cannot sustain either tye I or tye II arbitrage oortunities. In words, an investment that guarantees a future non-negative return, with a ossible ositive return, must have a ositive cost today, and, an investment that guarantees a non-negative return in the future must cost a non-negative amount today. Note that these two tyes of arbitrage oortunities are distinct in that the existence of tye I arbitrage does not in general imly the existence of tye II, and the existence of tye II in general does not imly the existence of tye I. Exercise 1.3 Give an examle economy in which there is a Tye I arbitrage oortunity but no Tye II arbitrage oortunity. Similarily, give an economy in which there is a Tye II arbitrage oortunity but no Tye I arbitrage oortunity. Exercise 1.4 If there exists a risk free asset with ositive value at t = 0 and t = T, show that any Tye II arbitrage ortfolio can be converted into a Tye I arbitrage ortfolio. Hint: consider adding a suitable amount of the risk free asset to the ortfolio]

6 1.4 The Positive Suorting Price Theorem 6 The revious exercise shows that when there is a risk free asset, the existence of Tye II arbitrage imlies the existence of Tye I arbitrage. Axiom 1.3 (The No Arbitrage Axiom) An economy in equilibrium cannot have arbitrage of Tye I or Tye II. 1.4 The Positive Suorting Price Theorem We are now ready for the main theorem, erhas one of the most fundamental theorems in the arbitrage ricing theory. The statement of the theorem may aear a little abstract, but as we shall soon see, it essentially tells us that we can rice an instrument at t = 0 (i.e., today) by taking an exectation over its future values at t = T (i.e., tomorrow). Theorem 1.4 (Positive suorting rice) There do not exist arbitrage oortunities of tye I or tye II if and only if There exists ψ > 0 such that S(0) = Z(T)ψ, or more exlicitly, ψ > 0 such that S 1 Z 11 Z Z 1K ψ 1 S 2. = Z 21 Z Z 2K ψ ψ i > 0 Z N1 Z N2... Z NK ψ K S N In the theorem, S refers to t = 0 and Z refers to t = T. Since this is the central theorem, we rovide an elementary roof using some fundamental tools from searating lane theory for convex sets. Before we move on to the roof, we elaborate on the imortant oints in the theorem, and examine some of the consequences of this theorem. First note that the statement of the theorem is an if and only if statement, thus if we find a ψ > 0 such that S = Zψ, then there is no arbitrage, and no arbitrage imlies the existence of such a ψ > 0. Second note that every comonent of ψ should be strictly ositive. Third, and most imortant, note that in the statement of the theorem, no where do the real world robabilities P come in. The beauty of this theorem is that it induces a linear relationshi (with ositive coefficients) between the current rices and the ossible future rices. In articular, if some subset of todays rices are not known, this linear relationshi may be sufficient to determine them assuming there is no arbitrage. Lets see the alication of this theorem to determine such rices in our simle stock-bond economy. it states that if there is no arbitrage in our little economy, then there exists ψ > 0 such that B(0) B(T) B(T) C(0) 0 50 [ ψ1 ψ 2 ],

7 1.4 The Positive Suorting Price Theorem 7 or that D(T) = (ψ 1 +ψ 2 ), 100 = 100ψ ψ 2, C(0) = 50ψ 2. Notice that the first two equations give us two equations in two unknowns, ψ 1,ψ 2. Solving for ψ 1,ψ 2 we find that ψ 1 = 3(D(T) 2/3), ψ 2 = 2(1 D(T)). For ψ > 0, it must therefore be that 2 < D(T) < 1. In this case C(0) = 100(1 D(T)), and 3 we have effortlessly reroduced all the results we ainstakingly derived before. In articular, because the theorem is an if and only if theorem, we have that if 2 < D(T) < 1 and 3 C(0) = 100(1 D(T)), then for ψ > 0 as secified above, S = Zψ and so there is no arbitrage. In general, if N > K, then there are fewer future states of the world than there are instruments. In that case, if K of the instruments have known rices today, then these K instruments can be used to extract the suorting vector ψ as follows. For concreteness suose it is the first K instruments whose rices today are known, and let S K be the vector consisting of the first K rices today. Similarly let Z K be the first K rows of the Z matrix, which are the known rices of the instruments in the future. Then, ψ = Z 1 K S K is the only set of rices which can satisfy the requirements of the ositive suorting rice theorem (roviding Z K is invertible). The first thing to check is that ψ > 0; if not then there is arbitrage, as there is no ψ > 0 which can satisfy S K = Z K ψ, let alone S = Zψ. If ψ > 0, then S = ZZ 1 K S K rices all the instruments today. If some of those instruments have rices already, they must agree with these rices or else there is arbitrage. If some of those instruments do not have rices, then these are the only ossible arbitrage-free rices. This formula reresents all the rices today as a function of the current and future rices of the first K instruments. Exercise 1.5 One roblem can arise in the above rescrition if the matrix Z K is not invertible. In this case some row of Z K is a linear combination of the other rows. Show that in this event, the current rice of that instrument should be this exact linear combination of the current rices of the other instruments, or else there is arbitrage. Thus, this instrument is redundant and can be removed from the economy. Hence we are justified in assuming that Z K is invertible.

8 1.4 The Positive Suorting Price Theorem 8 Exercise 1.6 [The Two Stock, Bond Economy] Proof of the Positive Suorting Price Theorem We will now give the formal roof of the main theorem. First, suose that there is a set of ositive suorting rices ψ > 0 such that S = Zψ. We will show that there is no arbitrage. Suose that there is a ortfolio θ such that θ T S 0 with θ T Z 0. In this case θ T Zψ > 0 since ψ > 0; but since S = Zψ, this means that θs > 0, contradicting θ T S 0, and so there cannot be a Tye I arbitrage. Suose that there is a ortfolio θ such that θ T S < 0 with θ T Z 0. In this case θ T Zψ 0 since ψ > 0; but since S = Zψ, this means that θs 0, contradicting θ T S < 0, and so there cannot be a Tye II arbitrage. Thus, if S = Zψ for ψ > 0, then there is no arbitrage. We now consider the converse, which is the hard art of the roof. Suose that there is no arbitrage of Tye I or II. We will introduce two sets M and C as follows. Consider all ossible ortfolios, θ and to each ortfolio define the K + 1 dimensional vector x(θ) R K+1 by [ ] θt S x(θ) =. Z T θ The first comonent is the value of the ortfolio at t = 0, and the remaining comonents are the ossible future values of the ortfolio in the K ossible future states of the world. The set M is the set of all ossible vectors x(θ) induced by all ossible θ R N, M = { x(θ) R K+1 : θ R N,x(θ) = [ θt S Z T θ Since the zero ortfolio θ = 0 is a erfectly valid ortfolio, for which x(0) = 0, and so 0 M. Note that x( θ) = x(θ), and so if x M, then x M. Two imortant observations about M are that it is closed and convex. ]}. Exercise 1.7 Prove that M is closed and convex.

9 1.4 The Positive Suorting Price Theorem 9 A set is closed if its comlement is oen. A set is oen U is oen if for every x U there is a ball B small enough and centered at x such that the entire ball is in U, i.e. B U. A set U is convex if for every x,y U and any 0 α 1, αx+(1 α)y U. We define the ositive orthant cone C to be the set of vectors with all coordinates nonnegative, U = { y R K+1 : y i 0 }. Note that 0 C, and so 0 M C. Just like M, C is also closed and convex. Exercise 1.8 Prove that C is closed and convex. The no arbitrage condition is equivalent to the condition that the only intersection oint of C and M is 0. Lemma 1.5 There is no arbitrage of Tyes I and II if and only if the only intersection oint of C and M is 0. The roof of this lemma is relatively straightforward, and we leave it as an exercise. Essentially the oints in M which are in the ositive orthant are arbitrage oortunities, and since there are no such oortunities, there must be no oints in M which are in C\{0}. Exercise 1.9 Prove Lemma 1.5. Thus, M and C are closed and convex and intersect only at the oint 0. We now invoke a very owerful theorem from linear rogramming, a searating hyerlane theorem for convex sets, [?]. Lemma 1.6 (Searating Hyerlane Lemma) Let M, C be two closed convex sets having a single oint of intersection x 0. Then there is a searating hyerlane (w 0,w) assing through which searates M and C. Secifically, x M w 0 +w T x 0, y C \x 0 w 0 +w T y > 0.

10 1.5 The Equivalent Martingale (Risk Neutral) Measure 10 The roof of this lemma is beyond the scoe of our discussion, however, we give a ictorial roof of the lemma below, M C Since 0 M, we conclude that w 0 0. Thus, w 0 because otherwise w 0 +w T y 0 for all y C, which is a contradiction. Suose that w 0 < 0, then choose a y C \ 0 such that y < w 0. In this case w wt y w y < w 0 from which we deduce that w T y < w 0. Hence w 0 +w T y < 0 which is a contradiction, hence the only ossibility is that w 0 = 0, from which we have that for some w R K+1, x M w T x 0, y C \x 0 w T y > 0. We now argue that w > 0. Suose, to the contrary, that some w i 0, and select y such that y i > 0 and y j = 0 for j i. Then y C \ 0 and w T y = 0 which is a contradiction, hence w > 0. We now claim that for all x M,w T x = 0. Indeed, suose to the contrary that for some x, w T x < 0. Then, since x M, and w T x > 0 which is a contradiction. Hence x M, w T x = 0. Let w = [β 0,β] T. Then the statement w T x = 0 for all x M is equivalent to the statement Let ψ = β/β 0, then since w > 0, ψ > 0, and θ R N θ T Sβ 0 +θ T Zβ = 0. θ R N θ T (Zψ S) = 0. Since the above equation holds for all θ R N, i.e., Zψ S is orthogonal to every vector in R N, it must be that Zψ S = 0, concluding the roof of the theorem. 1.5 The Equivalent Martingale (Risk Neutral) Measure The essential content of the ositive suorting rice theorem is that if there is no arbitrage, then a ositive rice vector ψ, exists such that the rices today are given by a weighted

11 1.5 The Equivalent Martingale (Risk Neutral) Measure 11 sum of the rices tomorrow, where the weightings are given by the rice vector ψ. The weightings are the same for each instrument. This looks susiciously like an exectation over the ossible rices tomorrow taken over the distribution ψ. To fully areciate this, we need to massage the theorem into a more suitable form. First, notice that only the relative rices should matter. To this end, we ick an instrument with resect to which we rice all other instruments. This instrument is tyically called the numeraire, which for our uroses is just a fancy word for reference instrument. Often, one chooses the numeraire to be the risk free instrument if one exists 2. Without loss of generality, assume that the first instrument is chosen as numeraire. Rewriting (1.4), we have, 1 1 S 2 /S 1. = Z 21 /Z 11. S N /S 1 Z N1 /Z 11 1 Z 22 /Z Z N2 /Z 12 1 ψ 1 Z 11 /S 1 Z 2K /Z 1K ψ 2 Z 12 /S 1.. Z NK /Z 1K ψ K Z 1K /S 1 Notice that for each instrument, we are measuring its rice in a given state relative to the rice of the numeraire in that state. Define a vector P by i = Z 1i S 1 ψ i Then, since ψ > 0, we see that P > 0. Further, from the first row in (1), i P i = 1, thus P is a robability vector. Further, the remaining rows in (1) give that (1) S i S 1 = K j=1 P j Z ij Z 1j We have now arrived at an equivalent formulation of the ositive suorting rice theorem, Theorem 1.7 (Equivalent Martingale Measure) There do not exist arbitrage oortunities of tye I or II if and only if there exists a robability vector P, called an equivalent martingale measure such that S i (0) S 1 (0) = E P [ ] Si (T) S 1 (T) In other words, S i /S 1 is a martingale 3 under the measure P. The measure P is also often referred to as the risk-neutral measure, or the risk-adjusted robabilities. If there is a risk free asset, one usually chooses this as the numeraire, in which case [ ] S1 (0) S i (0) = E P S 1 (T) S i(t) = E P [D(T)S i(t)] = D(T)E P [S i(t)] 2 It should be clear that there cannot exist two risk free instruments offering different returns. 3 X is a martingale if X(0) = E[X(T)].

12 1.5 The Equivalent Martingale (Risk Neutral) Measure 12 where D(T) factors out of the exectation because the numeraire is a risk free asset and thus is constant in every state. If the numeraire is not a risk free asset, then this ste is not valid and the discount factor must remain inside the exectation. Thus, the rice of an instrument today is given by the discounted exectation of its future rice, where the exectation is with resect to the risk neutral measure, S i (0) = PV(E P [S i(t)]). We emhasize that this is a comletely natural rocess, and had we started the discussion by saying that to comute the rice of an instrument you take its exected future value and comute the resent value of this exected future value one might have robably dismissed it as obvious. However, this is not what we are saying. What we are saying is that the real world robabilities are irrelevant. There is no mention of the real world robabilities P in the theorem. The theorem says that doing the intuitive thing (resent value of exected future values) is correct for some robability measure P. This robability measure P in general has no relationshi with the real world robabilities. Also we emhasize that the rices of all instruments are obtained by takine the resent value of exected future values with resect to the same Martingale Measure P. One can view the Martingale Measure as defining some other fictitious economy in which the robabilities are P in which rices are given by resent value of future exected values. This fictitious world is usually denoted the risk neutral world for the obvious reason that risk does not seem to affect the rice. To illustrate how this theorem is alied, we continue with the examle that was resented earlier. By theorem 1.7, there is a robability vector P > 0 such that /B(0) = 100/B(T) C(0)/B(0) /B(T) 50/B(T) ] [ Pdown u The first two rows yield that P u = 2(1/D(T) 1) and P down = 1 P u, hence, because 0 < P u < 1, we see that 2/3 < D(T) < 1. The third row then yields C(0) = 100(1 D(T)). We have thus (mechanically) reroduced this result for third time now. In general, if the number of states in the economy is large, the risk neutral measure will not be unique. More generally, suose that the rank of Z is k, which reresents the number of indeendent instruments. Then, we see that if the number of states (K) in the next eriod is less than or equal to k, we can choose K linearly indeendent rows of Z to solve uniquely for the risk neutral robabilities (by matrix inversion). If all of these risk neutral robabilities are not real robabilities, then an arbitrage oortunity exists. If the risk neutral robabilities exist, then they can be used to rice all the instruments. If, on the other hand, k < K, then there are either infinitely many ossible risk neutral robabilities(in which case, there are many ricings that are consistent with no arbitrage) or there are none (in which case, there is an arbitrage oortunity). As a further note, the the risk neutral robabilities can be obtained without aying any attention to the actual robabilities with which the future states occur. One simly needs to agree on what states are ossible.

13 1.5 The Equivalent Martingale (Risk Neutral) Measure 13 Exercise 1.10 Consider the following two eriod economies. Determine (to within reasonable recision) which ones have arbitrage oortunities. If you think that an economy has an arbitrage oortunity, give the ortfolio that results in the arbitrage oortunity. If you believe that there is no arbitrage oortunity, then give the risk neutral robabilities (or the Martingale measure). Note, sometimes the risk neutral robabilities may not be unique. In all cases, justify your answer. We use S to denote the current instrument rices, and Z to denote the future instrument rice matrix. (a) (i) S 1 = (ii) S 2 = Z 1 = Z 2 = (b) (i) S 1 = (ii) S 2 = [ ] [ ] 8 13 Z 1 = Z 2 = [ ] [ ] Change of Numeraire In an economy where the risk neutral robabilities are well defined, they still need not be unique. Itisossiblefordifferenteoletodisagreeastothechoiceofnumeraire. Thechoice of numeraire is arbitrary, and a articular numeraire is chosen in order to make subsequent calculations easier (for examle it may be easier to derive the risk neutral measure for a articular numeraire). However, indeendent of the choice of numeraire, rovided that the correct exectation is comuted, the rices obtained for the assets at t = 0 are unique. It is ossible that the risk neutral measure is more easily obtained with a articular numeraire, but that the exectations are more conveniently comuted with resect to another numeraire.

14 1.6 Continuous State Economies 14 If this is the case, then one needs to change the risk neutral robabilities to the relevant numeraire. SuosethatwithnumeraireS i onehascomutedthemartingalemeasure P, but that one wishes to comute the exectations with S j as numeraire. By direct substitution into (1.7) it can be verified that the new martingale measure P that reflects this change of numeraire is given by α = S i/z iα S j /Z jα α In articular, all one needs to know are the discount factors for the two instruments in the various future states. Such issues are discussed in more detail in [?]. In general, there is a risk free asset and it is usually most convenient to choose it as the numeraire. 1.6 Continuous State Economies We have restricted our attention to a finite state economy. We will here resent a very brief discussion of the continuous state (but finite instrument) economy. More details can be found in some of the references. The future state can be indexed by a random variable s which we assume to have a robability density function π(s). Then the rice matrix at t = T becomes Z( s), a random vector. Given a ortfolio Θ, its value at t = T is a random variable ṼΘ(s) = Θ T Z(s). Then, tye I arbitrage would require the existence of a ortfolio that satisfies Θ T S(0) 0, Ṽ Θ 0, P[ṼΘ > 0] > 0 and tye II arbitrage requires the existence of a ortfolio that satisfies Θ T S(0) < 0, Ṽ Θ 0 The analog of theorem 1.4 is that under suitable regularity conditions, the absence of arbitrage oortunities is equivalent to the existence of a ositive integrable function ψ(s) such that, S(0) = ds Z(s)ψ(s), ψ(s) > 0 Further, choosing a numeraire (instrument 1) and redefining π(s) = ψ(s) Z 1 (s)/s 1 (0), we have that S1 (0) S i (0) = Z i (s) π(s) = E π [D(T)(s) Z i ] Z 1 (s) The revious section suggests an aroach to ricing which we summarize here. Let K be the number of ossible future states for the economy, and let the number of indeendent instruments be N = K +n. Without loss of generality, we assume that the dynamics of the first K (underlying) instruments are comletely secified, including their current rices. The remaining n instruments are derivatives of the first K instruments, in that their values in the ossible future states are known, given the values of the underlying instruments. It is necessary to obtain the correct rices of the n derivatives at t = 0.

15 2 Stock and Bond Binomial Tree Model Choose a numeraire (reference instrument), and call this S Solve the set of K simultaneous linear equations in the K unknowns i > 0, i = 1...K, S i K Z ij = j for i = 1,...,K S 1 Z 1j j=1 to obtain the martingale measure P. 3. Obtain the risk neutral rices of all the instruments as follows [ ] Si (T) S i (0) = S 1 (0)E P = E P S 1 (T) [D(T)S i(t)] (a) = D(T)E P [S i(t)] where D(T) = S 1 (0)/S 1 (T) and (a) only follows if S 1 is a risk free instrument. 4. These rices (if the measure exists) are unique if K > K. If such a measure cannot be found, then there exist arbitrage oortunities. The connection with Monte Carlo techniques should now be clear. It is often ossible to obtain the risk neutral measure, but due to the comlexity of derivatives, it is often not ossible to comute the desired exectation analytically. One still needs to rice the derivatives efficiently, and an ideal tool for obtaining the necessary exectation is a Monte Carlo simulation. The following examles will illustrate the technique and how Monte Carlo simulations can be useful. 2 Stock and Bond Binomial Tree Model Our little stock-bond economy which we have used all along is very simle, however a suitable generalization of it leads to a very owerful model of a stock and bond economy. Suose the economy contains a stock and a risk free asset whose value grows according to a comounding interest rate in the one time eriod model B Be rδt λ + S S λ S δt

16 2 Stock and Bond Binomial Tree Model 16 This is a two state two instrument two eriod economy with a risk free instrument, retty much the simlest non-trivial economy that we could conceive of. This economy (and its dynamics) can be fully secified by giving S(0), Z(δt) and P. In this dynamics, the stock, at time δt can exist in an u state, λ + S, or a down state, λ S, where we assume without loss of generality that λ + > λ, S(0) = [ B S ] [ ] Be rδt Be Z(δt) = rδt Sλ + Sλ Setting u and solving (1) for u, we have [ ] 1 [ 1 1 S(0) B(0) = λ + S Be rδt λ S Be rδt ] [ ], 1 P = [ ] where is the Martingale Measure. We have one non-trivial equation in one unknown, which imlies that S = e rδt ( λ + S +(1 )λ S), = erδt λ λ + λ (2) The requirement of no arbitrage gives that the stock rice today is the linear combination of two different rices tomorrow. This in an of itself is a vacuous statement until we add the additional constraint that 0 < < 1, which means that λ < e rδt. This is a very simlistic model of the stock rice, and it is only a good aroximation in the limit δt 0. We will see later that this limit is ultimately equivalent to a very oular model for the stock rice, namely the Geometric Brownian Motion. Pricing Derivatives The most imortant asect of the fictitious risk neutral (Martingale) world is that this is where we can now rice any instrument, secifically any derivative of S, as long as its values in the next time ste are known. Pricing is done by taking the resent value of the exected future value where the exectation is with resect to the risk neutral (Martingale) robability. In articular, lets consider as an examle the call otion with strike K with maturity δt. It s value at time δt is max{0,s(δt) K} = [S(δt) K] +. (We use the notation [ ] + as shorthand for taking the ositive art.) Let C(K) be the rice of this otion, then taking the resent value of the exected future value, we have for the rice C(K) = e rδt( [λ + S K] + +(1 )[λ S K] +). In general, a derivative f is a function which secifies the cashflow at the future time δt. In our case there are two ossible cashflows to secify, f(λ + S) and f(λ S). Let F be the rice of this derivative, then F == e rδt ( f(λ + S)+(1 )f(λ S)).

17 2.1 Multi-eriod Binomial Tree Dynamics 17 Exercise 2.1 What is the rice P(K) at time 0 of the ut otion with strike K and maturity δt. Exercise 2.2 [Three State Stock Dynamics.] In this dynamics, the stock can exist in an unchanged, u or down state in the next time eriod. [ ] [ B Be rδt Be S(0) = Z(δt) = rδt Be rδt ] P u P = P S Sλ + Sλ 0 Sλ 0 P d For this economy, obtain the risk neutral robabilities (how many ossibilities are there?) 2.1 Multi-eriod Binomial Tree Dynamics We now extend our analysis to the multieriod setting, in which we move forward in time using an indeendent binomial two eriod model at each time ste. We will assume that the model is stationary, so that at each time ste, the stock rice can go u by a factor λ + or down by a factor λ. This results in a recombining binomial tree model, which has very imortant consequences on the efficiency of ricing algorithms. Each time ste is indeendent in the real world, and so we have the following icture for the dynamics, which we illustrate to time 4δt.

18 2.1 Multi-eriod Binomial Tree Dynamics 18 B Be rδt Be 2rδt Be 3rδt Be 4rδt λ 4 +S λ 3 +S λ 2 +S λ 3 +λ S S λ+s λ+λ S λ 2 +λ S λ 2 +λ 2 S λ S λ+λ 2 S λ 2 S λ+λ 3 S λ 3 S δt 2δt 3δt 4δt t λ 4 S One imortant roerty of the recombining nature of the tree (more aroriately a lattice 4 ) is that its size does not grow exonentially. Exercise 2.3 For the n-ste tree, how many ossible final stock rices are there? What is the robability of a articular final stock rice? As we will rove later, the risk neutral (Martingale) world in this multieriod setting is also an indeendent binomial tree dynamics, with the robability which is exactly the same which we obtained int he two eriod binomial model. The Martingale world, illustrated below, is the exact analogue of the real world with relaced by, where = erδt λ λ + λ. 4 Such binary lattices are often used to model economies (see for examle [?]).

19 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 19 B Be rδt Be 2rδt Be 3rδt Be 4rδt λ 4 +S λ 3 +S λ 2 +S 1 λ 3 +λ S S λ+s 1 1 λ+λ S λ 2 +λ S 1 λ 2 +λ 2 S 1 λ S 1 λ+λ 2 S 1 λ 2 S 1 λ+λ 3 S 1 λ 3 S 1 δt 2δt 3δt 4δt t The general statement of the Martingale Measure theorem for ricing is that the rice of an unriced derivative can be obtained by obtaining the exected resent value of future cashflows in this fictitious risk neutral world. This fictitious risk neutral world is a binomial multieriod model with robability. To illustrate, we consider a two eriod derivative which ays $4 if the stock goes u. If the stock goes down, and then u, the cashflow is $2 and if it goes down twice the cashflow is $2 (i.e., you owe $2). The ossible cashflows are illustrated below, $4 λ 4 S $4 S $2 S 1 $2 Real World $2 1 Risk Neutral Pricing World The real world dynamics of this derivative are not relevant to the ricing. All the ricing goes on in the risk neutral world. We can comute the rice of this derivative by comuting the exected resent values of the cash flows in the risk neutral world. We obtain Price = e rδt 4+(1 ) e 2dδt 2 (1 ) 2 e 2dδt 2. The rescrition for ricing is thus relatively straightforward in the risk neutral world. In fact it is as simle as taking an exectation over discounted cash flows. There is of course a gaing hole in that we have not roved that this is the right thing to do, and that it results in the arbitrage free rice. For the moment we will just take for granted that this is the right thing to do, and exlore more generally the rocess of ricing derivatives. $2 2.2 Pricing Derivatives in the Multieriod Risk Neutral World Assumethatwehaveneriods, δt,2δt,...,nδt. Ariceathforthestockcanbereresented by a binary vector, q {0,1} n. Each 1 reresents an increase in the stock rice by a factor

20 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 20 λ + and each 0 in the ath a decrease by a factor λ. For examle, the vector q = [1,0,0,1] corresondsto the stock ath S λ + S λ + λ S λ + λ 2 S λ 2 +λ 2 S. Exercise 2.4 As a function of n, how many ossible aths are there? We may now define a derivative formally as a function which assigns each ath (there are 2 n aths) to a cashflow at time nδt. We denote a derivative function by f : {0,1} n R. Technically seaking, the derivative we considered in the revious section is not defined in this way. For examle there was a cash flow which was received at time δt if the stock went u. Thus, in general one might exect a derivative to release cashflows along a stock rice ath, as oosed to only at the end. However, thanks to the existence of a risk free instrument, we can convert any cashflows along a ath to an equivalent single cashflow at time nδt. Exercise 2.5 Consider the following two eriod derivatives, and assume the bond grows in rice at every time ste by e rδt. f 1 f 2 $4e rδt $4 S $2 S $4e rδt $2 $2 $2 Show using an arbitrage argument that the rice of these two derivatives must be the same. Hence f 1 and f 2 are essentially the same derivative, excet that in f 2 all cashflows come on the last timeste. In the revious exercise, f 2 is an examle of a ath deendent derivative different stock rice aths leading to the same final rice have different ayoffs. This is the most general

21 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 21 tye of derivative that we will consider. The Asian otion, the barrier otion, the minimum otion, the American otion, etc. are all examles of ath deendent otions. In general, thesearethehardesttyesofotionstorice, aswewillseebelow(intermsofcomutational comlexity). The revious exercise shows that the derivative f 1 is equivalent to a derivative f 2 in which all cash flows occur at the last time ste. In general this is the case, and so our notion of a derivative is general enough to accomodate intermediate cashflows. Exercise 2.6 Show that in an economy with a risk free instrument, a derivative f 1 with cashflows occuring at intermediate times during a ath is equivalent to some derivative f 2 with all cashflows occuring at the last time ste. There is one subtlety not addressed in our notion of a derivative, which is the notion of exercise, in the sense that the definition of a derivative assumes that to every ath q, a well defined redetermined cash flow f(q) can be comuted. However for an otion like an American otion, the cashflow along a articular ath will deend on when the otion is exercised, i.e., it will deend on the articular exercise strategy which is emloyed. Thus, technically seaking, an otion with an exercise asect is only a well secified derivative given a articular exercise strategy. To obtain the rice of such an otion, we may assume that the holder of the otion will exercise otimally, and so to rice something like an American otion, one needs to comute the rice of the resulting cashflow stream from otimal exercise. In such a case we have to simultaneously comute what the otimal exercise function is in order to obtain the cash flow stream. We will discuss the American otion in deth later. For the moment, we will simly focus on derivatives with redetermined cashflows. In the risk neutram framework, the ricing of an arbitrary derivative is concetually simle, given by the exectation of the discounted cashflow. secifically, for a ath q let q = i q i be the number of u moves in the ath. Then the robability of a ath in the risk neutral world is given by P[q] = q (1 ) n q. Let C(f) be the rice of the derivative f. Then C(f) = e nrδt q = e nrδt q P[q]f(q), q (1 ) n q f(q). There are several roblems with this formula, starting with the secification of the derivative function f itself. It must be secified for every ossible ath, hence there are 2 n numbers to be secified which is O(2 n ) in memory requirement. Even if the function f could be secified comactly as a function of the n inuts q, the comutation in the summation involves 2 n

22 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 22 terms which is an O(2 n ) comutational comlexity. For the general derivative, there is no way around this comutational hurdle, and so there are only two ways to roceed. The first is to restrict the class of derivatives, the second is to consider aroximate comutations of this rice. This rice can be estimated to a very high accuracy very accurately using Monte Carlo techniques based on the fact that it is the exectation of some quantity according to some well defined robability distribution. We will discuss such issues later. For the moment, we will consider simler classes of derivatives. State Deendent Derivatives. Aside from the Asian otion which is essentially ath deendent, all the other common derivatives are state deendent. For this reason, Asian otions are notoriously hard to rice. A state deendent derivative is one whose cashflow function f is a function of only the states of the stock visited along the ath. Thus, a state deendent derivative is fully secified by a cash flow function f(s,t), which secifies a cashflow f(s,t) at the time t if the stock is at rice s. The cashflow for a ath q is the sum of all the cashflows accrued over the ath. f(q) = n e (n i)rδt f(s i (q),iδt), i=1 where s i (q) is the rice at time iδt. The term e (n i)rδt is the term required to convert a cashflow at time iδt to one at time nδt. At time ste i, the ossible rices are s i {λ i S,λ + λ i 1 S,λ 2 +λ i 2 S,...,λ j +λ i j S,...,λ i +S}. In other words, s i = λ j +λ i j S, where j is the number of u moves in the ath till time i. The state of the stock is summarized in the value j, and so we will use the more comact notation f(j,i) to denote f(s(j,i),iδt) where s(j,i) = λ j +λ i j S. The state of the rocess can be reresented by the air (j,i). Let q i be the refix of the ath q u to time i, then j = q i. In this notation, we have that f(q) = n e (n i)rδt f( q i,i), i=1

23 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 23 and so for the rice of the derivative, we get that C(f) = e nrδt q q (1 ) n q f(q), = e nrδt q = q n = (a) = i=1 i=1 n q (1 ) n q e (n i)rδt f( q i,i), i=1 n q (1 ) n q e irδt f( q i,i), e irδt q where (a) follows from next exercise. j=0 i=1 q (1 ) n q f( q i,i), n i ( ) i e irδt j (1 ) i j f(j,i), j Exercise 2.7 Show that q {0,1} n q (1 ) n q g( q i ) = for any function g defined on R. i j=0 ( ) i j (1 ) i j g(j), j [Hint: Break a ath into two arts, q = [q i,q l ] and break u the sum over all aths into i+1 sums where in each sum you only consider those aths in which q i = j for j = 0,1,...,i. Show that ( ) i q (1 ) n q = j (1 ) i j. j q {0,1} n : q i =j In terms of efficiency, we see that state deendent derivatives are easy to reresent, requiring only the secification of f(j,i) for i = 1...,n and j = 0,...i which is O(n 2 ) memory. We also see that ricing a state deendent derivative only involves the comutation of a double summation which is also O(n 2 ).

24 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 24 Exercise 2.8 Comute exactly how many values of f are required to reresent a state deendent derivative. Comute exactly how many terms are in the summation for the rice of a state deendent derivative. Exercise 2.9 Assuming that basic arithmetic oerations (multilication, division, exonentiation) are constant time oerations (which is not quite true) and that f(j, i) can be accessed in constant time, construct an efficient quadratic time algorithm to comute C(f) for a state deendent derivative. Exercise 2.10 We consider an alternative derivation of the ricing formula for a state deendent derivative based on the facts that the rice is an exectation of the discounted cashflow, and that the cashflow for a ath is the sum of the cashflows along the ath. 1. Show that the rice C(f) = i=1:n e irδt c i, where c i is the exected cashflow at timeste i. [Hint: Write f(q) as i f i(q), where f i is the cashflow for ath q at time ste i, and use the fact that the exectation of the sum of the cashflows is the sum of the exected cashflows.] 2. Show that c i = i j=0 ( ) i j (1 ) i j f(j,i). j

25 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 25 Exercise 2.11 We consider the Euroean call otion with maturity nδt and strike K (a) Show that this otion is a state deendent derivative with { 0 i < n, f(j,i) = (λ j + λi j S K)+ i = n. [Hint: Assume otimal exercise.] (b) Show that the rice of this derivative is given by n ( n C(S,K,n) = e i) nrδt i=0 i (1 ) n i (λ i +λ n i S K)+. One-Shot State Deendent Derivatives By and large most derivatives are one-shot in the sense that over the ath of a stock s rice, either there will be no cash flow (for examle if an otion is never exercised), or ther will be only one cash flow (for examle after an otion is exercised, there is an immediate cashflow, and no further cashflow). For each ath, there is only one oint at which a cash flow will be realized and it is only state deendent, i.e., will equal f(j,i). This will be the first state along the ath for which the cashflow is non-zero. All aths which have the state (j,i) as their first non-zero cashflow state will realize the cashflow f(j,i) at time iδt (resent value e irδt f(j,i)) and no further cashflow. In order to secify such a derivative, it is only necessary to secify the cashflows for the set of nodes which comrise the ossible first oints in aths for which there is a non-zero cashflow. One can show that such sets of nodes are minimal cuts in the lattice and are hence of linear size, hence to secify such a derivative, one only needs to secify O(n) non-zero cashflows and examle is the Euroean otion. It is then ossible to comute the rice of such an otion in linear time. We will not elaborate more generally on algorithms for one-shot derivatives excet to consider some examles. Examle: The Euroean Call Otion As described in an earlier exercise, the rice of the Euroean call otion is given by n ( n C(S,K,n) = e i) nrδt i=0 i (1 ) n i (λ i +λ n i S K) +. Clearly there are only a linear number of terms in the summation. Denote each summand by x i = i (s i K) +, where i = ( n) i (1 ) n i, s i i = λ i +λ n i S; The first summand

26 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 26 x 0 = 0 (s 0 K) +, with 0 = (1 ) n and s 0 = λ n S. Since exonentiation can be done in logarithmic time, s 0, 0 and hence x 0 can be comuted in O(logn) time. We now show how to get (s i+1, i+1 ) from (s i, i ) in constant time. The following simle recursion which we leave as an exercise establishes this fact, Exercise 2.12 Show that s i+1 = λ + s i, λ ( )( ) n i i+1 = i, i+1 1 Exercise 2.13 Exlain why the recursion in the revious exercise imlies that the rice of the Euroean call otion can be comuted in linear time. Exercise 2.14 The rice for the algorithm obtained in the revious section results in a numerically very unstable algorithm because it invloves taking very large exonents. For examle 0 = (1 ) n which for large n will be zero to machine recision. Thus it is numerically more stable to comute log i and logs i and use these to construct the summands x i. (a) Give the aroriate recursions for log i and logs i. (b) Using the answer to the revious art, give a linear time algorithm to rice the Euroean call otion.

27 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 27 Examle: The Barrier Otion We give one final examle to illustrate some techniques for constructing linear time algorithms to rice one-shot otions The Multieriod Risk Neutral Pricing Framework We will now rove that the multieriod risk neutral world in which rices of derivatives are obtained is exactly the indeendently iterated binomial model with the robability exactly as calculated in the two eriod case. We would like to argue that the risk neutral robabilities may be treated like ordinary robabilities. As such, they may be multilied to obtain the robability of indeendent events. We illustrate with an examle using the three eriod economy. Alying the risk neutral ricing at t = 0 we see that S = e rδt (S u +S d (1 )) Alying the risk neutral ricing at t = δt we see that S u = e rδt (S uu + S ud (1 )) and S d = e rδt (S ud +S dd (1 )) giving that S = e 2rδt (S uu 2 +2S ud (1 )+S dd (1 ) 2 ) Further, using S uu = e rδt (S uuu +S uud (1 )), S ud = e rδt (S uud +S udd (1 )) and S dd = e rδt (S udd +S ddd (1 )), we find that S = e 3rδt (S uuu 3 +3S uud 2 (1 )+3S udd (1 ) 2 +S ddd (1 ) 3 ) By insecting this formula, we can ick out that u 3 = 3, u 2 d = 3 2 (1 ), ud 2 = 3 (1 ) 2 and d 3 = (1 ) 3. It is aarent that the risk neutral robabilities at the N th time eriod will be given by a binomial distribution of order N P [ S(N) = S(0)u k d N k] ( ) N = k (1 ) N K k We do not rove this here, but a roof induction should be lausible at this oint. The Euroean call otion with exiry at time T and strike at K gives the holder the right (but not the obligation) to by the stock at time t = T at the strike rice K. Clearly the value of this otion (and hence its rice) at time T is given by (S(T) K)Θ(S(T) K), where Θ(x) = 0 for x < 0 and Θ(x) = 1 otherwise. Suose that we consider the time interval [0,T] broken into N smaller intervals of size δt = T/N. If δt is small enough, it is reasonable to assume that the stock could only evolve into an u or down state during the time interval. In which case, the multi-eriod binary lattice dynamics would aly and the rice of the call otion at time t = 0, C(0), is given by taking the discounted exectation with resect to the risk neutral measure given in (2.2.1). Thus, C(0) = C(S(0),K,r,σ,µ,T) = e rt E P [(S(T) K)Θ(S(T) K)] N ( ) N (S(0)λ = e rt k k +λ N k K ) k (1 ) N K Θ ( S(0)λ k +λ N k K ) k=0

28 2.2 Pricing Derivatives in the Multieriod Risk Neutral World 28 If we insist on using a discrete time reresentation of our market, then there is no otion but to comute this exectation numerically. The natural aroach would be to use Monte Carlo simulations for more comlex dynamics. A key issue is the efficiency of the comutation. Givenµ,σ andassumingthataroxyfortheinterestrater exists, oneonlyneedstocomute a single exectation. Often µ and σ are not known. In fact, one needs to calibrate the model to the observed otion data to extract µ and σ. This usually involves an otimization in the µ and σ sace, which is extremely comutationally intensive. We note that the µ deendence has disaeared due to some lucky cancellations. We could have exected this, as the risk neutral dynamics (??) do not contain µ deendence. U to now, we have not invoked any stochastic calculus. The limit of the dynamics in (??) can be reresented by the Ito stochastic differential equation ds = rsdt+σsdw where (informally) dw is a random variable with variance dt. A standard alication of Ito s lemma (see for examle [?]) immediately yields (??), from which (??) follows by taking the risk neutral exectation. There are many aroaches to obtaining the rice of the Euroean call for the dynamics that we have chosen. For other dynamics, such closed form solutions are generally not ossible, and one can resort to obtaining the rices using Monte Carlo techniques to take risk neutral exectations. Exercise 2.15 Consider the following stock dynamics according to our binomial model with robability. t S λ + S λ S t λ 2 +S λ + λ S λ λ + S λ 2 S A bond has values B, Be r t and Be 2r t at the three times, where e r t = 5 4. There is a variant of the call otion, instrument I, with current rice C. The strike is K = S 4 and if the stock initially goes down, you must

29 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 29 exercise, getting a cashflow λ S K at time t. If the stock initially goes u, then you must wait and exercise at the end of the next eriod. The stock dynamics are given by = 1 2, λ + = 3 2, λ = 1 2. There are two ossible rices (P, P) that one could select for instrument I. P is the exected value of the discounted future cashflows according to the real dynamics,. P is the exected value of the discounted future cashflows according to the risk neutral (fictitious ricing) dynamics, = er t λ λ + λ. (a) Comute P, P. (b) If the rice is actually P, then then show that there is an arbitrage oortunity i.e., construct the arbitrage oortunity. (c) What haens to your arbitrage oortunity if the rice were P. (d) Show that if the rice is not P, then there is an arbitrage oortunity. [Hint: You may want to consider what the rice of I should be at time t if the stock went u. Use this knowledge to construct an arbitrage strategy. Note: that in multi-eriod economies, we have to generalize the concet of an arbitrage rotfolio to an arbitrage strategy, where one is allowed to change the ortfolio according to the stock ath. As long as at any time, there is no nett investment using the strategy, and the strategy always yields, at time 2 T, non-negative and sometimes ositive return, then there is arbitrage. ] 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion We now return to the binomial model with a view towards linking it to the real stock rice world in which observations can be made about the real world dynamics. These measurements should in turn imly the arameters of the binomial model, which corresonds to callibrating the binomial world dynamics to the real world dynamics. We begin by moving into an equivalent binomial model, namely the log-rice world, illustrated below, λ + S logs +logλ + S logs δt λ S δt logs +logλ

30 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 30 Note that these two models yield identical stock rice dynamics. The main difference is that the log-rice world is easier to analyze because it is additive. Imagine now iterating forward for n time stes to some time increment t = nδt. The idea is that δt is infinitessimal and will tend to zero, δt 0 with t fixed, and so n = t. At each time ste i, the log δt rice logs will change by some amount x i, so logs( t) = logs + n x i, i=1 where the x i are indeendent identically distributed random variables with { logλ + w.., x i = logλ w... Exercise 2.16 Show that E[x i ] = logλ + +()logλ, Var[x i ] = ()(logλ + logλ ) 2. Thus, logs = n i=1 x i is the sum of a large number of iid random variables each of whose mean and variance is given by the revious exercise. We will now argue informally, assuming that the conditions for the central limit theorem hold(which in reality is true). By the central limit theorem, 1 n logs = 1 n n i=1 x i converges in distribution to a Normal random variable 5 with mean E[x i ] and variance Var[x i ]. We thus have that where η has a Normal distribution, S( t) = Se η, η N(µ,σ 2 ), 5 A random variable X has a normal distribution with mean µ and variance σ 2, written X N(µ,σ 2 ) is its robability density function (df) is given by. f X (x) = 1 e (x µ)2 2σ 2 2πσ 2

31 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 31 with µ = lim n n(logλ + +()logλ ), = lim δt (logλ + +()logλ ); σ 2 = lim n()(logλ + logλ ) 2, n δt 0 t t = lim δt 0 δt ()(logλ + logλ ) 2. Clearly these exressions only make sense when these limits exist, which means that for these limits to exist, λ ± should have a articular deendence on δt in this limit. Exercise 2.17 Suose that if in the limit we require that µ µ R t and σ 2 σ 2 R t. Show that the deendence of λ ± on δt is given by λ + = e µ Rδt+σ R δt, λ = e µ Rδt σ R δt. The stock dynamics S(t+ t) = S(t)e η, where η N(µ R t,σr 2 t) is known as a Geometric Brownian Motion (GBM) with drift arameter µ R and volatility arameter σ R. We see that in the limit as δt 0 the binomial model aroaches a geometric Brownian motion. The geometric Brownian motion is a very oular model for a stock s dynamics. In articular, it imlies that the stock rices at some future time are lognormally distributed. This is a reasonable aroximation to reality when it comes to ricing, however it has been widely observed that rices have a fatter tail than the lognormal distribution. This has imortant imlications for tasks such as risk analysis, however we will continue using the geometric Brownian motion as our model of a stocks dynamics for ricing uroses. Tyically the GBM arameters of the real world s stock rice dynamics (µ R,σ R ) can be measured by observing the return time series of a stock. Assume that µ R,σ R are given. The result of the revious exercise shows how to calibrate the arameters of the real world binomial model so that in the δt 0 the real world s GBM is recovered with arameters µ R,σ R. In articular, we see that λ + = e µ Rδt+σ R δt, λ = e µ Rδt σ R δt. Note that the real world binomial model has three arameters (three degrees of freedom),,λ ±, and as can be seen in the above formulas, in the δt 0 limit, the GBM has only two

32 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 32 degrees of freedom, and so one of the binomial model s degrees of freedom is surious. Thus we are free to choose one of the arameters more or less arbitrarily, for examle we could set = 1 2. Exercise 2.18 Let µ R = 0.07 and σ R = 0.25 and set δt = Starting with an initial stock rice S 0 = 1, simulate forward to time T = 1 the stock rice for the binomial model in which we set = 1 2, reeating many times to obtain the distribution for the stock rice at time 1. Now reeat for a choice of = 2 3 and comare the resulting two distributions. [Answer: dt=0.01 =1/2 =2/3 Continuous GBM dt=0.001 =1/2 =2/3 Continuous GBM dt= =1/2 =2/3 Continuous GBM Probability Density Probability Density Probability Density S S S ] The result of the revious exercise be convincing that one arameter, for examle is redundant and could be secified arbitrarily. We now consider the risk neutral ricing world, in which λ ± are unchanged, but = (e rδt λ )/(λ + λ ) and 1 = (λ + e rδt )/(λ + λ ). We now comute the geometric Brownian motion corresonding to this risk neutral world. We need to comute We first consider µ. 1 µ = lim δt 0 δt ( logλ + +(1 )logλ ); σ 2 = lim δt (1 1 )(logλ + logλ ) 2. δt 0 1 µ = lim δt 0 δt ( log λ + +logλ ), λ 1 = lim δt 0 δt ( ) σ R δt +µ R δt σ R δt ().

33 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 33 Exercise 2.19 Show that = 1 = (r 1 2 σ2 R µ R)δt+σ R δt+o(δt 3/2 ), () σ R δt σ R δt (r 1 2 σ2 R µ R)δt+O(δt 3/2 ), () σ R δt (1 ) = σ2 R δt+o(δt3/2 ). σr 2 δt () Using the result of the revious exercise, we conclude that 1 µ = lim δt 0 δt = r 1 2 σ2 R. ( (r 1 2 σ2 R)δt+O(δt 3/2 )) Using the result of the revious exercise, we erform a similar analysis for σ 2 to obtain σ 2 = lim δt (1 1 λ + )log2, δt 0 λ = lim δt (1 1 ) σr 2δt δt 0 (), = lim δt 0 1 = σ 2 R. δt (σ2 Rδt+O(δt 3/2 )), Summary We are now ready to summarize our conclusions. Assume that in the real world, the stock dynamics is given by a geometric Brownian motion, S(t+ t) = S(t)e η R, where η R is Normally distributed, η R N(µ R,σR 2) (µ R,σR 2 ) are tyically measured from the data. This geometric Brownian motion can be reresented as the δt 0 limit of the real world binomial model, ),

34 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 34 λ + S S λ S δt in which the arameters λ ± are given by λ + = e µ Rδt+σ R δt, λ = e µ Rδt σ R δt. and is arbitrary. The risk neutral world is given by a similar dynamics, with the same arameters for λ ±, but is no longer arbitrary. Instead, = (e rδt λ )/(λ + λ ). Taking the δt 0 limit of the binomial model for the risk neutral world, we obtain the geometric Brownian motion corresonding to the risk neutral ricing world, namely S(t+ t) = S(t)e η where η N(r 1 2 σ2 R,σ2 R ). Notice that the drift in the risk neutral world is indeendent of the drift µ R in the real world. Pricing in the Risk Neutral World We may use either the binomial version of the risk neutral world or the geometric Brownian motion version of the risk neutral world for ricing. In either case, we need to comute the exectation of the discounted cashflows for a derivative. In many cases, this can be done efficiently in the binomial model, esecially for state deendent derivatives. Tyically the algorithm will be O(n 2 ). For one-shot derivatives this may often be imroved to O(n). One should take n to be as large as is numerically feasible which for state deendent derivatives in the general case is means that one cannot go much beyond n = 10,000 if the rice is required in a matter of seconds. For one-shot derivatives for which one can construct an efficient linear time algorithm, one can take n very large, even in the million range and still get a rice in a matter of seconds. For n aroaching a million, one essentially has the exact rice. In general, the rice converges to the GBM rice with an error that is O( 1). n In general in the binomial world it is not ossible to get an analytic closed form solution to the rice of a derivative. However, in the GBM risk neutral world, if the otion is simle enough, it may be ossible to get the closed form solution. We will illustrate with two examles. Note that for uroses of closed form solutions, the error function erf(x) = 2 x ds π 0 e s2 or the standard normal cumulative distribution function φ(x) = 1 x ds 2π e 1 2 s2 are considered closed form.

35 2.3 The Real World, The Risk Neutral World and Geometric Brownian Motion 35 Examle: Analytic Price of The Euroean Call Otion in the GBM Model The cashflow is given by f(s,t) = (s K) + for t = T and 0 for t < T. We need to comute the exected discounted cashflow in the risk neutral world, which is E[e rt (S(T) K) + ]. In the risk neutral world, S(T) = Se η, with η N((r 1 2 σ2 R )T,σ2 RT), so we have that C(S,K,T) = E[e rt (S(T) K) + ], = e rt dη = e rt log K S dη 1 e (η (r 1 2 σ R 2 )T)2 2σ R 2 T (Se η K) +, 2πσ 2 R T 1 e (η (r 1 2 2πσ 2 R T 2σ 2 R T σ 2 R )T)2 (Se η K). After some algebraic maniulation (left as an exercise) using changes of variables in the final integration, we arrive at the celebrated Black-Scholes formula for the rice of the Euroean call otion. Exercise 2.20 Maniulate the integral for the rice of the Euroean call otion to write it using the standard normal distribution function to obtain the Black-Scholes formula, C(S,K,T;r,σ R ) = Sφ(d 1 ) Ke rt φ(d 2 ), where d 1 = log S K +(r σ2 R )T σr 2T d 2 = log S K +(r 1 2 σ2 R ) σ 2R T. [Hint: Useachangeofvariablestou = (η (r 1 2 σ2 R )T)/σ T andsearate the integral into two terms. You will then need to comlete the square in 1 one of the terms. Also note that 2π x ds e 1 2 s2 = 1 φ(x) = φ( x). ] Black and Scholes arrived at this formula through a much more difficult ath to arbitrage free ricing using stochastic differential equations and solving the resulting PDE that the rice of the Euroean call otion should satisfy. Given the risk neutral ricing framework, our aroach is concetually much simler. Nevertheless the PDE aroach is interesting and leads to efficient algorithms, which we will discuss later. One imortant thing to notice about this ricing formula is that there is no deendence on the real world drift, only on the real world volatility. In this sense, seculative trading of otions is sometimes refered to as trading volatility if I have a better estimate of the volatility than my counterarty, then I can make money off them because I will have the more accurate rice.

36 3 The Monte Carlo Aroach to Pricing Derivatives 36 Estimating the Real World Parameters µ R,σR 2. We briefly mention how to estimate µ R and σr 2. Given a historical rice time series of a stock, S 0,S 1,S 2,...,S n at times t 0,t 1,t 2,...,t n, we consider the logarithmic differences, logs i = log S i S i 1 and the time increments t i = t i t i 1 for i = 1,...,n. We develo a maximum likelihood aroach to estimating µ R,σR 2. We have that logs i arei.i.d. randomsamleswhereeach logs i hasanormaldistribution, logs i N(µ R t i,σr 2 t i). We comute the log-likelihood as [ n ] 2 logp[{ logs i } n i=1 µ R,σR] 2 1 = log e ( logs i µ R t i) σ R 2 t i, 2πσ 2 R t i = ( i=1 1 σ 2 R ) n ( logs i µ R t i ) 2 + n t i 2 logσ2 R +constant, i=1 In order to obtain the maximum likelihood estimates of µ R,σR 2, we ma maximize the loglikelihood, which may be accomlished by setting the artial derivatives with resect to µ R and σr 2 to zero. The final result is given as an exercise. Exercise 2.21 Show that the maximum likelihood estimates of µ R and σ 2 R are given by µ R = logs n logs 0, t n t 0 σr 2 = 1 n (logs i logs i 1 ) 2 (logs n logs 0 ) 2. n t i t i 1 n(t n t 0 ) i=1 3 The Monte Carlo Aroach to Pricing Derivatives As we have seen, in some cases, it is ossible to derive the rice of a derivative analytically by taking the exectation of the discounted cashflows with resect to the risk neutral measure in the geometric brownian motion model. We gave the Euroean call otion and the barrier otion as examles. In general, howeve, this is not ossible, and Monte Carlo simulation is an alternative aroach to the roblem. The fact that the rice is the exected discounted cashflows in the risk neutral world rovides a very concetually simle algorithm for aroximating the rice by virtue of the fact that a samle average over indeendently generated samles rovides a good randomized aroximation to the exectation. Secifically, if we could generate samle aths indeendently according to the robabilities in the risk neutral world, then for a single ath, the

37 3 The Monte Carlo Aroach to Pricing Derivatives 37 cash flows can be comuted as a deterministic cash flow stream. We know how to comute the resent value of a deterministic cash flow stream. So by generating a ath we generate a samle (according to the risk neutral robabilities) of the discounted cash flows. Reeating this many times and taking the average yields a Monte Carlo estimate of the exected discounted cashflows. This Monte Carlo ricing algorithm is very general and can be alied to any derivative whose cash flow function has been secified. The high level view of the algorithm is summarized below. 1: Obtain µ R,σR 2 for the real world geometric Brownian motion. 2: Comute the Risk Neutral World dynamics. 3: for i = 1 to Naths do 4: Generate a stock rice ath i according to the Risk Neutral World dynamics. 5: Comute the discounted cash flows along ath i. 6: Udate average discounted cash flow. 7: end for 8: return Average discounted cash flow over the N aths aths. There are two ways to generate a stock rice ath. The first is to obtain the risk neutral binomialmodelforsomediscretetimesteδt. Thesecondistousetheriskneutralgeometric Brownian motion and use some time resolution t to generate stock rices where for i 1, with η i N((r 1 2 σ2 R ) t,σ2 R t). S(0),S( t),s(2 t),..., S(i t) = S((i 1) t)e η i, Exercise 3.1 Consider the following otions on a stock S (in all cases S 0 is the initial stock rice): 1. C(S 0,K,T): the Euroean Call otion with strike K and exercise time T. 2. B(S 0,K,B,T): the Barrier otion with ayoff K, barrier B and horizon T. If and when the rice hits the barrier B the holder may buy at B and sell at K. 3. A(S 0,T): the average strike Asian call otion with exiry T a Euroean call otion with strike at the average value of the stock rice over [0,T]. 4. M(S 0,T): the minimum strike call otion a Euroean call otion with strike at the minimum rice over [0,T].

38 4 Stochastic Differential Equations and Alternate Reresentations of the Geometric Brownian Motion 38 Use Monte Carlo simulation to rice the 4 otions. Assume that S 0 = 1, µ = 0.07, r = 0.03, σ = 0.2, T = 2. For each case, use each of the three modes above, and comute the rice using each of the three time discretizations, t = 0.1, 0.01, In all cases, make some intelligent choice for the number of Monte Carlo samles that you need to take to get an accurate rice. [Hint: first take a few samles to get an estimate of the variance of the Monte Carlo samle values.] (a) Comute C(1,1,2) as efficiently as you can and comare with the analytic formula. (b) Comute B(1,1,0.95,2) (c) Comute A(1, 2), for three ossible definitions of average : the harmonic, arithmetic, and geometric means. Exlain the relative ordering of these rices. (d) Comute M(1,2). 4 Stochastic Differential Equations and Alternate Reresentations of the Geometric Brownian Motion Exercise 4.1 Assume the initial stock rice is S 0 and it follows real and risk neutral dynamics given by S = µs t+σs W S = r S t+σ S W. Write a rogram that takes as inut µ, r, σ, S 0, T, T and simulates the stock rice from time 0 to T in time stes of t for the risk neutral world, using each of the following modes: (a) Binomial mode I: comute λ ± from µ,σ assuming that = 1 2, and then comuting. (b) Binomial mode II: comute λ ± from µ,σ assuming that = 2 3, and then comuting. (c) Continuous mode: using the continuous risk neutral dynamics r, σ generate at time ste t as if the discrete model were taken to the limit dt 0.

39 5 The PDE Aroach to Arbitrage Free Pricing 39 For each of the three methods, give lots of reresentative rice aths for S 0 = 1, µ = 0.07, r = 0.03, σ = 0.2, T = 2 using t = 0.1,0.01, The PDE Aroach to Arbitrage Free Pricing