On Numéraires and Growth Optimum Portfolios an expository note

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1 On Numéraires and Growth Optimum Portfolios an expository note Theo K. Dijkstra Department of Economics, Econometrics & Finance University of Groningen Groningen, The Netherlands working paper July 2013 Abstract: As is well-known, models of financial markets are free of arbitrage opportunities when and only when the price processes, with prices expressed in units of a numéraire, are martingales after a change of measure. We show that the numéraire is the growth optimum portfolio under the new measure 1. This characterizes in a very simple way the various measures used to valuate financial claims. 1. Introduction The fundamental folk theorem of the theory of finance links the notion of no-arbitrage to the existence of equivalent martingale measures: a securities market model is arbitrage free if and only if the securities price processes, normalized by a numéraire, admit an equivalent martingale measure. Uniqueness of the measure, given the numéraire, is equivalent to completeness of the market model 2. A numéraire can be chosen in many ways. The dominant choice is the money market account whose value grows with the instantaneous, short term, interest rate. The induced equivalent martingale measure is traditionally known as the risk-neutral measure. Another numéraire, a zero coupon bond maturing at a convenient date T, leads to the T -forward measure ; it is particularly useful for the valuation of contingent claims on the term structure. A numéraire that is less well-known perhaps is the growth-optimum portfolio, the GOP, that maximizes the expected value of the logarithm of final In memory of Meije Smink, [ , ], wonderful colleague, without whom no version of this paper would have been written. The GOP plays a major role in (Smink, 1995). The first version dates from December The present version is a major revision of the second one from October This result is rather easy to establish, but to the best of my knowledge it was reported first in the (1996) version of this working paper. 2 See (Duffie, 1988, 2001) or (Baxter & Rennie, 1997) for a detailed technical account; a thorough yet accessible treatment is (Wiersema, 2008). 1

2 wealth, or equivalently, the growth rate of wealth invested in the securities market. For the log-utility investor there is no need to specify the investment horizon, the optimal strategy is myopic, even for market models where the related optimal power-utility strategies are not myopic (Hakansson & Ziemba, 1995). There is also no need to change the measure when the GOP is used as a numéraire: discounting by the GOP deals with impatience and risk preferences simultaneously. See (Long, 1990) who first introduced the GOP as numéraire. Below we show essentially that every numéraire maximizes the expected capital grwoth with respect to the martingale measure it induces. The paper is structured as follows. Section 2 contains a discrete (time, state) presentation of some partly well-known results and insights central to the contingent claim valuation literature. The discrete setup allows us to develop the relevant concepts without distraction by measure theoretic technicalities. Everything is build up from sets of scenarios for asset prices, subject to no-arbitrage conditions. In section 3 we restate and partly rederive the results in the context of a continuous-time market, with formulae that are potentially useful. Section 4 concludes. 2. A crash-course in contingent claim valuation in a discrete (timestate) setting 2.1 The framework The starting point of the analysis is a finite set of scenarios, describing the possible evolutions of the prices of a finite number of financial assets through time. Each scenario is a sequence of prices indexed by discrete (calendar) time. A useful geometric device is to represent the collection of scenarios by a tree. For example: t = 0 t = 1 t = 2 scenario [ 1, 1 ] [ 1 + u, 1 + r ] [ 1 d, 1 + r ] [ (1 + u) 2, (1 + r)(1 + r u ) ] [ (1 + u)(1 d), (1 + r)(1 + ru ) ] [ (1 d)(1 + u), (1 + r)(1 + rd ) ] [ (1 d) 2, (1 + r)(1 + r d ) ] ω 1 ω 2 ω 3 ω 4 Here we have two assets that cost one euro now (t = 0). The first asset 2

3 will be called the stock. Scenario ω 2 e.g. specifies that its price increases by 100 u% in the first period, and that it decreases by 100 d% in the second period 3. The other asset is a locally risk-free asset, a money market account, or cash bond. We know now for sure that the rate of interest for the first period equals r, but that we will have to wait till t = 1 to be sure of the rate for the second period. It could be r u or r d depending on the scenario actually followed. All conceivable possibilities as seen from time zero are collected in a set Ω := {ω 1, ω 2, ω 3, ω 4 }. At time one we will know whether the true scenario belongs to {ω 1, ω 2 } or to {ω 3, ω 4 }. So we have a partition at time one, P 1, whose constituent sets are called atoms, and similarly for other points in time. For time two we have P 2 := {{ω 1 }, {ω 2 }, {ω 3 }, {ω 4 }}, when everything is revealed. Note that through time the partitions become finer: each atom from P t is the union of atoms from P t+1. In continuous (time, state)-finance the concept needs refinement for purely technical reasons, and one uses filtration instead of partition sequence, but conceptually little if anything is lost when one sticks to the partition interpretation. The peculiar structure of the tree with two assets and each atom splitting into two new atoms allows us to introduce simple claims, also known as Arrow-Debreu securities. They pay one euro when and only when at a certain date a specified atom is realized. For example, we could have at time zero a claim on (1, {ω 1, ω 2 }), that will pay one euro at time one when the stock price has gone up, and nothing otherwise. This claim s payoff can be replicated exactly by a portfolio of stocks and bonds (we have two equations for the payoff and two unknowns, the number of stocks and the number of bonds). Another simple claim on (2, {ω 2 }) e.g. would pay one euro at time two when the stock price has shown an increase followed by a decrease along its path, and nothing otherwise. This claim can be generated by a portfolio strategy in stocks and bonds. Conversely, the original assets can now be seen as portfolios of simple claims. The stock e.g. can be represented by a portfolio that contains 1+u units of a simple claim on (1, {ω 1, ω 2 }) and 1 d units of a simple claim on (1, {ω 3, ω 4 }). Alternatively, we could express it in simple claims on complete scenarios. Trees as described here are examples of market models ( complete markets ), where it is immaterial whether we analyze valuation problems in terms of assets or in terms of simple claims. So far the numbers u, d, r, r u and r d may have appeared to be arbitrary, but they are not. We do not want to value claims on the basis of a market model that allows of arbitrage, with opportunities to get something 3 The size of the move may vary with the time or the trajectory followed, but the added complexity does not seem worthwhile in this largely expository paper. 3

4 for nothing. A portfolio strategy with a positive inflow at one (date, state) and no outflows ever after should not be possible. A little reflection will show that it is sufficient and necessary to exclude arbitrage for every oneperiod submarket. This is guaranteed when and only when all simple claims have positive prices. It requires in particular that 1 d < 1 + r < 1 + u (and similarly for r u and r d ) which is eminently reasonable, since the market cannot support a stock whose random return always outperforms, or always underperforms, the known and certain return of another asset. Now it is crucial to observe that the price of a claim on a scenario is the product of the prices of the claims on its sub-scenarios. As an example, suppose it costs 0.5 euro now to receive one euro at time one when the stock price has gone up, and nothing otherwise. And let it cost 0.8 euro at time one, after an observed price rise at time one, to get one euro at time two if the stock price subsequently goes down, and nothing otherwise. Then a simple claim on (2, {ω 2 }) can be replicated by buying 0.8 of a claim on (1, {ω 1, ω 2 }). This will cost = 0.4 euro. When at time one the stock price has in fact risen, the fraction of the claim we bought yields just enough to buy the claim on the remaining part of the scenario. For further reference we record this as follows: π(ω) = π (t, A) π (ω t, A) (1) where π(ω) is the price at time zero of a claim on (T, {ω}) with T representing the final date; A, with ω A, is an atom of P t ; π (t, A) is the price at time zero of a claim on (t, A) and π (ω t, A) is the price of a claim on (T, {ω}), conditional on the occurrence of state A at time t. 2.2 A general valuation formula We will leave the simple tree and turn to general sets of scenarios. Each ω Ω is a possible evolution of a finite set of asset prices through time. At each date t we have a partition P t of Ω that collects the atoms that specify the possible trajectories of the prices up to and including time t. In the sequence (Ω, P 1, P 2,..., P T ) every partition is a refinement of its predecessor, and the last one is the finest possible, so it contains each {ω} as atoms. We assume the model is arbitrage free: every simple claim has a positive price. We also take the model to be complete, which means that for every one period submarket we have as many assets as we have states, and it is immaterial whether we use simple claims or securities for valuation purposes. Let V be a financial claim defined by its payoff V (T, ω) at time T for each scenario ω Ω. It could be the final value of a self-financing portfolio 4

5 strategy, a plain vanilla call, a lookback option on a basket et cetera. Its price at time zero, V 0, should be V 0 = ω Ω π(ω)v (T, ω) (2) since we can replicate V s payoff at time T exactly by buying V (T, ω) units of the simple claim on (T, {ω}) for every ω Ω. In general the sum of the state prices will not be one, so they cannot be interpreted as probabilities. But there are various ways of rescaling them. For example, when the market supports a cash bond or money market account we can use that { } T π(ω) (1 + r (t, ω)) = 1 (3) ω Ω t=1 where r (t, ω) is the positive interest rate on the cash bond for the period that starts at t 1 and ends at t. As in the two-period example, r (t, ω) will have the same value for every ω A P t 1, so the cash bond is locally risk free. To see that (3) is correct, observe that the lefthand side is the value of a claim now whose payoff agrees along every scenario with the payoff of a one unit investment in the cash bond. Hence it must be worth one euro in a market that is arbitrage free. Using (3) we can write V 0 as the expected value of the discounted final payoff { } V 0 = ω Ω π(ω) T (1 + r (t, ω)) t=1 V (T, ω) T (4) t=1 (1 + r (t, ω)). Here the discounting is done via the cash bond and expected refers to the artificial probabilities obtained by scaling the state prices, also using the cash bond. Note that in continuous (time, state)-space we would replace T t=1 (1 + r (t, ω)) by exp T r 0 tdt. The literature refers to the artificial probabilities as risk-neutral, apparently because under this measure every claim has an expected rate of return equal to the short term interest rate (see also below). A better name would be futures measure, since, as one may verify the expected value of any claim V (T,.) with respect to this measure [ ] T π(ω) (1 + r (t, ω)) V (T, ω) (5) ω Ω t=1 is just its futures price 4. As we will show, another characterization of these 4 For crystal clear discussions of futures and forwards see in particular (Cox & Ingersoll & Ross, 1981b). (Duffie & Stanton, 1992) extend this to continuously resettled futures options and complete the argument in continuous time. 5

6 probabilities is that the portfolio strategy that maximizes the expected growth rate is simply to invest exclusively in the money market account or cash bond. Another way to scale the state prices is to divide them by their sum. The latter is just the value of a contingent claim that pays one euro at time T no matter which scenario is followed. It is the value of a default free zero coupon bond maturing at time T. So with B(0, T ) := ω Ω π(ω) we have trivially V 0 = B(0, T ) ω Ω π(ω) V (T, ω). (6) B(0, T ) But now V 0 is the discounted expected final value as opposed to the expected discounted final value before. An appropriate name, the only serious candidate in fact, for the new measure would be T -forward measure, since the expectation π(ω) V (T, ω) (7) B(0, T ) ω Ω is the time zero forward price for V (T,.). In a world governed by these probabilities the zero coupon bond maximizes the expected growth rate. When we have reached time t in atom A of P t we know the value of V, V (t, A). It must be that V (t, A) = ω A π (ω t, A) V (T, ω) (8) = ω A π(ω) V (T, ω) (9) π(t, A) since all simple claims on scenarios that do not belong to A have become worthless, and recall (1). Now consider an arbitrary numéraire N, a portfolio containing positive real multiples of simple claims on every (T, {ω}). Its starting value N 0 is one euro. Naturally, N(t, A) = ω A π(ω) N(T, ω). (10) π(t, A) If we divide V (t, A) by N(t, A) and rearrange we obtain the valuation formula V (t, A) N(t, A) = ( ) π (ω) N(T, ω) V (T, ω) ω A ν A π (ν) N(T, ν) N(T, ω). (11) Define numéraire dependent probabilities for ω Ω: p N (ω) := π(ω) N(T, ω) (12) 6

7 (they sum to one since N 0 = 1). So the value of a claim in units of the numéraire at time t when A is the trajectory followed so far, is the conditional expectation, with respect to p N, of its final value in units of the numéraire. In other words, V/N is a p N -martingale. Clearly, at time zero we have V 0 = ω Ω p N (ω) V (T, ω) N(T, ω). (13) We close this section with two remarks. The first concerns (12). It is an utterly trivial observation that the ratio of the probabilities p N 1 (ω) p N 2 (ω) as induced by two numéraires is just the ratio N 1 (ω) N 2 (ω), but is quite useful in continuous time finance. The second remark concerns the cash bond as numéraire. Take T = t + 1. Since t+1 ( ) N(t + 1, ω) = 1 + r(s, ω) s=1 (14) is constant on atom A P t, we can write with a slight abuse of notation N(t + 1, ω) = N(t, A) (1 + r(t + 1, A)). Multiplication of both sides of (11) for T = t + 1 by N(t + 1, ω) yields V (t, A) (1 + r(t + 1, A)) = ω A ( π (ω) N(t + 1, ω) ν A π (ν) N(t + 1, ν) ) V (t + 1, ω). (15) And so with respect to the cash bond induced probabilities we can say that the expected one period return of any claim is just the short term interest rate. In a one period model this valuation can be called risk neutral. In a multi-period setting the natural risk free benchmark would appear to be the default free zero coupon bond, see also below. 2.3 The growth optimum portfolio Suppose we have one euro to invest at time zero, and want a strategy that maximizes the expected rate of growth at time T. Equivalently, the strategy should maximize the expected logarithm of final wealth. So we need to know how many simple claims, W (ω), we should buy on each (T, {ω}) so that all wealth is invested, 1 = ω Ω π(ω)w (ω), and ω Ω p(ω) log (W (ω)) is as large as possible. Here p (ω) is the true likelihood of scenario ω. It is easily checked that the optimal number, GOP (ω), of simple claims on (T, {ω}) is just GOP (ω) = p(ω) π(ω). (16) 7

8 In principle one could device a self-financing portfolio strategy in stocks and bonds with a final payoff that is identical with the GOP as a portfolio of simple claims. Here we will stick to the latter for simplicity. In continuous time it is more convenient and easier to work with market securities directly, see below. We will show the well-known fact that the growth optimum strategy is myopic in the sense that the investment horizon is irrelevant. Suppose we are in atom A of P 1. The value of the growth optimum portfolio of simple claims equals π (ω 1, A) GOP (ω) (17) ω A = π(ω) p(ω) (18) π (1, A) π(ω) ω A 1 = p (ω) (19) π (1, A) = p (1, A) π (1, A) ω A (20) where p(1, A) denotes the likelihood of atom A at time one. But this value would also have been obtained when a one period optimal growth strategy had been adopted, with p (1, A) π (1, A) claims bought at time zero on (1, A) for each atom of P 1. Observe that an investment of the amount available at time one in simple claims on (T, {ω}) for every ω A, aimed at maximization of the expected growth rate for the time remaining till T, yields p (1, A) p (ω 1, A) π(1, A) π (ω 1, a) = p(ω) π(ω) (21) claims on those scenarios. This is exactly what the T -period optimal portfolio contains at time one in atom A. Note that we do not need independent returns. Short term optimization of the expected growth rate is optimal from a long term perspective also. This property may very well characterize the logarithm 5. 5 Let υ be the inverse of the derivative of a utility function. When long term expected utility optimization yields the same result as short term optimization, we must have ( ) ( ) π(ω) π (1, A) π(ω)υ λ T = π (1, A) υ λ 1 (22) p(ω) p(1, A) ω A where λ 1 and λ T are Lagrange multipliers for the all wealth is invested -constraint. For the log-utility the λ s equal one and υ (x) equals 1/x, verifying (22). The log would appear to be necessary as well. 8

9 If we use the GOP as a numéraire then the price of a claim is the expected final value with respect to the true probabilities in units of the GOP: V 0 = ω Ω π(ω)v (T, ω) (23) = V (T, ω) π(ω)gop (ω) GOP (ω) ω Ω (24) = V (T, ω) p(ω) GOP (ω). ω Ω (25) Every other numéraire will require a change of measure. However, after the change the numéraire has become the GOP with respect to the new probabilities: the growth optimum portfolio requires us to buy p N (ω) π(ω) simple claims on (T, {ω}) for every ω, but since p N (ω) = π(ω) N (T, ω) the number of simple claims is precisely N (T, ω). 3. Stochastic calculus and growth-optimum portfolios 3.1 A market model and the GOP Here we will consider a securities market model in continuous time with n + 1 price processes of tradable assets, a cash bond and n securities (stocks and bonds, possibly). This context allows the derivation of explicit formulae for growth-optimum portfolios and contingent claim valuation. We will not pay tribute to the customary formalities and regularity conditions, but refer to any of the standard books (see e.g. (Duffie, 1988 or 2001), (Baxter & Rennie, 1997) or (Wiersema, 2008)). We have the following stochastic differential equation on [0, T ] ds 0,t = r t S 0,t dt (26) ds i,t = S i,t [µ i,t dt + σ i.,t dz t ], i = 1, 2,..., n. (27) Here S 0 refers to the price of the cash bond. Z = [Z 1, Z 2,..., Z n ] is n- dimensional Brownian Motion. The short term interest rate r t, the relative drift rates µ i,t and the relative volatility matrix σ t with i th row σ i.,t may all depend on the trajectory of the prices up to and including time t 6. The inverse of σ t is assumed to exist always and everywhere. Basically, it is 6 The wording is mathematically a bit sloppy, to enhance readability. I know this statement is an anathema to mathematicians, but I believe no harm is done, and the formulae to be derived do have some instrumental value. 9

10 assumed that the market model is arbitrage free as well as complete. As before we ignore dividends. For the sake of completeness we will derive in the customary heuristic way the well-known growth optimum portfolio. Let wealth now, W 0, be one euro, and let x be any suitable n-dimensional weight process (x t may depend on the trajectory of the prices up to and including time t, but obviously not beyond). Wealth, invested fully in a self-financing portfolio of the n+1 assets in the market, changes by [ n ( ) ] ds i,t n dw t = W t x i,t + r t 1 x i,t dt. (28) S i,t With Ito s lemma we find log W T = i=1 T + 0 T 0 i=1 [r t + x t (µ t r t ) 12 x t σ t σ t x t ] dt (29) x t σ t dz t. (30) Since the expected value of the term with the Brownian Motion is zero 7, and the integration with respect to time does not involve wealth, the weights of the growth-optimum portfolio maximizes the quadratic form in square brackets. The optimal x t is therefore Define the market price of risk (σ t σ t ) 1 (µ t r t ). (31) φ t := σ 1 t (µ t r t ). (32) For the value of the optimally invested wealth at time t, denoted by GOP t one easily finds [ t GOP t = exp (r s + 12 ) t ] φ sφ s ds + φ sdz s. (33) We also note that 0 dgop t = GOP t [(r t + φ t φ t ) dt + φ t dz t ] (34) which expresses the link between excess return and relative volatility for the growth optimum portfolio. 7 T E 0 0 x t σ t dz t = T 0 E 0x t σ t dz t = T 0 E 0E t x t σ t dz t = 0 since E t x t σ t dz t = x t σ t E t dz t = x t σ t

11 We will verify that the prices in units of the GOP are martingales. To this end consider first a numéraire N, a strictly positive price process with N t determined by the history of the securities price processes up to and including time t, satisfying dn t = N t (υ t dt + τ t dz t ). (35) N 0 = 1, as always. With Ito s lemma we have for an arbitrary security price process S i d ( ) S i = ds i dn S i N ds ( ) 2 i dn dn S i N + (36) N N S i N = (µ i υ σ i. τ + τ τ) dt + (σ i. τ ) dz. (37) If N = GOP then υ = r + φ φ and τ = φ. This implies for the cash bond with µ 0 := r and σ 0. := 0 d ( ) S 0 GOP) = (r r φ φ 0 + φ φ) φ dz = φ dz (38) ( S0 GOP and for i = 1, 2,..., n d ( ) Si GOP) = ( µ i r σ i. σ 1 (µ r) ) dt + (σ i. φ ) dz (39) ( Si GOP = (µ i r µ i + r) dt + (σ i. φ ) dz (40) = (σ i. φ ) dz. (41) And so the price processes normalized by the GOP are indeed martingales with respect to the true measure. Consequently, for any contingent claim, with price process V, that can be replicated by a self-financing portfolio strategy, we have V t V T = E t. (42) GOP t GOP T This equation was presented before in (Merton, 1990), (Long, 1990) and (Johnson, 1996). It is worthwhile to note that the GOP is the only numéraire with this property. For a general N it is clear that the normalized cash bond price process will have zero drift rate when and only when υ = r + τ τ. And the drift rates of the other processes are zero when and only when µ = r+στ. We must therefore have τ = φ. For an arbitrary numéraire the general theory of contingent claim valuation says that the normalized price processes can be turned into martingales by changing the probability measure. The new equivalent measure induces a 11

12 change in the drift of the Brownian Motion Z. I.e. there exists an essentially unique process α, say, of the same dimension as Z with α t determined by the price processes up to and including time t, such that dz t α t dt is Brownian Motion with respect to the new measure. If we denote the new relative drift rate of the securities by µ N and the new relative drift rate of N by υ N we have µ N = r + στ (43) υ N = r + τ τ. (44) And α = τ φ. These results follow from the fact that N is the GOP relative to the new measure. To prove the last statement, let V be the price process of any self-financing portfolio (V 0 = 1). Recall that for positive real x we have that log (x) x 1 with equality when and only when x = 1, and so ( ) ( ) VT VT log + 1 (45) N T is a non-positive random variable. Its non-positive expectation with respect to the new equivalent martingale measure, indicated by the superscript N, equals ( E (log N VT N ( ) T = E N VT log N T N T ) ( VT N T ) ) + 1 (46) V 0 N (47) = E N log V T E N log N T 0. (48) In other words, E N log V T V T = N T ( almost surely ). is always smaller than E N log N T, except when 3.2 Three possibly well-known facts that may be of some interest to recall First, observe that the price of a zero coupon bond at time t that matures at time T, B(t, T ), must satisfy B(t, T ) GOP t = E t 1 GOP T. (49) 12

13 So for a general contingent claim with price process V we have V t V T = E t (50) GOP t GOP T ( 1 = E t GOP T = B(t, T ) GOP t ) ) 1 E t (V T ) + cov t (V T, GOP ( T 1 E t (V T ) + cov t V T, GOP T (51) ). (52) Or ) 1 V t = B(t, T ) E t (V T ) + GOP t cov t (V T,. (53) GOP T In other words, the value of a claim equals its discounted expected payoff with regard to the true probablity measure, plus a hedging premium 8. The latter is positive when the claim tends to do well when the GOP does not. The hedging premium is absent for all claims when and only when taking expectations yields in fact forward prices. In that world, the zero coupon bond is the best capital growth investment. A second observation is related to the expected excess return that is earned by an arbitrary claim. Let dc = C (µ c dt + σ c dz). (54) Since C normalized by the GOP must be a martingale relative to the true measure, the relative drift rate of C GOP must be zero. Hence, as is easily verified, the excess expected return of any claim in a market which is arbitrage free, is not free, but is determined by its relative volatility and the market prices of risk: µ c r = σ c σ 1 (µ r). (55) Finally, we recall that CAPM is alive and kicking in terms of the GOP, as observed by (Merton, 1990) and analyzed extensively by (Johnson, 1996). It is in fact a simple identity µ t = r t + beta t (S, GOP ) E t ( dgopt GOP t ) r t dt (56) when we define, with ds as the n-vector with S ith element ds i S i for i = 1, 2,..., n ( ds beta t (S, GOP ) := cov t S, dgop ) ( ) t dgopt var t. (57) GOP t GOP t 8 To the best of my knowledge this easy result was reported first in the 1996 version of this working paper. See (Korn & Schäl, 2009) who refer to this as (Dijkstra, 1998) where the year 1998 is a mistake. 13

14 Of course, the original formulation is in terms of the market portfolio of risky assets, and that works also provided the market maximizes the expected growth rate. If we define for convenience the market portfolio price process as M := n i=1 S i then its relative volatility must be σ S/M, and so the expected return of the risky assets S relative to the martingale measure induced by M is just µ M := r + σσ S/M. (58) 3.3 An application 9 Consider the option to swap the second security for the first at time T. Both securities are assumed here to be strictly positive always. The payoff of the option is max (S 1,T S 2,T, 0) = (S 1,T S 2,T ) 1 (S1,T S 2,T). (59) The price of the option at time zero equals { } { } V 0 := E 1 S1,T 1 N (S1,T S 1,T 2,T) E 2 S2,T 1 N (S1,T S 2,T 2,T) (60) where we are free to choose the numéraires (E s superscript indicates the martingale measure induced by the numéraire). Choices that present themselves are N 1,t := S 1,t S 1,0 and N 2,t := S 2,t S 2,0 that make the expectations simple probabilities. We obtain V 0 = Q 1 S 1,0 Q 2 S 2,0 (61) where Q i is the chance as evaluated at time zero that the option ends in the money at time T in a world where the i th asset is the best capital growth investment. Since the option is linearly homogeneous in (S 1, S 2 ) the probabilities are deltas : Q 1 = V/ S 1 and Q 2 = V/ S 2, both evaluated at time zero. Note that with respect to the martingale measure induced by N 1 ds i = S i [r + σ i. σ 1.] dt + S i σ i. dz 1 (62) 9 The field of interest rate options is particularly rich in apllications. An important case in point is the seminal (Vasicek, 1977) who values bonds using a numéraire portfolio of a long term zero coupon bond and the cash bond, relative to the true measure. The numéraire is in fact the GOP in his setup, but apparently was not recognized as such. 14

15 for i = 1, 2,..., n and analogously for N 2 (a superscript indicates the relevant Brownian Motion of the martingale measure). Of course one would calculate Q 1 using ( ) ( ) S2 S2 d = (σ 2. σ 1. ) dz 1 (63) S 1 S 1 and similarly for Q 2. With a bit of effort one can extend the approach to options where the third security is swapped for the best of the first two, with payoff max (max (S 1,T, S 2,T ) S 3,T, 0) (64) et cetera. For more (and perhaps more realistic) examples see (Wiersema, 2008) or (Brockhaus et al, 1999). 3.4 Risk-neutral probabilities, an additional remark When we use the cash bond as numéraire the expected short term return of any claim with respect to the induced martingale measure is the instantaneous interest rate. Risk neutral investors would then be indifferent between the locally risk free cash bond and all other financial claims and assets. Since in a multi-period or dynamic setting with random interest rates risk does not add but compounds (Cox & Ingersoll & Ross, 1981a), it is not evident that investors would remain indifferent. In fact one could argue that in equilibrium the probability measure that deserves the adjective risk-neutral is the one that yields forward prices for expected future values instead of futures prices. A heuristic argument that might support this position is based on a slight adaptation of (Bick, 1990). Here one assumes the existence of a representative agent/investor with a strictly increasing utility function U, and market equilibrium. For any contingent claim on the market portfolio M T in zero net supply with price C t, we must have [ U d dε E t ( M T + ε (C T C t exp when evaluated at ε = 0, or ( ( T E t [U (M T ) C T C t exp t ( T t ))] r s ds = 0 (65) ))] r s ds = 0. (66) This follows since the representative investor holds the market and does not trade. Adding ε non-zero units of a claim on the market, financed by an 15

16 investment in the cash bond is not optimal, hence the condition. Rewriting it yields E t [U (M T ) C T ] C t = ( )] E t [U T (67) (M T ) exp r t s ds [ ] U (M T ) = B(t, T ) E t E t U (M T ) C T. (68) (Take C T = 1, identically, to get B(t, T )). If the representative investor is risk neutral, so that marginal utility is a positive constant, then C t = B(t, T ) E t (C T ) (69) and the corresponding probability measure is the forward measure (E t (C T ) is the T -forward price of C T at time t). 4. Conclusion A shortcoming of this paper under many perhaps is that dividends are ignored. But that does not seem to be a major issue. We also gladly overlooked transaction costs, an issue of some relevance in practice. We assumed market completeness, and are happy to refer to (Korn & Schäl, 2009) for a detailed, subtle and complete analysis in a discrete time setting. We also ignored model and parameter uncertainty. When the main task is the valuation of contingent claims, consistent with other market prices, in the presence of a highly liquid market in standard options, this is perhaps also tolerable. When however one has to construct optimal portfolio strategies, it is of paramount importance to take it into account. A bit of reflection on this issue inevitably makes the paper end on a sombre note. (Barberis, 2000) has shown how dramatically large the impact can be on portfolio advice of a proper incorporation of parameter uncertainty. His approach has been extended in many directions, including non-stationary dynamics and both model and parameter uncertainty by (Brandt & Santa-Clara & Stroud, 2005). Nevertheless, as pointed out emphatically by (Vickers, 1994), time is never really present in economic models, in spite of its nominal appearance, because of its inherently open nature. Roughly, all models, no matter how sophisticated just extrapolate from the past. The economic events of the last 25 years or so, the last five years in particular, offer spectacular material for contemplation on the unpredictability of economic and political events. The interaction between events, perceptions, expectations and explanatory stories may make the generation of scenarios with a fair chance that one 16

17 of them is close to the real unfolding of events virtually impossible 10. For support and an elaboration of this view, see (Shiller, 2005) and (Akerlof & Shiller, 2010); of course, (Keynes, 1936), in particular chapter 12 on the state of the long-term expectation, has not lost its relevance at all. References [1] Akerlof, G. A. & Shiller, R. J. (2010). Animal Spirits. Princeton University Press, Princeton [2] Baxter, M. & Rennie, A. (1997). Financial Calculus. Cambridge University Press, Cambridge [3] Bick, A. (1990). On Viable Diffusion Price Processes of the Market Portfolio. Journal of Finance, 45(2), [4] Brandt, M. W. & Goyal, A. & Santa-Clara, P. & Stroud, J. R. (2005). A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning about Return Predictability. The Review of Financial Studies, 18(3), [5] Brockhaus, O. & Ferraris, A. & Gallus, C. & Long, D. & Martin, R. & Overhaus, M. (1999). Modelling and Hedging Equity Derivatives. Risk Books, London [6] Cox, J. & Ingersoll, J. & Ross, S. (1981a). A Re-examination of Traditional Hypotheses about the Term Structure of Interest Rates. Journal of Finance, 36(4), [7] Cox, J. & Ingersoll, J. & Ross, S. (1981b). The Relation between Forward Prices and Futures Prices. The Journal of Financial Economics, 9, [8] Duffie, D. (1988). Security Markets. Academic Press, San Diego Ca. [9] Duffie, D. & Stanton, R. (1992). Pricing Continuously Resettled Contingent Claims. Journal of Economic Dynamics and Control, 16, [10] Duffie, D. (2001). Dynamic Asset Pricing Theory (third ed.). Princeton University Press, Princeton 10 The second version of this paper, of 1999, offered the same lament. Subsequent events have not softened this scepticism. 17

18 [11] Hakansson, N. H. & Ziemba, W. T. (1995). Capital Growth Theory. In: Jarrow, R. & Maksimovic, V. & Ziemba, W. T. (eds.). Handbook in Operations Research & Management Science, volume 9, Finance, chapter 3, Elsevier North-Holland, Amsterdam [12] Johnson, B. E. (1996). The Pricing Property of the Optimal Growth Portfolio: Extensions and Applications. Department of Engineering- Economic Systems, Stanford University [13] Keynes, J. M. (1936). The General Theory of Employment, Interest and Money. Cambridge University Press, Cambridge [14] Korn, R. & Schäl, M. (2009). The numéraire portfolio in discrete time: existence, related concepts and applications. In: Albrecher, H. & Runggaldier, W. J. & Schachermayer, W. (eds.). Advanced Financial Modelling. Radon Series Comp. Appl. Math., 8, de Gruyter, Berlin [15] Long, J. B. Jr. (1990). The numéraire portfolio. Journal of Financial Economics, 26, [16] Merton, R. (1990). Continuous-Time Finance. Basil Blackwell, Oxford [17] Shiller, R. J. (2005). Irrational Exuberance. Currency Doubleday, New York [18] Smink, M. (1995). Asset Liability Management in Life Insurance. PhD thesis, University of Groningen, The Netherlands [19] Vickers, D. (1994). Economics and the Antagonism of Time. The University of Michigan, Ann Arbor [20] Wiersema, U. F. (2008). Brownian Motion Calculus. John Wiley, 18