On Numéraires and Growth Optimum Portfolios an expository note


 Shavonne Jefferson
 1 years ago
 Views:
Transcription
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 wellknown, 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 noarbitrage 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 riskneutral 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 wellknown perhaps is the growthoptimum 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 logutility investor there is no need to specify the investment horizon, the optimal strategy is myopic, even for market models where the related optimal powerutility 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 wellknown 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 noarbitrage conditions. In section 3 we restate and partly rederive the results in the context of a continuoustime market, with formulae that are potentially useful. Section 4 concludes. 2. A crashcourse 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 riskfree 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 ArrowDebreu 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 subscenarios. 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 selffinancing 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 twoperiod 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 riskneutral, 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 multiperiod 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 selffinancing 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 wellknown 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 logutility 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 growthoptimum 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 growthoptimum 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 wellknown growth optimum portfolio. Let wealth now, W 0, be one euro, and let x be any suitable ndimensional 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 selffinancing 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 growthoptimum 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 selffinancing 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 selffinancing 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 nonpositive random variable. Its nonpositive 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 wellknown 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 nvector 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 Riskneutral 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 multiperiod 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 riskneutral 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 nonzero 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 nonstationary dynamics and both model and parameter uncertainty by (Brandt & SantaClara & 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 longterm 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. & SantaClara, 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 Reexamination 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 NorthHolland, 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). ContinuousTime 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
The BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationIntroduction to Mathematical Finance
Introduction to Mathematical Finance Martin Baxter Barcelona 11 December 2007 1 Contents Financial markets and derivatives Basic derivative pricing and hedging Advanced derivatives 2 Banking Retail banking
More informationOptions pricing in discrete systems
UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationFour Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationINTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE
INTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE J. DAVID CUMMINS, KRISTIAN R. MILTERSEN, AND SVEINARNE PERSSON Abstract. Interest rate guarantees seem to be included in life insurance
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationPutCall Parity. chris bemis
PutCall Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationLECTURE 10.1 Default risk in Merton s model
LECTURE 10.1 Default risk in Merton s model Adriana Breccia March 12, 2012 1 1 MERTON S MODEL 1.1 Introduction Credit risk is the risk of suffering a financial loss due to the decline in the creditworthiness
More informationLecture 4: The BlackScholes model
OPTIONS and FUTURES Lecture 4: The BlackScholes model Philip H. Dybvig Washington University in Saint Louis BlackScholes option pricing model Lognormal price process Call price Put price Using BlackScholes
More informationMidterm Exam:Answer Sheet
Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a riskfree and risky asset, with returns given by r f and E(r p ), respectively. Let y be the
More informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationStocks paying discrete dividends: modelling and option pricing
Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the BlackScholes model, any dividends on stocks are paid continuously, but in reality dividends
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationUnderstanding N(d 1 ) and N(d 2 ): RiskAdjusted Probabilities in the BlackScholes Model 1
Understanding N(d 1 ) and N(d 2 ): RiskAdjusted Probabilities in the BlackScholes Model 1 Lars Tyge Nielsen INSEAD Boulevard de Constance 77305 Fontainebleau Cedex France Email: nielsen@freiba51 October
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationThe Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees
The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow HeriotWatt University, Edinburgh (joint work with Mark Willder) Marketconsistent
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationAn Introduction to Modeling Stock Price Returns With a View Towards Option Pricing
An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted
More informationForwards, Swaps and Futures
IEOR E4706: Financial Engineering: DiscreteTime Models c 2010 by Martin Haugh Forwards, Swaps and Futures These notes 1 introduce forwards, swaps and futures, and the basic mechanics of their associated
More informationForward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.
Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero
More informationAn Introduction to Exotic Options
An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationOptimal proportional reinsurance and dividend payout for insurance companies with switching reserves
Optimal proportional reinsurance and dividend payout for insurance companies with switching reserves Abstract: This paper presents a model for an insurance company that controls its risk and dividend
More informationFair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Fair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationThe Valuation of Currency Options
The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationThe Intuition Behind Option Valuation: A Teaching Note
The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationThe Discrete Binomial Model for Option Pricing
The Discrete Binomial Model for Option Pricing Rebecca Stockbridge Program in Applied Mathematics University of Arizona May 4, 2008 Abstract This paper introduces the notion of option pricing in the context
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationRisk/Arbitrage Strategies: An Application to Stock Option Portfolio Management
Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and HansFredo List Swiss Reinsurance Company Mythenquai 50/60, CH8022
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationt = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3
MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate
More informationCFA Examination PORTFOLIO MANAGEMENT Page 1 of 6
PORTFOLIO MANAGEMENT A. INTRODUCTION RETURN AS A RANDOM VARIABLE E(R) = the return around which the probability distribution is centered: the expected value or mean of the probability distribution of possible
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationOptions 1 OPTIONS. Introduction
Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or
More informationAmerican Capped Call Options on DividendPaying Assets
American Capped Call Options on DividendPaying Assets Mark Broadie Columbia University Jerome Detemple McGill University and CIRANO This article addresses the problem of valuing American call options
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationASimpleMarketModel. 2.1 Model Assumptions. Assumption 2.1 (Two trading dates)
2 ASimpleMarketModel In the simplest possible market model there are two assets (one stock and one bond), one time step and just two possible future scenarios. Many of the basic ideas of mathematical finance
More informationIntroduction to Binomial Trees
11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths
More informationValuation of the Minimum Guaranteed Return Embedded in Life Insurance Products
Financial Institutions Center Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products by Knut K. Aase SveinArne Persson 9620 THE WHARTON FINANCIAL INSTITUTIONS CENTER The Wharton
More informationMath 194 Introduction to the Mathematics of Finance Winter 2001
Math 194 Introduction to the Mathematics of Finance Winter 2001 Professor R. J. Williams Mathematics Department, University of California, San Diego, La Jolla, CA 920930112 USA Email: williams@math.ucsd.edu
More informationCaps and Floors. John Crosby
Caps and Floors John Crosby Glasgow University My website is: http://www.johncrosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February
More information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationEC824. Financial Economics and Asset Pricing 2013/14
EC824 Financial Economics and Asset Pricing 2013/14 SCHOOL OF ECONOMICS EC824 Financial Economics and Asset Pricing Staff Module convenor Office Keynes B1.02 Dr Katsuyuki Shibayama Email k.shibayama@kent.ac.uk
More informationBuy a number of shares,, and invest B in bonds. Outlay for portfolio today is S + B. Tree shows possible values one period later.
Replicating portfolios Buy a number of shares,, and invest B in bonds. Outlay for portfolio today is S + B. Tree shows possible values one period later. S + B p 1 p us + e r B ds + e r B Choose, B so that
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends)
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More informationOption Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25
Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationDetermination of Optimum Fair Premiums in PropertyLiability Insurance: An Optimal Control Theoretic Approach
Determination of Optimum Fair Premiums in PropertyLiability Insurance: An Optimal Control Theoretic Approach by Amin Ussif and Gregory Jones ABSTRACT Dynamic valuation models for the computation of optimum
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationEXP 481  Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481  Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationAdvanced Fixed Income Analytics Lecture 1
Advanced Fixed Income Analytics Lecture 1 Backus & Zin/April 1, 1999 Vasicek: The Fixed Income Benchmark 1. Prospectus 2. Models and their uses 3. Spot rates and their properties 4. Fundamental theorem
More informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Putcall
More informationLECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS
LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50
More informationLecture 12. Options Strategies
Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same
More information