Mathematics of financial markets / Robert J. Elliott and P. Ekkehard Kopp.—2nd ed. . Texts for this market have multiplied, as the rapid growth of the Springer. Library of Congress Cataloging-in-Publication Data Elliott, Robert J. (Robert James), – Mathematics of financial markets / Robert J. Elliott and P. Ekkehard. In this introductory course we review some of the basic concepts of Math- ematical Finance. the general theory of semi-martingale models of financial markets.
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Its content is suitable particularly for graduate students in mathematics who have a DRM-free; Included format: PDF; ebooks can be used on all reading devices securities, such as options, futures and swaps, in modern financial markets. 𝗣𝗗𝗙 | Introduction to the Economics and Mathematics of Financial Markets fills the longstanding need for an accessible yet serious textbook treatment of. Request PDF on ResearchGate | On Jan 1, , Robert J. Elliott and others published Mathematics of Financial Markets.
He is a member of the Editorial Board of Springer Finance.
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This second edition contains a significant number of changes and additions …. The target audience is readers with sound mathematical background on elementary concepts from measure-theoretic probability …. It should be an equally valuable resource to practitioners interested in the mathematical tools …. A new chapter on coherent risk measures for instance reflects the recent trend in research and applications in the area of risk management.
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The new edition adds substantial material from current areas of active research, notably: Show all. In fact, we were able to construct the equivalent martingale measure Q for S directly and showed that in this model there is a unique equivalent martingale measure.
Theorem 4. Then, by 2. Conversely, suppose that the market model is viable but not complete, so that there exists a non-negative random variable X that cannot be generated by an admissible trading strategy. However, X would be attained by this strategy. We have therefore constructed an equivalent martingale measure distinct from Q. Thus, in a viable incomplete market, the EMM is not unique. This completes the proof of the theorem.
We saw in Section 2. We explore the content of the martingale representation result Proposition 4. In such simple cases, a direct proof of the martingale representation theorem is almost obvious and does not depend on the nature of the sample space, since the Rt contain all the relevant information. We follow the proof given in , Now suppose that 4. Recall from Proposition 4. Exactly as in the proof of Proposition 4.
Exercise 4. Use formula 4. This idea is closely related to the concept of extremality of a probability measure among certain convex sets of martingale measures, and in this setting, the ideas also extend to continuous-time models see , . Outline of Proof. In other words, that the matrix of price increments has linearly independent columns. But we have already seen that non-singularity of the matrix of price increments is equivalent to completeness in the single-period model. The proof may now be completed by pasting together the steps to construct the unique EMM.
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Example 4. We already know that the binomial random walk model is complete by virtue of the uniqueness of the EMM. Our present interest is in the splitting index. It can be shown much as we did in Chapter 2 that the multifactor Black-Scholes model is a limit of multifactor random walk models and is complete. Consequently, it is possible to have a complete continuous-time model that is a limit in some sense of incomplete discrete models.
See ,  for details. Hence Q is not the only EMM in the model, which contradicts completeness.
This severely restricts its practical applicability, as Kreps [, p. The Arbitrage Interval We return to the general setup of extended securities market models that was introduced in Section 2. We wish to examine the set of possible prices of a European contingent claim H that preclude arbitrage.
Since H is itself a tradeable asset, we need to include it in the assets that can be used to produce trading strategies. It was shown in Theorem 2. We need the following result. P Proof. The same is true for the upper bound if the sets are bounded above. This is left to the reader as an exercise in using the fact that the EMM can always be chosen to have bounded density relative to the given reference measure.
Our proof follows that in . Let H be a European claim in a securities market model. Remark 4. Note that Theorems 4. Having done this, and also having characterised the attainability of claims, we can now go much further in identifying the class of complete market models more fully.
We shall demonstrate, after the fact, that the argument provided to prove Theorem 4. With the more advanced tools now at our disposal, the proof of this far-reaching result is elementary for general market models. Throughout, we only need to work with bounded claims: First consider the single-period case i. In particular, as already observed in Chapter 2, the indicator 1A of any set in F is an attainable claim.
Hence Q A is also uniquely determined for each A, so that P is a singleton. Again by Theorem 4. Thus every contingent claim is automatically bounded, hence attainable. Turning now to the multi-period case, we argue by induction on T. Lemma 4. There are various other characterisations of completeness, notably in terms of the set of extreme points of P , which are better adapted to their continuous-time analogues. We refer to  and  for details.
The only constraint is that the option ceases to be valid at time T and thus cannot be exercised after the expiry date T. The pricing problem for American options is more complex than those considered up to now, and we need to develop appropriate mathematical concepts to deal with it.
Exercise 5. Recall that the Ft increase with t. These often require care about measurability problems. Hedging Constraints Hedge portfolios also require a little more care than in the European case since the writer may face the liability inherent in the option at any time in T. As considered in Section 2.
This raises several questions for the given claim f: These questions are examined in this chapter. This extension requires us to establish results about N martingale convergence, continuous-time versions of which will also be needed in later chapters.
The idea of stopping times for stochastic processes, while intuitively obvious, provides perhaps the most distinguishing feature of the techniques of probability theory that we use in this book. Remark 5. Note, however, that this depends on the countability of N.
Nevertheless, many of the basic results about stopping times, and their proofs, are identical in both setups, and the exceptions become clear from the following examples and exercises. Example 5.
The continuoustime analogue of this result is proved in Theorem 6.
Convince yourself that a virtually identical statement and proof applies here. The next two results, which we will extend considerably later, use the fact that stopping a martingale is essentially a special case of taking a martingale transform. They are used extensively in the rest of this chapter. Theorem 5. A Hence the result follows, again using Exercise 5.
To complement Theorem 5. We deal with the supermartingale case. The martingale case is then obvious. The counterparts of these results in the continuous-time setting are outlined in Chapter 6. Prove that if C is UI, then it is bounded in L1 , but the converse is false. A useful additional hypothesis is domination in L1: See, e. To illustrate why uniform integrability is so important for martingales, consider the following.
Proposition 5. First we need an important inequality, which we will use frequently. Thus the family U is Lp -bounded and hence UI. The same is true for L1. Here the integrability of Nt , which is required for the application of 5. Similar results follow upon applying 5. The role of uniform integrability is evident from the following proposition. Suppose Xn is a sequence of integrable random variables and X is integrable.
Mathematics of financial markets
See  or  for the proof of this standard result. Since a. Corollary 5. Thus, to prove that a UI martingale converges in L1 -norm, the principal task is showing a. We outline here the beautifully simple treatment given in , to which we refer for details.
Now suppose that M is a supermartingale. Then 5. Thus the a. This proves 5. Whenever 5. To summarise, we have the following. We extend Theorems 5.
Let M be a UI martingale. In discrete time, this is easily accomplished; remarkably there is also such a decomposition in continuous time the Doob-Meyer decomposition, see , .
The Doob decomposition is unique in the following sense. Such a process must be constant, as we saw in Chapter 2. When X is a submartingale, equation 5. This increasing predictable process A therefore has an a. Observe, using 5. We have shown that an L2 -bounded martingale has integrable quadratic variation.
In Chapters 6 to 8, we make fuller use of the preceding results in the continuous-time setting. This enables one to use countable dense subsets to approximate the path behaviour and use the results just presented; see ,  for details. We assume this in Chapter 6 and beyond. If X is a UI martingale, this is automatic from Theorem 5. Note that since M is a martingale, 5. Thus 5. We now examine the properties of the process Z. From 5. To see that it is the smallest such supermartingale, we argue by backward induction.
Therefore X0 and Z0 are a. Characterisation of Optimal Stopping Times We are now able to describe how the martingale property characterises optimality more generally. By Proposition 5. This proves condition 1 above. Again because Z is a supermartingale, we also have, by Theorem 5. From Proposition 5.
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It is clearly T-valued and therefore bounded. This followed from our assumption that F0 contained only null sets and their complements, and it led us to establish 5.
We need to ensure that we obtain this supremum as an F-measurable function, even for an uncountable family. There exists a unique Fmeasurable function g: Proofs of this result can be found in , . The idea is simple: When X is UI, we can still use the optional stopping results proved earlier in this context. The martingale characterisation of optimal stopping times can be extended as well; see  or  for details. Given an American option ft in this model e.
Hence 5. It follows that x is the minimum initial investment for which a hedging strategy can be constructed. This means that the second condition in Proposition 5. Note that the same considerations apply to the option writer: We have proved the following theorem. We showed by an arbitrage argument in Chapter 1 that American options are more valuable than their European counterparts in general but that for a simple call option there is no advantage in early exercise, so that the American and European call options have the same value.
Using the theory of optimal stopping, we can recover these results from the martingale properties of the Snell envelope. This shows that the value process of the American call option dominates that of the European call option. The constraint 5. In summary, we have the following.
Writing Z for the Snell envelope of X, the supermartingale Z dominates Xand can be used as the process U in the previous discussion. We have proved the following. Chapter 6 Continuous-Time Stochastic Calculus 6. We denote the time parameter set by T in each case. Remark 6. Just as in the discrete case, Ft represents the history of some process or processes up to time t.
However, all possible histories must be allowed. Many of the properties of stopping times carry over to this setting, however. We continue with some basic properties of stopping times.
Proposition 6. The proof is identical to that given in Example 5. Exercise 6. One then can establish the following compare with Exercise 5. Theorem 6. Suppose S, T are stopping times. Each of the three sets on the right-hand side is in Ft: X could represent the evolution of the price of oil or the price of a stock over time. Equivalence of Processes A natural question is to ask when two stochastic processes model the same phenomenon. It turns out that if we consider the process as a map X: This is the famous Kolmogorov extension theorem; see [, Theorem 2.
Then A is called evanescent if 1A is indistinguishable from the zero process. Finally, we recall the following. Then X is said to be adapted to Ft if Xt is Ft -measurable for each t. A martingale is a purely random process in the sense that, given the history of the process so far, the expected value of the process at some later time is just its present value. Brownian Motion The most important example of a continuous-time martingale is a Brownian motion. This process is named for Robert Brown, a Scottish botanist who studied pollen grains in suspension in the early nineteenth century.
He observed that the pollen was performing a very random movement and thought this was because the pollen grains were alive. We now know this rapid movement is due to collisions at the molecular level.
We can immediately establish the following. Indeed, the third statement characterises a Brownian motion. Notation 6. Write M for the set of uniformly integrable martingales. For example, C might be the bounded processes, or the processes of bounded variation.
This result, which is the analogue of Theorem 5. A complete proof of this result in continuous time can be found in [, Theorem 4. Note that our discussion of the discrete case in Chapter 5 showed how the extension from bounded to more general stopping times required the martingale convergence theorem and conditions under which a supermartingale is closed by an L1 -function.
This condition is also required in the following. The following is a consequence of the optional stopping theorem. Lemma 6. Using the optional stopping theorem Theorem 6. Then, for 1 Also, for 1 Proof. Also, for 1 As in Section 5. Therefore, Theorem 6. Corollary 6. Consequently, the preceding value process is a martingale transform. The natural extension to continuous time of such a martingale 4t transform is the stochastic integral 0 Hs dS s.
We work initially on the time interval [0, T ]. Suppose H is a simple process. Outline of the Proof. Then there is a unique linear map I from H into the space of continuous Ft -martingales on [0, T ] such that: The right-hand side can be made arbitrarily small, so I H t is an Ft martingale. The remaining results follow by continuity and from the density in H of simple processes.
Taking limits, the result follows. From Theorem 6. Because 4t Hs is adapted, 0 Hs2 ds is adapted and Tn is an Ft -stopping time. The continuity of the operator I, is therefore established. The result follows from Lemma 6. Choosing a sub1. This decomposition is entirely analogous to the Doob decomposition described in Section 5.
Mathematics of Financial Markets.pdf - Index of
For details, see the development in [, Chapter 10] or [, Chapter 3]. Consequently, 6. For a proof see . Furthermore, formula 6. We noted in 6. Then Wt is a Brownian motion. Iterating 6. Each X n is a continuous process, so X is a continuous process. Using the bounded convergence theorem, we can take the limit in 6. Therefore, X is the unique solution of the equation 6. See Kunita . If Xs x, t is the solution of 6. Before establishing the Markov property of solutions of 6.
Write PY for the probability law of Y. Suppose Z is any A-measurable random variable. This identity is true for all such Z; the result follows. Suppose X0 x, t is the solution of 6. Suppose g: As in Lemma 6. Consequently, the augmented process is Markov and, for any bounded Borel function f: Chapter 7 Continuous-Time European Options In this chapter, we shall develop a continuous-time theory that is the analogue of that in Chapters 1 to 3. The simple model will consist of a riskless bond and a risky asset, which can be thought of as a stock.
The dynamics of our model are described in Section 7. The following two sections present the fundamental results of Girsanov and martingale representation. These are then applied to discuss the hedging and pricing of European options. Recall that the Black-Scholes pricing formula for a European call was derived in Section 2. We suppose the market contains a riskless asset, or bond, whose price at time t is St0 , and a risky asset, or stock, whose price at time t is St1.
Let r be a non-negative constant that represents the instantaneous interest rate on the bond. This instantaneous interest rate should not be confused with the interest rate over a period of time in discrete models. In particular, log St1 is a normal random variable, which 1 is often expressed from 7. It is immediate and 7. The set of martingales for which convergence results hold is the set of uniformly integrable martingales.
Consistent with Notation 6. In continuous time, versions of martingales are considered that are right-continuous and have left limits. Lemma 7. Xt Mt is a local martingale under P if and only if Xt is a local martingale under Q. We prove the result for martingales. The extension to local martingales can be found in [, Proposition 3. Recall from Theorem 6. We show how Bt behaves under a change of measure.
Remark 7. Equivalently, from Lemma 7. Hitting Times of Brownian Motion We shall need the following results on hitting times of Brownian motion. Theorem 7.
Ta in 7. Then, from the optional stopping theorem Theorem 6. Now if Ta 7. Corollary 7. We now have the following result. Suppose K is a stable subspace of H2. See [, page ]. Write HT2 for the space of square integrable Ft -martingales on [0, T ].
From Remark 7. We now extend Theorem 7. By considering stopping times and pasting, the representation results apply to locally square integrable martingales. Then, from Theorem 7. Here the asterisk denotes the transpose. A special semimartingale is a semimartingale that is the sum of a local martingale and a predictable process of locally integrable variation. The decomposition is unique when it exists see [, Theorem Now N is a special semimartingale, so the decompositions given in 7.
As there is no bounded variation term in 7. The quantity Ht0 resp. Ht1 denotes the amount of the bond St0 resp. The intuitive meaning of 7. We consider European claims with an expiration time T. Consequently, for 7.
Notation 7. We can then establish the following result, whose discrete-time analogue was discussed in Section 2. If there are no contributions or withdrawals, the corresponding wealth process is given in 7. Suppose more generally as we did for discrete-time models in Section 5.
Let these be modelled by 7.
A European contingent claim is a positive random variable h, measurable with respect to FT. European put option. Let us summarise these observations in the following result. We can then establish the following lemma. For any European claim fT , we must therefore do the following: The discussion in Section 2.
From 7. Then, from 7. From the martingale representation result, Theorem 7. The condition 7. In summary, we have shown that the following results hold. Suppose that fT represents a European claim, which can be exercised at time T. P- is sometimes called a risk-neutral measure. We now extend these ideas to multifactor Brownian pricing models. The prices discounted prices. Equation 7.
Mathematics of Financial Markets.pdf - Index of
Three cases can arise: Consequently, there is no arbitrage if 7. Then Ht0 , Ht1 ,. In the case where 7. In Chapters 3 and 4, we derived the two fundamental theorems of asset pricing in discrete time models.
The extensive literature on this topic began with two papers by Harrison and Pliska ,. The reader is referred especially to the papers by Stricker  and Delbaen and Schachermayer . We require some integrability properties of f: Now, from 7.
As the left-hand side of 7. Noting that proved the following theorem. This is also the rational price for the option at time t.
Specialising the above results, we recover the Black-Scholes pricing formula 2. From Theorem 7. Now, from Theorem 7. Exercise 7. To ensure the claim is attainable, the number of sources of 7. As in Section 7. Applying Theorem 7. However, from 7. The following two exercises serve to introduce two further options closely related to the call and put.
A chooser option gives the holder the right to choose either the call or the put at time t. What is the rational price at time 0 of such a chooser option for the above call and put? The result follows from 7. The result follows from Lemma 7.
The proof is similar to that of Lemma 7. The price at any time t 7. An up and out call option gives the holder the right but not the obligation to download S 1 for strike price K at time T provided that the price St1 does not rise above H in which case the option ceases to exist.
An up and in call option gives the holder the right but not the obligation to download S 1 at time T for strike price K provided that at some time before T the price St1 becomes greater than H; otherwise, the option does not yet exist.
This is often called the BlackScholes equation. As the solution 7. It is of interest to recall the original derivation of equation 7. This approach has become widely known as delta-hedging. The terminology will become clear shortly. Suppose, as above, that Vt,T S represents the value at time t of a European call with expiration time T when the value of the underlying S 1 at time t is given by S.
This is the Black-Scholes equation 7. To solve it, and hence derive the Black-Scholes formula, one can apply a sequence of transformations to reduce this inhomogeneous linear parabolic equation to the well-known heat equation. We indicate the main steps in the solution. Therefore 7.
From formula 7. There are analogous representations for the other barrier options.This approach has become widely known as delta-hedging. In fact, there are two reasons why someone who sells a bond might experience a loss. Stochastic analysis. Theorem 6. The book provides a rigorous overview of the subject, while its flexible presentation makes it suitable for use with different levels of undergraduate and graduate students. The identity 2. We shall need it several times in this chapter, as well as in Chapter 9.
It was shown in Theorem 2.
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