# incomplete markets

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The Concepts and Practice of Mathematical Finance by Mark S. Joshi

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A closed-form formula as an infinite sum can be developed for the price of a call or put option in a jump-diffusion model. The market consisting of a stock evolving to a jump-diffusion model and a riskless bond is incomplete. In an incomplete market an option does not have a unique price. When changing measure in a jump-diffusion world, we can change the drift, the intensity of the jumps and the jump distribution but we cannot change the volatility of the underlying. 386 Incomplete markets and jump-diffusion processes Increasing jump-intensity always increases the price of a European option, which has a convex final payoff. For a digital option increasing jump-intensity can either increase or decrease the price of an option. In an incomplete market it is the market which chooses the measure. It is possible to hedge in a jump-diffusion model using options provided we assume that the market does not change its choice of measure.

Our hedging argument has shown that only prices between zero and five can be non-arbitrageable, whilst the risk-neutral argument shows that prices between zero and five are not arbitrageable. We therefore conclude that the set of arbitrage-free prices for the option is the set of prices between zero and five. The three-world universe is an example of an incomplete market, that is, a market where portfolios cannot be arranged to give precisely the desired pay-off, and it is characteristic of incomplete markets that the price of an option can only be shown to lie in an interval rather than being forced to take a precise value. The market price of such an option would then be determined within the range of possible prices by the risk-preferences of traders in the market rather than mathematics. , 3.3 Multiple time steps 3.3.1 More realism At this point, option pricing is not looking very successful - clearly we will want to price options on assets that can have more than two values in the future.

The bank can still make money by selling put options but it is doing so by taking on risk, rather than by charging for the cost of hedging, as in the Black-Scholes framework. The purchase of the put option is therefore a transference of risk from the fund manager to the bank. The market price will settle on a point where the 361 362 Incomplete markets and jump-diffusion processes banks feel that they are being adequately compensated for taking on the extra risk. A major determinant of the price is therefore risk preferences rather than arbitrage. Once we have moved to an incomplete market, there are two different issues to be addressed. The first is how to use arbitrage to bound the prices of vanilla options. The second is to determine prices for exotic options which are compatible with both the model and the prices of the vanilla options traded in the market.

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

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In other words, all these measures coincide on In addition, note that theoretical problems also arise for the case of random r. 5.11.2 Pricing for an incomplete market Mean-variance hedging Similarly to the case of the discrete time market, Definition 5.43 leads to superreplication for incomplete markets. Clearly, it is not always meaningful. Therefore, there is another popular approach for an incomplete market. Definition 5.65 (mean-variance hedging). The fair price of the option is the initial wealth X(0) such E|X(T)−ψ|2 is minimal over all admissible self-financing strategies. In many cases, this definition leads to the option price e−rTE*ψ, where E* is the expectation for a risk-neutral equivalent measure that needs to be chosen by some optimal way, since this measure is not unique for an incomplete market. This measure needs to be found via solution of an optimization problem.

Find the fair price of the option with payoff F(S1,…, ST)=max(ST−1, 0). Solution. We have For incomplete markets, Definition 3.49 leads to super-replication. That is not always meaningful. Therefore, there is another popular approach for incomplete markets. Definition 3.55 (mean-variance hedging). The fair price of the option is the initial wealth X0 such that E|XT−ψ|2 is minimal over all admissible self-financing strategies. © 2007 Nikolai Dokuchaev Discrete Time Market Models 41 In many cases, this definition leads to the option price calculated as the expectation under a risk-neutral equivalent measure which needs to be chosen by some optimal way, since a risk-neutral equivalent measure is not unique for an incomplete market. 3.11 Increasing frequency and continuous time limit In reality, prices may change and be measured very frequently.

If an equivalent risk-neutral measure is not unique then, by Theorem 5.35, the market cannot be complete, i.e., there are claims ψ that cannot be replicable. In this section, we assume that r is non-random and constant. Let be the filtration generated by the process S(t). (For this case of non-random general case, the filtration For the generated by the process (w(t), η(t) is larger than 5.11.1 Examples of incomplete markets An example with a≡r Let a(t)≡r(t), and let σ=σ(t, η), where η is a random process (or a random vector, or a random variable), independent from the driving Wiener process w(t) (for instance, η may represent another Wiener process). Clearly, any original probability measure P=Pη is a risk-neutral measure (note that for any η). Any probability measure is defined by the pair (w, η), therefore it depends on the choice of η.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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Oxford Science Publications Lewis, A Series of articles in Wilmott magazine September 2002 to August 2004 Merton, RC 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-44 What is Meant by “Complete” and “Incomplete” Markets? Short Answer A complete market is one in which a derivative product can be artificially made from more basic instruments, such as cash and the underlying asset. This usually involves dynamically rebalancing a portfolio of the simpler instruments, according to some formula or algorithm, to replicate the more complicated product, the derivative. Obviously, an incomplete market is one in which you can’t replicate the option with simpler instruments. Example The classic example is replicating an equity option, a call, say, by continuously buying or selling the equity so that you always hold the amountΔ = e−D(T −t)N (d1), in the stock, where and Long Answer A slightly more mathematical, yet still quite easily understood, description is to say that a complete market is one for which there exist the same number of linearly independent securities as there are states of the world in the future.

People can therefore disagree on the probability of a stock rising or falling but still agree on the value of an option, as long as they share the same view on the stock’s volatility. In probabilistic terms we say that in a complete market there exists a unique martingale measure, but for an incomplete market there is no unique martingale measure. The interpretation of this is that even though options are risky instruments we don’t have to specify our own degree of risk aversion in order to price them. Enough of complete markets, where can we find incomplete markets? The answer is ‘everywhere.’ In practice, all markets are incomplete because of real-world effects that violate the assumptions of the simple models. Take volatility as an example. As long as we have a lognormal equity random walk, no transaction costs, continuous hedging, perfectly divisible assets,..., and constant volatility then we have a complete market.

This is because there are now more states of the world than there are linearly independent securities. In reality, we don’t know what volatility will be in the future so markets are incomplete. We also get incomplete markets if the underlying follows a jump-diffusion process. Again more possible states than there are underlying securities. Another common reason for getting incompleteness is if the underlying or one of the variables governing the behaviour of the underlying is random. Options on terrorist acts cannot be hedged since terrorist acts aren’t traded (to my knowledge at least). We still have to price contracts even in incomplete markets, so what can we do? There are two main ideas here. One is to price the actuarial way, the other is to try to make all option prices consistent with each other. The actuarial way is to look at pricing in some average sense.

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Derivatives Markets by David Goldenberg

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Unfortunately, contrary to intuition, the no-arbitrage condition is not enough machinery to uniquely price derivative securities. That is, no-arbitrage is not a sufficient condition for uniquely pricing derivative securities. There are generally many linear, positive pricing mechanisms, all consistent with no-arbitrage, for non-replicable ﬁnancial claims! OPTION PRICING IN DISCRETE TIME, PART 1 451 Another way to say this is that in an incomplete market, there can be many equally valid (no-arbitrage) pricing mechanisms. We already have an inkling of this through our study of ROP. Rational option pricing, which is exclusively based on the foundation of the no-arbitrage principle, produces pricing relationships (such as European Put-Call Parity) and pricing bounds or ranges into which rational option prices must fall. ROP does not produce speciﬁc, unique prices.

Therefore, no risk premium for pricing the option is needed; a risk premium is built into the pricing of the underlying stock, St (see Chapter 17). There is no immediate and completely adequate empirical ﬁx for the constant assumption, except to throw out Black–Scholes’ assumption of a stationary log-normal diffusion, and search for a viable (smile-consistent) underlying stochastic process among the vast set of alternatives, many of which will lead to incomplete markets. Black–Scholes and its modiﬁcations, however, still have tremendous appeal, especially among traders, who use Black–Scholes calibrated to an implied volatility surface. Traders use ATM options to imply volatility, since these are the most liquid, and therefore most informative about future volatility. Furthermore, there are exotic and American options for which the lognormal GBM remains the workhorse.

Because the non-hedgeable risks are precisely those that cannot be diversiﬁed away (hedged out) by attempting to replicate the contingent claim, they would RISK-NEUTRAL VALUATION 601 command risk premia in the contingent claim price. These would, of necessity, show up in arbitrage-free pricing formulas for the contingent claim. This also renders such contingent claim prices not preference-free. We can summarize this as follows. In incomplete markets, non-replicable claims could be priced in a manner that is arbitrage-free (EMMs exist), and yet not preference-free. Furthermore, there would be multiple arbitrage-free, non-preference-free valuations of non-replicable claims. We know this is true, because FTAP2 tells us that a claim is replicable if and only if there is a unique EMM for the discounted price process. We will also see this in another, more practical manner below. 17.1.5 Black–Scholes’ Contribution A few more observations about Black–Scholes are in order.

Monte Carlo Simulation and Finance by Don L. McLeish

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SOME BASIC THEORY OF FINANCE While it is not too diﬃcult to solve this system in this case one can see that with more branches and more derivatives, this non-linear system of equations becomes diﬃcult very quickly. What do we do if we observe market prices for only two derivatives defined on this stock, and only two parameters can be obtained from the market information? This is an example of what is called an incomplete market, a market in which the risk neutral distribution is not uniquely specified by market information. In general when we have fewer equations than parameters in a model, there are really only two choices (a) Simplify the model so that the number of unknown parameters and the number of equations match. (b) Determine additional natural criteria or constraints that the parameters must satisfy. In this case, for example, one might prefer a model in which the probability of a step up or down depends on the time, but not on the current price of the stock.

SOME BASIC THEORY OF FINANCE Theorem 9 shows that this exponentially tilted distribution has the property of being the closest to the original measure P while satisfying the condition that the normalized sequence of stock prices forms a martingale. There is a considerable literature exploring the links between entropy and risk-neutral valuation of derivatives. See for example Gerber and Shiu (1994), Avellaneda et. al (1997), Gulko(1998), Samperi (1998). In a complete or incomplete market, risk-neutral valuation may be carried out using a martingale measure which maximizes entropy or minimizes cross-entropy subject to some natural constraints including the martingale constraint. For example it is easy to show that when interest rates r are constant, Q is the risk-neutral measure for pricing derivatives on a stock with stock price process St , t = 0, 1, ... if and only if it is the probability measure minimizing H(Q, P ) subject to the martingale constraint 1 St+1 ]. 1̄ + r St = EQ [ (2.23) There is a continuous time analogue of (2.22) as well which we can anticipate by inspecting the form of the solution.

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Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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PRICING STATE-CONTINGENT CLAIMS Theorem 5.4.1 If there are M complex securities (M = N ) and the payoff matrix Y is non-singular, then markets are complete. Proof Suppose the optimal trade for consumer i state j is xij − eij . Then can invert Y to work out optimal trades in terms of complex securities. Q.E.D. An (N + 1)st security would be redundant. Either a singular square matrix or < N complex securities would lead to incomplete markets. So far, we have made no assumptions about the form of the utility function, written purely as u (x0 , x1 , x2 , . . . , xN ) , where x0 represents the quantity consumed at date 0 and xi (i > 0) represents the quantity consumed at date 1 if state i materialises. 5.4.1 Completion of markets using options Assume that there exists a state index portfolio, Y , yielding different non-zero payoffs in each state (i.e. a portfolio with a different payout in each state of nature, possibly one mimicking aggregate consumption).

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The Euro: How a Common Currency Threatens the Future of Europe by Joseph E. Stiglitz, Alex Hyde-White

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,” IMF Staff Discussion Note 11/08, April 8, 2011, available at https://www.imf.org/external/pubs/ft/sdn/2011/sdn1108.pdf; and Jonathan D. Ostry, Andrew Berg, and Charalambos G. Tsangarides, “Redistribution, Inequality, and Growth,” IMF Staff Discussion Note 14/02, February 2014, available at https://www.imf.org/external/pubs/ft/sdn/2014/sdn1402.pdf. 14 Bruce C. Greenwald and Joseph E. Stiglitz, “Externalities in Economies with Imperfect Information and Incomplete Markets,” Quarterly Journal of Economics 101, no. 2 (1986): 229–64. 15 I became particularly engaged in “the economics of crises” during my time at the World Bank and wrote extensively on the subject, both alone and with my colleagues at the World Bank. A popular account is provided in Globalization and Its Discontents. See also my articles “Lessons from the Global Financial Crisis,” in Global Financial Crises: Lessons from Recent Events, ed.

Neyman [Berkeley: University of California Press, 1951], pp. 507–32; and Gerard Debreu, “Valuation Equilibrium and Pareto Optimum,” Proceedings of the National Academy of Sciences 40, no. 7 [1954]: 588–92; and Debreu, The Theory of Value [New Haven, CT: Yale University Press, 1959.]) The circumstances that they identified where markets did not lead to efficiency were called market failures. Subsequently, Greenwald and Stiglitz showed that whenever information was imperfect and markets incomplete—essentially always—markets were not efficient (“Externalities in Economies with Imperfect Information and Incomplete Markets”). Of course, even earlier, Keynes had emphasized that markets do not by themselves maintain full employment. 34 See James Edward Meade, The Theory of International Economic Policy, vol. 2, Trade and Welfare (London: Oxford University Press, 1955); and Richard G. Lipsey and Kelvin Lancaster, “The General Theory of Second Best,” Review of Economic Studies 24, no. 1 (1956): 11–32. 35 See David Newbery and J.

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The Irrational Economist: Making Decisions in a Dangerous World by Erwann Michel-Kerjan, Paul Slovic

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K. Hsee (2000). “The Affection Effect in Insurance Decisions.” Journal of Risk and Uncertainty 20, no. 2: 149-159. Kunreuther, H., and M. V. Pauly (2000). NBER Reporter, March 22. Pagán, J. A., and M. V. Pauly (2006). “Community-Level Uninsurance and Unmet Medical Needs of Insured and Uninsured Adults.” Health Services Research 41, no. 3: 788-803. Schlesinger, H., and N. Doherty (1985). “Incomplete Markets for Insurance: An Overview.” Journal of Risk and Insurance 52: 402-423. 17 The Hold-Up Problem Why It Is Urgent to Rethink the Economics of Disaster Insurance Protection W. KIP VISCUSI As other contributors to this book have suggested, how people make decisions involving risk and uncertainty and how economists think people should make these decisions are often quite different matters.

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Tools for Computational Finance by Rüdiger Seydel

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The Entrepreneurial State: Debunking Public vs. Private Sector Myths by Mariana Mazzucato

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Imperfections can arise for various reasons: the unwillingness of private firms to invest in areas, like basic research, from which they cannot appropriate private profits because the results are a ‘public good’ accessible to all firms (results of basic R&D as a positive externality); the fact that private firms do not factor in the cost of their pollution in setting prices (pollution as a negative externality); or the fact that the risk of certain investments is too high for any one firm to bear them all alone (leading to incomplete markets). Given these different forms of market failure, examples of the expected role of the State would include publicly funded basic research, taxes levied on polluting firms and public funding for infrastructure projects. While this framework is useful, it cannot explain the ‘visionary’ strategic role that government has played in making these investments. Indeed, the discovery of the Internet or the emergence of the nanotechnology industry did not occur because the private sector wanted something but could not find the resources to invest in it.

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Grave New World: The End of Globalization, the Return of History by Stephen D. King

A BORDERLESS WORLD A third option would be to dispense with borders altogether. The nearest we have got to this is perhaps the European Union – or, more specifically, the 19 members that make up the Eurozone. Yet the single currency project is only half-finished – and arguably only half-baked. The Eurozone has some aspects of nationhood: a single currency, a single monetary policy, a single (although incomplete) market and, for those who also happen to be members of Schengen, a common external border. Yet it lacks other aspects: there is no common fiscal policy and no common border force; the European Parliament is a weak and distant institution; and a common European defence policy has so far proved be more a matter of words than deeds. Moreover, the strategic direction of the EU is determined by a European Council that is not much more than a talking shop for the various European heads of state or government – in other words, it is a bit like a White House occupied not only by a president, but also by assorted state governors, each of whom is entitled to promote his or her legitimate point of view.

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Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

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Remark 3.6.8 in Karatzas and Shreve (1998). 11.6 Duality Approach For any stopping time τ ∈ S[0,T ] , we denote by τ (x) the set of portfolio/ consumption-rate processes triplets (π, C) for which (π, C, τ ) ∈ A(x). For ﬁxed τ ∈ S, we consider the utility maximization problem Vτ (x) sup (π ,C )∈ τ (x) J (x; π, C, τ ). (11.25) 301 11.6 Duality Approach Not allowing our agent to invest in the stock market (π(τ ,T ] ≡ 0) after retirement, creates an incomplete market in which the problem is difﬁcult to solve explicitly. However, under the additional Assumption (11.1) that the interest rate is locked after retirement, we can solve the optimization problem after retirement explicitly by pathwise optimization given the information available at retirement time Fτ . Equation 11.12 with π(τ ,T ] ≡ 0 implies that we have the following constraint ∞ γ (t)c(t)dt ≤ γ (τ )(X x,π ,C − ξ ), a.s. (11.26) τ In conjunction with Equation 11.15 this constraint leads to τ H (τ ) T E γ (t)c(t)dt + H (τ )ξ + H (s)c(s)ds γ (τ ) τ 0 τ x,π ,C (τ ) − ξ ) + H (τ )ξ + H (s)c(s)ds ≤ x. ≤ E H (τ )(X 0 The problem can be solved as usual, with the introduction of a Lagrange multiplier λ > 0: τ T −βt −βτ −βt e U1 (c(t))dt + e U2 (ξ ) + e U3 (c(t))dt J (x; π, C, τ ) = E τ 0 τ ≤E + e−βt Ũ1 (λeβt H (t))dt + e−βτ Ũ2 (λeβτ H (τ )) 0 T −βt e τ τ +λ·E 0 τ ≤E + τ −βt e H (τ ) H (t)c(t)ds + H (τ )ξ + γ (τ ) τ T γ (t)c(t)dt e−βt Ũ1 (λeβt H (t))dt + e−βτ Ũ2 (λeβτ H (τ )) 0 T βt H (τ ) Ũ3 λe γ (t) dt γ (τ ) βt H (τ ) Ũ3 λe γ (t) dt γ (τ ) + λx with equality if and only if ⎧ βt ⎨I1 (λe 0 < t < τ, H (t)) H (τ ) (11.27) c(t) = ⎩I3 λeβt γ (t) τ < t ≤ T. γ (τ ) T γ (s) βτ x,π ,C βs H (τ ) −ξ = ξ = I2 (λe H (τ )) and X I3 λe γ (s) ds, γ (τ ) τ γ (τ ) (11.28) 302 CHAPTER 11 The ‘‘Retirement’’ Problem τ E 0 H (τ ) H (t)c(t)dt + H (τ )ξ + γ (τ ) T τ γ (t)c(t)dt = x

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How Markets Fail: The Logic of Economic Calamities by John Cassidy

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For the past thirty or forty years, many of the brightest minds in economics have been busy examining how markets function when the unrealistic assumptions of the free market model don’t apply. For some reason, the economics of market failure has received a lot less attention than the economics of market success. Perhaps the word “failure” has such negative connotations that it offends the American psyche. For whatever reason, “market failure economics” never took off as a catchphrase. Some textbooks refer to the “economics of information,” or the “economics of incomplete markets.” Recently, the term “behavioral economics” has come into vogue. For myself, I prefer the phrase “reality-based economics,” which is the title of Part II. Reality-based economics is less unified than utopian economics: because the modern economy is labyrinthine and complicated, it encompasses many different theories, each applying to a particular market failure. These theories aren’t as general as the invisible hand, but they are more useful.

Making Globalization Work by Joseph E. Stiglitz

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I should be clear: while the intellectual groundings have been taken away from market fundamentalism, newspaper columnists and pundits—and occasionally, even a few economists—sometimes still invoke economic "science" in defense of their position. 3. This research was cited when I was awarded the Nobel Prize. 4. See Bruce Greenwald and Joseph E. Stiglitz, "Externalities in Economies with Imperfect Information and Incomplete Markets," Quarterly Journal of Economics, vol. 101, no. 2 (May 1986), pp. 229-64. 5. Joseph E. Stiglitz, The Roaring Nineties (New York: W W Norton, 2003). 6. An expression used by the philanthropist George Soros. 7. Matthew Miller, The Two Percent Solution: Fixing America's Problems in Ways Liberals and Conservatives Can Love (New York: PublicAffairs, 2003). This is how Miller phrases the issue in the prologue to his book: "We'll first step back and lay a little philosophical groundwork by examining the pervasive role of luck in 2 94 NOTES TO PAGES 1 1 –1 6 NOTES TO PAGES xvir-1 1 life, and how taking life's 'pre-birth lottery' seriously can bring the consensus we need to make progress." 8.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

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The model predicts a very low equity risk premium (well below 1%) due to the low observed volatility of consumption growth and low observed correlation between consumption growth and asset returns, unless an extremely high risk aversion coefficient is used. A huge academic literature has tried to reconcile this puzzle, using market frictions (borrowing constraints, limited market participation, incomplete markets, and idiosyncratic risk), non-standard utility functions (habit formation, recursive utility), modified consumption data (durable goods, luxury goods, long-term consumption risk), and biased sample explanations (survivorship bias among countries studied, absence of negative rare events in the sample, unexpected repricing of equities or bonds) as rational explanations for high observed equity outperformance—but there is little consensus to date.

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Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown by Philip Mirowski

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Neoliberalism has therefore expanded to become a comprehensive worldview, and has not been just a doctrine solely confined to economists.77 With regard to the crisis, one wing of neoliberals has appealed to natural science concepts of “complexity” to suggest that markets transcend the very possibility of management of systemic risk.78 However, the presumed relationship of the market to nature tends to be substantially different under neoliberalism than under standard neoclassical theory. In brief, neoclassical theory has a far more static conception of market ontology than do the neoliberals. In neoclassical economics, many theoretical accounts portray the market as somehow susceptible to “incompleteness” or “failure,” generally due to unexplained natural attributes of the commodities traded: these are retailed under the rubric of “externalities,” “incomplete markets,” or other “failures.” Neoliberals conventionally reject all such recourse to defects or glitches, in favor of a narrative where evolution and/or “spontaneous order” brings the market to ever more complex states of self-realization, which may escape the ken of mere humans.79 This explains why the NTC has rejected out of hand all neoclassical “market failure” explanations of the crisis. [4] A primary ambition of the neoliberal project is to redefine the shape and functions of the state, not to destroy it.

Adaptive Markets: Financial Evolution at the Speed of Thought by Andrew W. Lo

From the biological perspective, the limitations of Homo economicus are now obvious. Neuroscience and evolutionary biology confirm that rational expectations and the Efficient Markets Hypothesis capture only a portion of the full range of human behavior. That portion isn’t small or unimportant—it provides an excellent first approximation of many financial markets and circumstances, and should never be ignored—but it’s still incomplete. Market behavior, like all human behavior, is the outcome of eons of evolutionary forces. In fact, investors would be wise to adopt the Efficient Markets Hypothesis as the starting point of any business decision. Before launching a venture, asking why your particular idea should succeed, and why The Adaptive Markets Hypothesis • 177 someone else hasn’t already done it, is a valuable discipline that can save you a lot of time and money.