incomplete markets Concepts

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi · 24 Dec 2003

-terminal swaptions Lower bounds for Bermudan swaptions Upper bounds for Bermudan swaptions Factor reduction and Bermudan swaptions Interest-rate smiles Key points Further reading Exercises Incomplete markets and jump-diffusion processes 15.1 Introduction 15.2 Modelling jumps with a tree 15.3 Modelling jumps in a continuous framework 15.4 Market

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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

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a sum invested today. The markets implied by such models are said to be incomplete and there is a theory of no-arbitrage pricing in incomplete markets, which obtains bounds instead of unique prices. The issues surrounding the pricing of derivatives in a non-Black-Scholes world are still very current and

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states then we need k linearly independent instruments to hedge with. These observations are mainly of use when we wish to hedge complicated options in incomplete markets. If we develop a model of how vanilla options change in value, then we can use them to hedge the exotic option perfectly although the

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.2 + e-3r The instantaneous correlation is 0.5. What is covariance of fl and f2 over the time interval from 0 to 1? 15 Incomplete markets and jump-diffusion processes 15.1 Introduction We have mentioned the imperfections of the Black-Scholes model of stock-price evolution. In this chapter, we

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the Black-Scholes world. Our secondary purpose in this chapter is to discuss many of the issues, both philosophical and practical, involved in pricing in incomplete markets. In the equity world, the call and put option prices generally display a steeply downward-sloping smile. One explanation of this smile is that it

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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

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prices of options for all strikes and maturities. We shall return to the issue of hedging with options once we have developed more theory. 364 Incomplete markets and junzp-diffusion processes 15.3 Modelling jumps in a continuous framework Suppose we have a stock moving under geometric Brownian motion, with the added

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then have that the final distribution of log S is log(So) + (bt - 1 2) T + a N(0, 1) + N(T) log(J). 366 Incomplete markets and jummp-diffusion processes We can rewrite this as log(So) + (h - r)T + N(T) log(J) + I r - 1 a2 I T + a

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and finer, this number can only get bigger (by the triangle inequality). For a continuously differentiable function it will converge. If the limit exists, 368 Incomplete markets and jump-diffusion processes this number is called the first variation of f. However, for a Brownian motion path, this number will always diverge to

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will define an arbitrage. So the price of the option 0 must be less than or equal to that of any super-replicating portfolio. 370 Incomplete markets and jump-diffusion processes Similarly, if a self-financing portfolio R is such that R(T) < f(ST)=O(T) then we have that 0

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of our replicating portfolio will now always be CBS(t, St) - X. If there are subsequent jumps, X will of course increase, and X 372 Incomplete markets and jump-diffusion processes will, of course, grow at the risk-free rate. At time T the value of our portfolio will therefore be CBS

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slightly feckless but since whatever hedge one chooses, one will do badly in the case of a jump arguably it's pointless to try. 374 Incomplete markets and jump-diffusion processes If we decide to hedge only the diffusive part and bear the risk of jumps, then clearly we should Delta-hedge

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and that therefore no risk-aversion would be attached to them. This is equivalent to saying the pricing measure should be the one with jump Incomplete markets and jump-diffusion processes 376 intensity equal to the real world intensity. Unfortunately, whilst this view may have been accurate in 1976 when Merton wrote

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to decide the appropriateness of jump-diffusion models, one important is- sue is how well they reproduce the market prices of options. As we have Incomplete markets and jump-diffusion processes 378 0.34 0.29 -One Year -.-- Two Years - Three Years Four Years 0.14 0.09 TI 1 f_ I

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] has the same distribution if ) is replaced by the average value of , which we denote by T = T f).(s)ds. 0 (15.20) 380 Incomplete markets and jump-diffusion processes If we consider the evolution of the log, we see that the terminal distribution of the Brownian part of S is

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different? Any option that can be precisely strong statically replicated will have the same price. In practice, this means that the vanilla European options 382 Incomplete markets and jump-diffusion processes will have the same prices but not a lot more. In particular, recalling the results of Section 6.3, the risk

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call option or indeed any option with a convex pay-off is positive for any log-type model. Recall that a function is convex 384 Incomplete markets and jump-diffusion processes if the chord between any two points on the graph lies above the graph, and that convexity is equivalent to the

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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

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a formula for the price of a call option on a dividend-paying stock with constant dividend rate d under a jump-diffusion model. 388 Incomplete markets and jump-diffusion processes Exercise 15.10 Suppose we wish to price an Asian option by Monte Carlo using a jump-diffusion model with log

Unelected Power: The Quest for Legitimacy in Central Banking and the Regulatory State

by Paul Tucker · 21 Apr 2018 · 920pp · 233,102 words

Policy.” Remarks at the Federal Reserve Bank of St. Louis, October 11, 2002. Greenwald, Bruce, and Joseph Stiglitz. “Externalities in Economies with Imperfect Information and Incomplete Markets.” Quarterly Journal of Economics 101, no. 2, 1986. Greenwood, Robin, Samuel G. Hanson, Joshua S. Rudolph, and Lawrence H. Summers. “Government Debt Management at the

Frequently Asked Questions in Quantitative Finance

by Paul Wilmott · 3 Jan 2007 · 345pp · 86,394 words

Further Reading What is a Jump-Diffusion Model and How Does It Affect Option Values? References and Further Reading What is Meant by “Complete” and “Incomplete” Markets? References and Further Reading What is Volatility? References and Further Reading What is the Volatility Smile? References and Further Reading What is GARCH? References and

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-difference method? 136 31. What is a jump-diffusion model and how does it affect option values? 142 32. What is meant by ‘complete’ and ‘incomplete’ markets? 146 33. What is volatility? 151 34. What is the volatility smile? 157 35. What is GARCH? 164 36. How do I dynamically hedge? 170

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cost of setting up the static hedge; a model is not needed. (But then the option wasn’t exotic in the first place.) Superhedging In incomplete markets you cannot eliminate all risk by classical dynamic delta hedging. But sometimes you can superhedge meaning that you construct a portfolio that has a positive

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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

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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

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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

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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

Radical Uncertainty: Decision-Making for an Unknowable Future

by Mervyn King and John Kay · 5 Mar 2020 · 807pp · 154,435 words

stable structure of the world about which we could learn from past experience and use to extrapolate future behaviour. We live in a world of incomplete markets in which there are simply no price signals to guide us back to an efficient equilibrium. There are times when expectations have a life of

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

by Philip Mirowski · 24 Jun 2013 · 662pp · 180,546 words

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

Derivatives Markets

by David Goldenberg · 2 Mar 2016 · 819pp · 181,185 words

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

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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

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option prices. It should be noted that GE models typically involve risk premia due to the problem that not all risks may be fully hedgeable (incomplete markets). If they aren’t, we don’t get the strong preference-free results of partial equilibrium models like the BOPM or Black–Scholes. These models

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, 304; implicit floating-rate bond, valuation of 308 implicit short positions 340 In-the-Money calls 337 In-the-Money covered call writes 421–4 incomplete markets 450–1 independent securities and risks 600 index points 226 individual stock forwards: long position 38–9; short position 41–3 infinitesimal intervals 93 informational

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; fundamental theorem of asset pricing one (FTAP1) 450, 451, 452; fundamental theorem of asset pricing two (FTAP2) 452; general equilibrium (GE) 453; hedge ratio 455; incomplete markets 450–1; key concepts 466; law of one price (LOP) 452; model-independent vs. MBOP 370–1; no-arbitrage, principle of 448; objective of 437

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev · 24 Apr 2007

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

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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

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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

The Euro: How a Common Currency Threatens the Future of Europe

by Joseph E. Stiglitz and Alex Hyde-White · 24 Oct 2016 · 515pp · 142,354 words

://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

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, 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

Adaptive Markets: Financial Evolution at the Speed of Thought

by Andrew W. Lo · 3 Apr 2017 · 733pp · 179,391 words

’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

Adam Smith: Father of Economics

by Jesse Norman · 30 Jun 2018

French, 1953). Bruce Greenwald and Joseph Stiglitz importantly extended the same overall line of argument in their paper ‘Externalities in Economies with Imperfect Information and Incomplete Markets’, Quarterly Journal of Economics, 101.2, 1986. This showed that the effect of imperfect information was to break the link between equilibrium models and any

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presumption of efficiency. The formal conditions of market failure were thus all but inevitable in any real-world situation, with imperfect information, incomplete markets and a host of other frictional factors at work. In such cases—that is, in virtually every case—policy interventions could in principle be made