Financial economics
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Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade".[1] Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus:[2] asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance.
The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".
Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise the relationships identified. Mathematical finance is related in that it will derive and extend the mathematical or numerical models suggested by financial economics. Whereas financial economics has a primarily microeconomic focus,
Underlying economics
Fundamental valuation equation [5] |
Four equivalent formulations,[6] where:
|
Financial economics studies how
Present value, expectation and utility
Underlying all of financial economics are the concepts of present value and expectation.[6]
Calculating their present value allows the decision maker to aggregate the
An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, and respectively. [note 2]
This decision method, however, fails to consider risk aversion ("as any student of finance knows"[6]). In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich, the approach is to therefore "adjust" the weight assigned to the various outcomes ("states") correspondingly, . See
Choice under uncertainty here may then be characterized as the maximization of
The impetus for these ideas arise from various inconsistencies observed under the expected value framework, such as the St. Petersburg paradox and the Ellsberg paradox. [note 3]
Arbitrage-free pricing and equilibrium
JEL classification codes |
In the Journal of Economic Literature classification codes, Financial Economics is one of the 19 primary classifications, at JEL: G. It follows Monetary and International Economics and precedes Public Economics. For detailed subclassifications see JEL classification codes § G. Financial Economics.
The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries in v. 8, Subject Index, including Financial Economics at pp. 863–64. The below have links to entry abstracts of The New Palgrave Online for each primary or secondary JEL category (10 or fewer per page, similar to Google searches):
Tertiary category entries can also be searched.[10] |
The concepts of arbitrage-free, "rational", pricing and equilibrium are then coupled with the above to derive "classical"[11] (or "neo-classical"[12]) financial economics.
Economic equilibrium is, in general, a state in which economic forces such as supply and demand are balanced, and, in the absence of external influences these equilibrium values of economic variables will not change. General equilibrium deals with the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium. (This is in contrast to partial equilibrium, which only analyzes single markets.)
The two concepts are linked as follows: where market prices do not allow for profitable arbitrage, i.e. they comprise an arbitrage-free market, then these prices are also said to constitute an "arbitrage equilibrium". Intuitively, this may be seen by considering that where an arbitrage opportunity does exist, then prices can be expected to change, and are therefore not in equilibrium.[13] An arbitrage equilibrium is thus a precondition for a general economic equilibrium.
The immediate, and formal, extension of this idea, the fundamental theorem of asset pricing, shows that where markets are as described – and are additionally (implicitly and correspondingly) complete – one may then make financial decisions by constructing a risk neutral probability measure corresponding to the market.
"Complete" here means that there is a price for every asset in every possible state of the world, , and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for n (risk-neutral) probabilities, , given n prices. For a simplified example see Rational pricing § Risk neutral valuation, where the economy has only two possible states – up and down – and where and (=) are the two corresponding probabilities, and in turn, the derived distribution, or "measure".
The formal derivation will proceed by arbitrage arguments.[6][13] The analysis here is often undertaken assuming a representative agent, [14] essentially treating all market-participants, "agents", as identical (or, at least, that they act in such a way that the sum of their choices is equivalent to the decision of one individual) with the effect that the problems are then mathematically tractable.
With this measure in place, the expected,
Thus, continuing the example, in pricing a derivative instrument its forecasted cashflows in the up- and down-states, and , are multiplied through by and , and are then discounted at the risk-free interest rate; per the second equation above. In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with and combined. In general, this premium may be derived by the CAPM (or extensions) as will be seen under § Uncertainty.
The difference is explained as follows: By construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by "manufacturing" the instrument as a combination of the
(Correspondingly, mathematical finance separates into two analytic regimes: risk and portfolio management (generally) use physical (or actual or actuarial) probability, denoted by "P"; while derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q". In specific applications the lower case is used, as in the above equations.)
State prices
With the above relationship established, the further specialized Arrow–Debreu model may be derived. [note 4] This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. The Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods.
A direct extension, then, is the concept of a
In the above example, the state prices, , would equate to the present values of and : i.e. what one would pay today, respectively, for the up- and down-state securities; the
State prices find immediate application as a conceptual tool ("
Using the related stochastic discount factor - also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor , and then taking the expectation;[15] the third equation above. Essentially, this factor divides expected utility at the relevant future period - a function of the possible asset values realized under each state - by the utility due to today's wealth, and is then also referred to as "the intertemporal marginal rate of substitution".
Resultant models
DCF valuation formula, where the value of the firm, is its forecasted free cash flows discounted to the present using the weighted average cost of capital. For share valuation investors use the related dividend discount model. |
The capital asset pricing model (CAPM):
The |
The Black–Scholes equation:
spot price on an option price will (must) realize as growth at , the risk free rate, when the option is correctly "manufactured". |
The Black–Scholes formula for the value of a call option:
in the money value - i.e. a specific formulation of the fundamental valuation result. is the probability that the call will be exercised; is the present value of the expected asset price at expiration, given that the asset price at expiration is above the exercise price. |
Applying the above economic concepts, we may then derive various economic- and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information",[1][14] as will be seen below.
- Time: money now is traded for money in the future.
- Uncertainty (or risk): The amount of money to be transferred in the future is uncertain.
- Options: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.
- Information: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with future monetary value (FMV).
Applying this framework, with the above concepts, leads to the required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations.
Certainty
The starting point here is "Investment under certainty", and usually framed in the context of a corporation. The Fisher separation theorem, asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is the Modigliani–Miller theorem, which shows that, under certain conditions, the value of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as a benchmark for evaluating the effects of factors outside the model that do affect value. [note 5]
The mechanism for determining (corporate) value is provided by [25] [26]
These "certainty" results are all commonly employed under corporate finance; uncertainty is the focus of "asset pricing models", as follows. Fisher's formulation of the theory here - developing an intertemporal equilibrium model - underpins also [25] the below applications to uncertainty. [note 7] See [27] for the development.
Uncertainty
For "choice under uncertainty" the twin assumptions of rationality and market efficiency, as more closely defined, lead to modern portfolio theory (MPT) with its capital asset pricing model (CAPM) – an equilibrium-based result – and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitrage-free result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see Asset pricing § Interrelationship.
Briefly, and intuitively – and consistent with § Arbitrage-free pricing and equilibrium above – the relationship between rationality and efficiency is as follows.[28] Given the ability to profit from private information, self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient, prices: the efficient-market hypothesis, or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See earnings response coefficient.) The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the best guess of the future: the assumption of rational expectations. The EMH does allow that when faced with new information, some investors may overreact and some may underreact, but what is required, however, is that investors' reactions follow a normal distribution – so that the net effect on market prices cannot be reliably exploited to make an abnormal profit. In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news:[29] the random walk hypothesis. This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution. [note 8]
Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium. Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, [5] [29] [14] as based on currently available information. What is required though, is a theory for determining the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM.
In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are. [note 9] This result will be independent of the investor's level of risk aversion and assumed
Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of European-styled options. [note 10] The model is expressed as the Black–Scholes equation, a partial differential equation describing the changing price of the option over time; it is derived assuming log-normal, geometric Brownian motion (see Brownian model of financial markets). The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing (, the value, or price, of the option, grows at , the risk-free rate).[6][5] This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is the solution to the equation.) Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with § Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation. Itô's lemma provides the underlying mathematics, and, with Itô calculus more generally, remains fundamental in quantitative finance. [note 11]
As mentioned, it can be shown that the two models are consistent; then, as is to be expected, "classical" financial economics is thus unified. Here, the Black Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM.[37][12] The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the fundamental result above – further explaining, and if required demonstrating, this unity.[6] Here, the CAPM is derived by linking , risk aversion, to overall market return, and setting the return on security as ; see Stochastic discount factor § Properties. The Black-Scholes formula is found, in the limit, by attaching a
Extensions
More recent work further generalizes and extends these models. As regards asset pricing, developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models.
Portfolio theory
The majority of developments here relate to required return, i.e. pricing, extending the basic CAPM. Multi-factor models such as the Fama–French three-factor model and the Carhart four-factor model, propose factors other than market return as relevant in pricing. The intertemporal CAPM and consumption-based CAPM similarly extend the model. With intertemporal portfolio choice, the investor now repeatedly optimizes her portfolio; while the inclusion of consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return.
Whereas the above extend the CAPM, the single-index model is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization. The arbitrage pricing theory (APT) similarly differs as regards its assumptions. APT "gives up the notion that there is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns."[38] It returns the required (expected) return of a financial asset as a linear function of various macro-economic factors, and assumes that arbitrage should bring incorrectly priced assets back into line.[note 12]
As regards portfolio optimization, the Black–Litterman model[41] departs from the original
Derivative pricing
PDE for a zero-coupon bond:
Interpretation: Analogous to Black-Scholes, [43] arbitrage arguments describe the instantaneous change in the bond price for changes in the (risk-free) short rate ; the analyst selects the specific short-rate model to be employed. |
In pricing derivatives, the
Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "
Similarly, the various short-rate models allow for an extension of these techniques to fixed income- and interest rate derivatives. (The Vasicek and CIR models are equilibrium-based, while Ho–Lee and subsequent models are based on arbitrage-free pricing.) The more general HJM Framework describes the dynamics of the full forward-rate curve – as opposed to working with short rates – and is then more widely applied. The valuation of the underlying instrument – additional to its derivatives – is relatedly extended, particularly for hybrid securities, where credit risk is combined with uncertainty re future rates; see Bond valuation § Stochastic calculus approach and Lattice model (finance) § Hybrid securities. [note 15]
Following the Crash of 1987, equity options traded in American markets began to exhibit what is known as a "volatility smile"; that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus implied volatilities, than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here – as well as implications re the standard theory – are discussed in the next section.
After the
A related, and perhaps more fundamental change, is that discounting is now on the
Corporate finance theory
Mirroring the above developments, asset-valuation and decisioning no longer need assume "certainty".
Related to this, is the treatment of forecasted cashflows in
Other developments here include
"Corporate finance" as a discipline more generally, per Fisher above, relates to the long term objective of maximizing the value of the firm - and its return to shareholders - and thus also incorporates the areas of capital structure and dividend policy. [66] Extensions of the theory here then also consider these latter, as follows: (i) optimization re capitalization structure, and theories here as to corporate choices and behavior: Capital structure substitution theory, Pecking order theory, Market timing hypothesis, Trade-off theory; (ii) considerations and analysis re dividend policy, additional to - and sometimes contrasting with - Modigliani-Miller, include: the Walter model, Lintner model, and Residuals theory, as well as discussion re the observed clientele effect and dividend puzzle.
As described, the typical application of real options is to capital budgeting type problems. However, here, they are also applied to problems of capital structure and dividend policy, and to the related design of corporate securities; [67] and since stockholder and bondholders have different objective functions, in the analysis of the related agency problems. [57] In all of these cases, state-prices can provide the market-implied information relating to the corporate, as above, which is then applied to the analysis. For example, convertible bonds can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.[20][61]
Financial markets
The discipline, as outlined, also includes a formal study of financial markets. Of interest especially are market regulation and market microstructure, and their relationship to price efficiency.
Regulatory economics studies, in general, the economics of regulation. In the context of finance, it will address the impact of financial regulation on the functioning of markets and the efficiency of prices, while also weighing the corresponding increases in market confidence and financial stability. Research here considers how, and to what extent, regulations relating to disclosure (earnings guidance, annual reports), insider trading, and short-selling will impact price efficiency, the cost of equity, and market liquidity.[68]
Market microstructure is concerned with the details of how exchange occurs in markets (with
For both regulation [72] and microstructure,[73] and generally,[74] agent-based models can be developed[75] to examine any impact due to a change in structure or policy - or to make inferences re market dynamics - by testing these in an artificial financial market, or AFM. [note 17] This approach, essentially
These 'bottom-up' models "start from first principals of agent behavior",[76] with participants modifying their trading strategies having learned over time, and "are able to describe macro features [i.e. stylized facts] emerging from a soup of individual interacting strategies".[76] Agent-based models depart further from the classical approach — the representative agent, as outlined — in that they introduce heterogeneity into the environment (thereby addressing, also, the aggregation problem).
Challenges and criticism
As above, there is a very close link between (i) the random walk hypothesis, with the associated belief that price changes should follow a normal distribution, on the one hand, and (ii) market efficiency and rational expectations, on the other. Wide departures from these are commonly observed, and there are thus, respectively, two main sets of challenges.
Departures from normality
As discussed, the assumptions that market prices follow a random walk and that asset returns are normally distributed are fundamental. Empirical evidence, however, suggests that these assumptions may not hold, and that in practice, traders, analysts and risk managers frequently modify the "standard models" (see Kurtosis risk, Skewness risk, Long tail, Model risk). In fact, Benoit Mandelbrot had discovered already in the 1960s [77] that changes in financial prices do not follow a normal distribution, the basis for much option pricing theory, although this observation was slow to find its way into mainstream financial economics. [78]
Closely related is the volatility smile, where, as above, implied volatility – the volatility corresponding to the BSM price – is observed to differ as a function of strike price (i.e. moneyness), true only if the price-change distribution is non-normal, unlike that assumed by BSM. The term structure of volatility describes how (implied) volatility differs for related options with different maturities. An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure. These empirical phenomena negate the assumption of constant volatility – and
Within institutions, the function of Black-Scholes is now, largely, to communicate prices via implied volatilities, much like bond prices are communicated via YTM; see Black–Scholes model § The volatility smile.In consequence traders (and risk managers) now, instead, use "smile-consistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. non-quoted, strike/maturity combinations, or of non-European derivatives, and generally for hedging purposes. The two main approaches are local volatility and stochastic volatility. The first returns the volatility which is "local" to each spot-time point of the finite difference- or simulation-based valuation; i.e. as opposed to implied volatility, which holds overall. In this way calculated prices – and numeric structures – are market-consistent in an arbitrage-free sense. The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first calibrated to observed prices, and are then applied to the valuation or hedging in question; the most common are Heston, SABR and CEV. This approach addresses certain problems identified with hedging under local volatility.[81]
Related to local volatility are the
As discussed, additional to assuming log-normality in returns, "classical" BSM-type models also (implicitly) assume the existence of a credit-risk-free environment, where one can perfectly replicate cashflows so as to fully hedge, and then discount at "the" risk-free-rate. And therefore, post crisis, the various x-value adjustments must be employed, effectively correcting the risk-neutral value for
As mentioned at top, mathematical finance (and particularly financial engineering) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, James Rickards, amongst other critics [78] of financial economics, suggests that, instead, the theory needs revisiting almost entirely:
- "The current system, based on the idea that risk is distributed in the shape of a bell curve, is flawed... The problem is [that economists and practitioners] never abandon the bell curve. They are like medieval astronomers who believe the sun revolves around the earth and are furiously tweaking their geo-centric math in the face of contrary evidence. They will never get this right; they need their Copernicus."[84]
Departures from rationality
Market anomalies and economic puzzles |
|
As seen, a common assumption is that financial decision makers act rationally; see
Consistent with, and complementary to these findings, various persistent market anomalies have been documented, these being price or return distortions – e.g. size premiums – which appear to contradict the efficient-market hypothesis; calendar effects are the best known group here. Related to these are various of the economic puzzles, concerning phenomena similarly contradicting the theory. The equity premium puzzle, as one example, arises in that the difference between the observed returns on stocks as compared to government bonds is consistently higher than the risk premium rational equity investors should demand, an "abnormal return". For further context see Random walk hypothesis § A non-random walk hypothesis, and sidebar for specific instances.
More generally, and particularly following the
A related problem is systemic risk: where companies hold securities in each other then this interconnectedness may entail a "valuation chain" – and the performance of one company, or security, here will impact all, a phenomenon not easily modeled, regardless of whether the individual models are correct. See: Systemic risk § Inadequacy of classic valuation models; Cascades in financial networks; Flight-to-quality.
Areas of research attempting to explain (or at least model) these phenomena, and crises, include
On the obverse, however, various studies have shown that despite these departures from efficiency, asset prices do typically exhibit a random walk and that one cannot therefore consistently outperform market averages, i.e. attain
Relatedly, institutionally inherent limits to arbitrage – as opposed to factors directly contradictory to the theory – are sometimes proposed as an explanation for these departures from efficiency.See also
- Category:Finance theories
- Category:Financial models
- Deutsche Bank Prize in Financial Economics – scholarly award
- Finance § Financial theory
- Fischer Black Prize – economics award
- List of financial economics articles– Overview of finance and finance-related topics
- List of financial economists
- List of unsolved problems in economics § Financial economics
- Master of Financial Economics – graduate degree with a major in financial economics
- Monetary economics – Branch of economics covering theories of money
- Outline of economics – Overview of and topical guide to economics
- Outline of corporate finance – Overview of corporate finance and corporate finance-related topics
- Outline of finance – Overview of finance and finance-related topics
Historical notes
- Richard Witt in 1613, in his Arithmeticall Questions,[7] and was further developed by Johan de Witt in 1671 [8] and by Edmond Halley in 1705.[9]
- ^ These ideas originate with Blaise Pascal and Pierre de Fermat in 1654.
- ^ The development here is originally due to Daniel Bernoulli in 1738, which was later formalized by John von Neumann and Oskar Morgenstern in 1947.
- ^ State prices originate with Kenneth Arrow and Gérard Debreu in 1954.[16] Lionel W. McKenzie is also cited for his independent proof of equilibrium existence in 1954.[17] Breeden and Litzenberger's work in 1978[18] established the use of state prices in financial economics.
- ^ The theorem of Franco Modigliani and Merton Miller is often called the "capital structure irrelevance principle"; it is presented in two key papers of 1958,[23] and 1963.[24]
- ^ John Burr Williams published his "Theory" in 1938; NPV was recommended to corporate managers by Joel Dean in 1951.
- ^ In fact, "Fisher (1930, [The Theory of Interest]) is the seminal work for most of the financial theory of investments during the twentieth century… Fisher develops the first formal equilibrium model of an economy with both intertemporal exchange and production. In so doing, at one swoop, he not only derives present value calculations as a natural economic outcome in calculating wealth, he also justifies the maximization of present value as the goal of production and derives determinants of the interest rates that are used to calculate present value."[11]: 55
- ^ The EMH was presented by Eugene Fama in a 1970 review paper,[30] consolidating previous works re random walks in stock prices: Jules Regnault (1863); Louis Bachelier (1900); Maurice Kendall (1953); Paul Cootner (1964); and Paul Samuelson (1965), among others.
- ^ The efficient frontier was introduced by Harry Markowitz in 1952. The CAPM was derived by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) independently.
- ^ "BSM" – two seminal 1973 papers by Fischer Black and Myron Scholes,[32] and Robert C. Merton[33] – is consistent with "previous versions of the formula" of Louis Bachelier (1900) and Edward O. Thorp (1967);[34] although these were more "actuarial" in flavor, and had not established risk-neutral discounting.[12] Vinzenz Bronzin (1908) produced very early results, also.
- ^ Kiyosi Itô published his Lemma in 1944. Paul Samuelson[35] introduced this area of mathematics into finance in 1965; Robert Merton promoted continuous stochastic calculus and continuous-time processes from 1969. [36]
- ^ The single-index model was developed by William Sharpe in 1963. [39] APT was developed by Stephen Ross in 1976. [40] The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers.
- ^ The universal portfolio algorithm was published by Thomas M. Cover in 1991. The Black–Litterman model was developed in 1990 at Goldman Sachs by Fischer Black and Robert Litterman, and published in 1991.
- ^
The binomial model was first proposed by ISBN 013504605X), and in 1979 formalized by Cox, Ross and Rubinstein [44] and by Rendleman and Bartter. [45] Finite difference methods for option pricing were due to Eduardo Schwartz in 1977.[46] Monte Carlo methods for option pricing were originated by Phelim Boyle in 1977; [47] In 1996, methods were developed forAmerican[48] and Asian options. [49]
- ^
Oldrich Vasicekdeveloped his pioneering short-rate model in 1977. [50] The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in 1987. [51]
- ^ Simulation was first applied to (corporate) finance by David B. Hertz in 1964. Decision trees, a standard operations research tool, were applied to corporate finance also in the 1960s.[59] Real options in corporate finance were first discussed by Stewart Myers in 1977.
- ^ The Benchmark here is the pioneering AFM of the Santa Fe Institute developed in the early 1990s. See [76] for discussion of other early models.
- ^
An early anecdotal treatment is Benjamin Graham's "Mr. Market", discussed in his The Intelligent Investor in 1949.
See also John Maynard Keynes' 1936 discussion of "Animal spirits", and the related Keynesian beauty contest, in his General Theory, Ch. 12.
economic bubbles.
- ^ Burton Malkiel's A Random Walk Down Wall Street – first published in 1973, and in its 13th edition as of 2023 – is a widely read popularization of these arguments. See also John C. Bogle's Common Sense on Mutual Funds; but compare Warren Buffett's The Superinvestors of Graham-and-Doddsville.
References
- ^ a b William F. Sharpe, "Financial Economics" Archived 2004-06-04 at the Wayback Machine, in "Macro-Investment Analysis". Stanford University (manuscript). Archived from the original on 2014-07-14. Retrieved 2009-08-06.
- ^ Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, Journal of Portfolio Management. Summer 1999.
- ^ Robert C. Merton "Nobel Lecture" (PDF). Archived (PDF) from the original on 2009-03-19. Retrieved 2009-08-06.
- ^ a b See Fama and Miller (1972), The Theory of Finance, in Bibliography.
- ^ ISBN 1904339050
- ^ a b c d e f g h i j k Rubinstein, Mark. (2005). "Great Moments in Financial Economics: IV. The Fundamental Theorem (Part I)", Journal of Investment Management, Vol. 3, No. 4, Fourth Quarter 2005; ~ (2006). Part II, Vol. 4, No. 1, First Quarter 2006. See under "External links".
- ^ C. Lewin (1970). An early book on compound interest Archived 2016-12-21 at the Wayback Machine, Institute and Faculty of Actuaries
- ^ James E. Ciecka. 2008. "The First Mathematically Correct Life Annuity". Journal of Legal Economics 15(1): pp. 59-63
- ^ James E. Ciecka. 2008. "Edmond Halley’s Life Table and Its Uses". Journal of Legal Economics 15(1): pp. 65-74.
- ^ For example, http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00 Archived 2013-05-29 at the Wayback Machine.
- ^ a b See Rubinstein (2006), under "Bibliography".
- ^ a b c Emanuel Derman, A Scientific Approach to CAPM and Options Valuation Archived 2016-03-30 at the Wayback Machine
- ^ a b Freddy Delbaen and Walter Schachermayer. (2004). "What is... a Free Lunch?" Archived 2016-03-04 at the Wayback Machine (pdf). Notices of the AMS 51 (5): 526–528
- ^ S2CID 4506630.
- ^ a b See: David K. Backus (2015). Fundamentals of Asset Pricing, Stern NYU
- JSTOR 1907353.
- JSTOR 1907539.
- S2CID 153841737.
- ^ SSRN 3120028.
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- ^ a b Don M. Chance (2008). "Option Prices and State Prices" Archived 2012-02-09 at the Wayback Machine
- ^ a b See Luenberger's Investment Science, under Bibliography.
- JSTOR 1809766.
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- ^ a b The New School. "Finance Theory". Archived from the original on 2006-07-02. Retrieved 2006-06-28.
- ^ Mark Rubinstein (2002). "Great Moments in Financial Economics: I. Present Value". Archived from the original on 2007-07-13. Retrieved 2007-06-28.
- ^ Gonçalo L. Fonseca (N.D.). Irving Fisher's Theory of Investment. History of Economic Thought series, The New School.
- ^ For a more formal treatment, see, for example: Eugene F. Fama. 1965. Random Walks in Stock Market Prices. Financial Analysts Journal, September/October 1965, Vol. 21, No. 5: 55–59.
- ^ (PDF) from the original on 2015-04-12.
- ^ Fama, Eugene (1970). "Efficient Capital Markets: A Review of Theory and Empirical Work". Journal of Finance.
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- ^ a b Haug, E. G. and Taleb, N. N. (2008). Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula, Wilmott Magazine January 2008
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- ^ Merton, Robert C. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case." The Review of Economics and Statistics 51 (August 1969): 247-257.
- ^ a b Don M. Chance (2008). "Option Prices and Expected Returns" Archived 2015-09-23 at the Wayback Machine
- ^ The Arbitrage Pricing Theory, Chapter VI in Goetzmann, under External links.
- S2CID 55778045.
- ISSN 0022-0531.
- Journal of Fixed Income. September 1991, Vol. 1, No. 2: pp. 7-18
- ^ Guangliang He and Robert Litterman (1999). "The Intuition Behind Black-Litterman Model Portfolios". Goldman Sachs Quantitative Resources Group
- ^ For a derivation see, for example, "Understanding Market Price of Risk" (David Mandel, Florida State University, 2015)
- .
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- . Retrieved June 28, 2012.
- .
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- ^ David Heath, Robert A. Jarrow, and Andrew Morton (1987). Bond pricing and the term structure of interest rates: a new methodology – working paper, Cornell University
- ^ a b Didier Kouokap Youmbi (2017). "Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach". Bank of England – Prudential Regulation Authority
- ^ "Post-Crisis Pricing of Swaps using xVAs" Archived 2016-09-17 at the Wayback Machine, Christian Kjølhede & Anders Bech, Master thesis, Aarhus University
- ^ John C. Hull and Alan White (2014). Collateral and Credit Issues in Derivatives Pricing. Rotman School of Management Working Paper No. 2212953
- ^ Hull, John; White, Alan (2013). "LIBOR vs. OIS: The Derivatives Discounting Dilemma". Journal of Investment Management. 11 (3): 14–27.
- ^ ISBN 0137043775
- ^ a b Damodaran, Aswath (2005). "The Promise and Peril of Real Options" (PDF). NYU Working Paper (S-DRP-05-02). Archived (PDF) from the original on 2001-06-13. Retrieved 2016-12-14.
- (PDF) from the original on 2010-06-12. Retrieved 2017-08-17.
- ^ See for example: Magee, John F. (1964). "Decision Trees for Decision Making". Harvard Business Review. July 1964: 795–816. Archived from the original on 2017-08-16. Retrieved 2017-08-16.
- S2CID 158093964.
- ^ a b See Kruschwitz and Löffler under Bibliography.
- ^ "Capital Budgeting Applications and Pitfalls" Archived 2017-08-15 at the Wayback Machine. Ch 13 in Ivo Welch (2017). Corporate Finance: 4th Edition
- ISBN 0132905221
- ^ See Jensen and Smith under "External links", as well as Rubinstein under "Bibliography".
- .
- ISBN 978-1118808931
- ISBN 9780262072236
- ^ See for example: Hazem Daouk, Charles M.C. Lee, David Ng. (2006). "Capital Market Governance: How Do Security Laws Affect Market Performance?". Journal of Corporate Finance, Volume 12, Issue 3; Emilios Avgouleas (2010). "The Regulation of Short Sales and its Reform" DICE Report, Vol. 8, Iss. 1.
- ISBN 1-55786-443-8, p.1.
- ^ King, Michael, Osler, Carol and Rime, Dagfinn (2013). "The market microstructure approach to foreign exchange: Looking back and looking forward", Journal of International Money and Finance. Volume 38, November 2013, Pages 95-119
- ^ Randi Næs, Johannes Skjeltorp (2006). "Is the market microstructure of stock markets important?". Norges Bank Economic Bulletin 3/06 (Vol. 77)
- ^ See, e.g., Westerhoff, Frank H. (2008). "The Use of Agent-Based Financial Market Models to Test the Effectiveness of Regulatory Policies", Journal of Economics and Statistics
- ^ See, e.g., Mizuta, Takanobu (2019). "An agent-based model for designing a financial market that works well". 2020 IEEE Symposium Series on Computational Intelligence (SSCI).
- ^ a b For a survey see: LeBaron, Blake (2006). "Agent-based Computational Finance". Handbook of Computational Economics. Elsevier
- ^ a b Katalin Boer, Arie De Bruin, Uzay Kaymak (2005). "On the Design of Artificial Stock Markets". Research In Management ERIM Report Series
- ^ a b c LeBaron, B. (2002). "Building the Santa Fe artificial stock market". Physica A, 1, 20.
- doi:10.1086/294632.
- ^ a b Nassim Taleb and Benoit Mandelbrot. "How the Finance Gurus Get Risk All Wrong" (PDF). Archived from the original (PDF) on 2010-12-07. Retrieved 2010-06-15.
- ^ .
- ISBN 978-0976609704
- Wilmott Magazine(Sep): 84–108.
- ISBN 0-471-49922-6.
- ^ These include: Jarrow and Rudd (1982); Corrado and Su (1996); Brown and Robinson (2002); Backus, Foresi, and Wu (2004).
See: Emmanuel Jurczenko, Bertrand Maillet, and Bogdan Negrea (2002). "Revisited multi-moment approximate option pricing models: a general comparison (Part 1)". Working paper, London School of Economics and Political Science.
- ^ The Risks of Financial Modeling: VAR and the Economic Meltdown, Hearing before the Subcommittee on Investigations and Oversight, Committee on Science and Technology, House of Representatives, One Hundred Eleventh Congress, first session, September 10, 2009
- SSRN 2906887.
- Kenneth S. Rogoff, 2009. This Time Is Different: Eight Centuries of Financial Folly, Princeton. Description Archived 2013-01-18 at the Wayback Machine, ch. 1 ("Varieties of Crises and their Dates". pp. 3-20) Archived 2012-09-25 at the Wayback Machine, and chapter-preview links.
- ^ William F. Sharpe (1991). "The Arithmetic of Active Management" Archived 2013-11-13 at the Wayback Machine. Financial Analysts Journal Vol. 47, No. 1, January/February
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{{cite book}}
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