
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. 
[6] Calculating their present value allows the decision maker to aggregate the cashflows (or other returns) to be produced by the asset in the future to a single value at
the date in question, and to thus more readily compare two opportunities; this concept is the starting point for financial decision making. 
[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.

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”. 
Here, as under the certaintycase above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, [5] [30] [15]
as based on currently available information. 
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 underlying and a risk free “bond”; see Rational pricing § Delta hedging (and § Uncertainty below). 
[59] More traditionally, decision trees – which are complementary – have been used to evaluate projects, by incorporating in the valuation (all) possible events (or states)
and consequent management decisions;[60][58] the correct discount rate 
each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then meanvariance efficient portfolios can be formed simply
as a combination of holdings of the riskfree asset and the “market portfolio” (the Mutual fund separation theorem), with the combinations here plotting as the capital market line, or CML. 
Applied to derivative valuation, the price today would simply be: the fourth formula (see above regarding the absence of a risk premium here).

[note 5] The mechanism for determining (corporate) value is provided by [26] [27] John Burr Williams’ The Theory of Investment Value, which proposes that the value of an asset
should be calculated using “evaluation by the rule of present worth”. 
“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 statesoftheworld
can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for n (riskneutral) probabilities, , given n prices. 
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”. 
A direct extension, then, is the concept of a state price security (also called an Arrow–Debreu security), a contract that agrees to pay one unit of a numeraire (a currency
or a commodity) if a particular state occurs (“up” and “down” in the simplified example above) at a particular time in the future and pays zero numeraire in all the other states. 
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. 
[note 9] This result will be independent of the investor’s level of risk aversion and assumed utility function, thus providing a readily determined discount rate for corporate
finance decision makers as above,[32] and for other investors. 
Net present value (NPV) is the direct extension of these ideas typically applied to Corporate Finance decisioning.

managerial – actions that impact underlying value: by incorporating option pricing logic, these actions are then applied to a distribution of future outcomes, changing with
time, which then determine the “project’s” valuation today. 
The subject is thus built on the foundations of microeconomics and derives several key results for the application of decision making under uncertainty to the financial markets.

Black–Litterman instead starts with an equilibrium assumption, and is then modified to take into account the ‘views’ (i.e., the specific opinions about asset returns) of the
investor in question to arrive at a bespoke [44] asset allocation. 
State prices find immediate application as a conceptual tool (“contingent claim analysis”);[6] but can also be applied to valuation problems.

[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. 
It has two main areas of focus:[2] asset pricing and corporate finance; the first being the perspective of providers of capital, i.e.

Portfolio theory[edit] See also: Postmodern portfolio theory and Mathematical finance § Risk and portfolio management: the P world Plot of two criteria when maximizing return
and minimizing risk in financial portfolios (Paretooptimal points in red) Examples of bivariate copulæ used in finance. 
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. 
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 riskfree rate). 
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, . 
[6] [14] [11] The analysis here is often undertaken assuming a representative agent, [15] essentially treating all marketparticipants, “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. 
Whereas these “certainty” results are all commonly employed under corporate finance, uncertainty is the focus of “asset pricing models” as follows.

Uncertainty[edit] 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 equilibriumbased result – and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitragefree result. 
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;[16] the third equation above. 
“[40] It returns the required (expected) return of a financial asset as a linear function of various macroeconomic factors, and assumes that arbitrage should bring incorrectly
priced assets back into line. 
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.

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. 
Similarly, the various shortrate models allow for an extension of these techniques to fixed income and interest rate derivatives.

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. 
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. 
Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered when pricing,[55] and a credit valuation
adjustment, or CVA – and potentially other valuation adjustments, collectively xVA – is generally added to the riskneutral derivative value. 
For a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state price “density”.

For other results, as well as specific models developed here, see the list of “Equity valuation” topics under Outline of finance § Discounted cash flow valuation.

Thus, for a common stock, the “intrinsic”, longterm worth is the present value of its future net cashflows, in the form of dividends.

More formally, the resulting expected utility hypothesis states that, if certain axioms are satisfied, the subjective value associated with a gamble by an individual is that
individual’s statistical expectation of the valuations of the outcomes of that gamble. 
[21] Given the pricing mechanism described, one can decompose the derivative value – true in fact for “every security”[2] – as a linear combination of its stateprices; i.e.

[3][4] It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and
is concerned with deriving testable or policy implications from acceptable assumptions. 
states) and then rearranging for the terms corresponding to and , per the boxed description; see Binomial options pricing model § Relationship with Black–Scholes.

Real options valuation allows that option holders can influence the option’s underlying; models for employee stock option valuation explicitly assume nonrationality on the
part of option holders; Credit derivatives allow that payment obligations or delivery requirements might not be honored. 
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. 
Williams and onward allow for forecasting as to these – based on historic ratios or published dividend policy – and cashflows are then treated as essentially deterministic;
see below under § Corporate finance theory. 
The two concepts are linked as follows: where market prices do not allow for profitable arbitrage, i.e.

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. 
It thus also includes a formal study of the financial markets themselves, especially market microstructure and market regulation.


[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.

As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitragefree with respect to the more fundamental, equilibrium
determined, securities prices; see Asset pricing § Interrelationship. 
prices using all available information are identical to the best guess of the future: the assumption of rational expectations.

Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation.

This is because investors here can then maximize utility through leverage as opposed to pricing; see Separation property (finance), Markowitz model § Choosing the best portfolio
and CML diagram aside. 
Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of Europeanstyled options.

After the financial crisis of 2007–2008, a further development:[54] as outlined, (over the counter) derivative pricing had relied on the BSM risk neutral pricing framework,
under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. 
[39][13] The Black–Scholes theory, although built on Arbitragefree pricing, is therefore consistent with the equilibrium based capital asset pricing.

(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 riskneutral probability (or arbitragepricing probability), denoted by “Q”. 
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 § Arbitragefree pricing and equilibrium above. 
[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. 
(Some investors may in fact be risk seeking as opposed to risk averse, but the same logic would apply).

Resultant models Applying the above economic concepts, we may then derive various economic and financial models and principles.

Thus, continuing the example, in pricing a derivative instrument its forecasted cashflows in the up and downstates, and , are multiplied through by and , and are then discounted
at the riskfree interest rate; per the second equation above. 
The more general HJM Framework describes the dynamics of the full forwardrate curve – as opposed to working with short rates – and is then more widely applied.

• Information: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with future monetary value (FMV).

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”.
Works Cited
[‘Its history is correspondingly early: Fibonacci developed the concept of present value already in 1202 in his Liber Abaci. Compound interest was discussed in depth by 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]
2. ^ These ideas originate with Blaise Pascal and Pierre de Fermat in 1654.
3. ^ The development here is originally due to Daniel Bernoulli in 1738, which was later formalized by John
von Neumann and Oskar Morgenstern in 1947.
4. ^ State prices originate with Kenneth Arrow and Gérard Debreu in 1954.[17] Lionel W. McKenzie is also cited for his independent proof of equilibrium existence in 1954.[18] Breeden and Litzenberger’s
work in 1978[19] established the use of state prices in financial economics.
5. ^ 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,[24] and
1963.[25]
6. ^ John Burr Williams published his “Theory” in 1938; NPV was recommended to corporate managers by Joel Dean in 1951.
7. ^ 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.”[12]: 55
8. ^ The EMH was presented by Eugene
Fama in a 1970 review paper,[31] 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.
9. ^ 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.
10. ^ “BSM” – two seminal 1973 papers by Fischer Black and
Myron Scholes,[34] and Robert C. Merton[35] – is consistent with “previous versions of the formula” of Louis Bachelier (1900) and Edward O. Thorp (1967);[36] although these were more “actuarial” in flavor, and had not established riskneutral discounting.[13]
Vinzenz Bronzin (1908) produced very early results, also.
11. ^ Kiyosi Itô published his Lemma in 1944. Paul Samuelson[37] introduced this area of mathematics into finance in 1965; Robert Merton promoted continuous stochastic calculus and continuoustime
processes from 1969. [38]
12. ^ The singleindex model was developed by William Sharpe in 1963. [41] APT was developed by Stephen Ross in 1976. [42] The linear factor model structure of the APT is used as the basis for many of the commercial risk
systems employed by asset managers.
13. ^ 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.
14. ^
The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (ISBN 013504605X), and in 1979 formalized by Cox, Ross and Rubinstein [46] and by Rendleman and Bartter. [47] Finite difference methods for option pricing were
due to Eduardo Schwartz in 1977.[48] Monte Carlo methods for option pricing were originated by Phelim Boyle in 1977; [49] In 1996, methods were developed for American [50] and Asian options. [51]
15. ^ Oldrich Vasicek developed his pioneering shortrate
model in 1977. [52] The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in 1987. [53]
16. ^ 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.[61] Real options in corporate finance were first discussed by Stewart Myers in 1977.
17. ^ The Benchmark here is the pioneering AFM of the Santa Fe Institute developed in the early
1990s. See [78] for discussion of other early models.
18. ^ 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. Extraordinary Popular Delusions and the Madness of Crowds is a study of crowd psychology by Scottish journalist Charles Mackay, first published in 1841, with Volume I discussing
economic bubbles.
19. ^ 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 GrahamandDoddsville.
20. William F. Sharpe, “Financial Economics” Archived 20040604 at the Wayback Machine, in “MacroInvestment Analysis”. Stanford University (manuscript). Archived from
the original on 20140714. Retrieved 20090806.
21. ^ Jump up to:a b Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, Journal of Portfolio Management. Summer 1999.
22. ^ Robert C. Merton “Nobel Lecture” (PDF). Archived
(PDF) from the original on 20090319. Retrieved 20090806.
23. ^ Jump up to:a b See Fama and Miller (1972), The Theory of Finance, in Bibliography.
24. ^ Jump up to:a b c d e f Christopher L. Culp and John H. Cochrane. (2003). “”Equilibrium
Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management” Archived 20160304 at the Wayback Machine, in Modern Risk Management: A History. Peter Field, ed. London: Risk Books, 2003. ISBN 1904339050
25. ^
Jump up to: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”.)
26. ^ C. Lewin (1970). An early book on compound interest Archived 20161221 at the Wayback Machine, Institute and Faculty of Actuaries
27. ^ James E. Ciecka. 2008. “The First Mathematically Correct
Life Annuity”. Journal of Legal Economics 15(1): pp. 5963
28. ^ James E. Ciecka (2008). “Edmond Halley’s Life Table and Its Uses”. Journal of Legal Economics 15(1): pp. 6574.
29. ^ For example, http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00
Archived 20130529 at the Wayback Machine.
30. ^ Jump up to:a b c d Varian, Hal R. (1987). “The Arbitrage Principle in Financial Economics”. Economic Perspectives. 1 (2): 55–72. JSTOR 1942981.
31. ^ Jump up to:a b See Rubinstein (2006), under
“Bibliography”.
32. ^ Jump up to:a b c Emanuel Derman, A Scientific Approach to CAPM and Options Valuation Archived 20160330 at the Wayback Machine
33. ^ Jump up to:a b Freddy Delbaen and Walter Schachermayer. (2004). “What is… a Free Lunch?”
Archived 20160304 at the Wayback Machine (pdf). Notices of the AMS 51 (5): 526–528
34. ^ Jump up to:a b c d Farmer J. Doyne, Geanakoplos John (2009). “The virtues and vices of equilibrium and the future of financial economics” (PDF). Complexity.
14 (3): 11–38. arXiv:0803.2996. Bibcode:2009Cmplx..14c..11F. doi:10.1002/cplx.20261. S2CID 4506630.
35. ^ Jump up to:a b See: David K. Backus (2015). Fundamentals of Asset Pricing, Stern NYU
36. ^ Arrow, K. J.; Debreu, G. (1954). “Existence of
an equilibrium for a competitive economy”. Econometrica. 22 (3): 265–290. doi:10.2307/1907353. JSTOR 1907353.
37. ^ McKenzie, Lionel W. (1954). “On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems”. Econometrica. 22 (2):
147–161. doi:10.2307/1907539. JSTOR 1907539.
38. ^ Breeden, Douglas T.; Litzenberger, Robert H. (1978). “Prices of StateContingent Claims Implicit in Option Prices”. Journal of Business. 51 (4): 621–651. doi:10.1086/296025. JSTOR 2352653. S2CID
153841737.
39. ^ Jump up to:a b c Figlewski, Stephen (2018). “RiskNeutral Densities: A Review Annual Review of Financial Economics”. Annual Review of Financial Economics. 10: 329–359. doi:10.1146/annurevfinancial110217022944. S2CID 158075926.
SSRN 3120028.
40. ^ Jump up to:a b c See de Matos, as well as Bossaerts and Ødegaard, under bibliography.
41. ^ Jump up to:a b Don M. Chance (2008). “Option Prices and State Prices” Archived 20120209 at the Wayback Machine
42. ^ Jump up to:a
b See Luenberger’s Investment Science, under Bibliography.
43. ^ Modigliani, F.; Miller, M. (1958). “The Cost of Capital, Corporation Finance and the Theory of Investment”. American Economic Review. 48 (3): 261–297. JSTOR 1809766.
44. ^ Modigliani,
F.; Miller, M. (1963). “Corporate income taxes and the cost of capital: a correction”. American Economic Review. 53 (3): 433–443. JSTOR 1809167.
45. ^ Jump up to:a b The New School. “Finance Theory”. Archived from the original on 20060702. Retrieved
20060628.
46. ^ Mark Rubinstein (2002). “Great Moments in Financial Economics: I. Present Value”. Archived from the original on 20070713. Retrieved 20070628.
47. ^ Gonçalo L. Fonseca (N.D.). Irving Fisher’s Theory of Investment. History
of Economic Thought series, The New School.
48. ^ 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.
49. ^ Jump up
to:a b Shiller, Robert J. (2003). “From Efficient Markets Theory to Behavioral Finance” (PDF). Journal of Economic Perspectives. 17 (1 (Winter 2003)): 83–104. doi:10.1257/089533003321164967. Archived (PDF) from the original on 20150412.
50. ^
Fama, Eugene (1970). “Efficient Capital Markets: A Review of Theory and Empirical Work”. Journal of Finance.
51. ^ Jensen, Michael C. and Smith, Clifford W., “The Theory of Corporate Finance: A Historical Overview”. In: The Modern Theory of Corporate
Finance, New York: McGrawHill Inc., pp. 2–20, 1984.
52. ^ See, e.g., Tim Bollerslev (2019). “Risk and Return in Equilibrium: The Capital Asset Pricing Model (CAPM)”
53. ^ Black, Fischer; Myron Scholes (1973). “The Pricing of Options and Corporate
Liabilities”. Journal of Political Economy. 81 (3): 637–654. doi:10.1086/260062. S2CID 154552078. [1]
54. ^ Merton, Robert C. (1973). “Theory of Rational Option Pricing” (PDF). Bell Journal of Economics and Management Science. 4 (1): 141–183. doi:10.2307/3003143.
hdl:1721.1/49331. JSTOR 3003143. [2]
55. ^ Jump up to:a b Haug, E. G. and Taleb, N. N. (2008). Why We Have Never Used the BlackScholesMerton Option Pricing Formula, Wilmott Magazine January 2008
56. ^ Samuelson Paul (1965). “A Rational Theory
of Warrant Pricing” (PDF). Industrial Management Review. 6: 2. Archived (PDF) from the original on 20170301. Retrieved 20170228.
57. ^ Merton, Robert C. “Lifetime Portfolio Selection under Uncertainty: The ContinuousTime Case.” The Review of
Economics and Statistics 51 (August 1969): 247257.
58. ^ Jump up to:a b Don M. Chance (2008). “Option Prices and Expected Returns” Archived 20150923 at the Wayback Machine
59. ^ The Arbitrage Pricing Theory, Chapter VI in Goetzmann, under External
links.
60. ^ Sharpe, William F. (1963). “A Simplified Model for Portfolio Analysis”. Management Science. 9 (2): 277–93. doi:10.1287/mnsc.9.2.277. S2CID 55778045.
61. ^ Ross, Stephen A (19761201). “The arbitrage theory of capital asset pricing”.
Journal of Economic Theory. 13 (3): 341–360. doi:10.1016/00220531(76)900466. ISSN 00220531.
62. ^ Black F. and Litterman R. (1991). “Asset Allocation Combining Investor Views with Market Equilibrium”. Journal of Fixed Income. September 1991,
Vol. 1, No. 2: pp. 718
63. ^ Guangliang He and Robert Litterman (1999). “The Intuition Behind BlackLitterman Model Portfolios”. Goldman Sachs Quantitative Resources Group
64. ^ For a derivation see, for example, “Understanding Market Price of
Risk” (David Mandel, Florida State University, 2015)
65. ^ Cox, J. C.; Ross, S. A.; Rubinstein, M. (1979). “Option pricing: A simplified approach”. Journal of Financial Economics. 7 (3): 229. CiteSeerX 10.1.1.379.7582. doi:10.1016/0304405X(79)900151.
66. ^
Richard J. Rendleman, Jr. and Brit J. Bartter. 1979. “TwoState Option Pricing”. Journal of Finance 24: 10931110. doi:10.2307/2327237
67. ^ Schwartz, E. (January 1977). “The Valuation of Warrants: Implementing a New Approach”. Journal of Financial
Economics. 4: 79–94. doi:10.1016/0304405X(77)90037X.
68. ^ Boyle, Phelim P. (1977). “Options: A Monte Carlo Approach”. Journal of Financial Economics. 4 (3): 323–338. doi:10.1016/0304405x(77)900058. Retrieved June 28, 2012.
69. ^ Carriere,
Jacques (1996). “Valuation of the earlyexercise price for options using simulations and nonparametric regression”. Insurance: Mathematics and Economics. 19: 19–30. doi:10.1016/S01676687(96)000042.
70. ^ Broadie, M.; Glasserman, P. (1996). “Estimating
Security Price Derivatives Using Simulation” (PDF). Management Science. 42 (2): 269–285. CiteSeerX 10.1.1.196.1128. doi:10.1287/mnsc.42.2.269. Retrieved June 28, 2012.
71. ^ Vasicek, O. (1977). “An equilibrium characterization of the term structure”.
Journal of Financial Economics. 5 (2): 177–188. CiteSeerX 10.1.1.164.447. doi:10.1016/0304405X(77)900162.
72. ^ 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
73. ^ Jump up to:a b Didier Kouokap Youmbi (2017). “Derivatives Pricing after the 20072008 Crisis: How the Crisis Changed the Pricing Approach”. Bank of England – Prudential Regulation Authority
74. ^ “PostCrisis
Pricing of Swaps using xVAs” Archived 20160917 at the Wayback Machine, Christian Kjølhede & Anders Bech, Master thesis, Aarhus University
75. ^ John C. Hull and Alan White (2014). Collateral and Credit Issues in Derivatives Pricing. Rotman School
of Management Working Paper No. 2212953
76. ^ Hull, John; White, Alan (2013). “LIBOR vs. OIS: The Derivatives Discounting Dilemma”. Journal of Investment Management. 11 (3): 14–27.
77. ^ Jump up to:a b c d e Aswath Damodaran (2007). “Probabilistic
Approaches: Scenario Analysis, Decision Trees and Simulations”. In Strategic Risk Taking: A Framework for Risk Management. Prentice Hall. ISBN 0137043775
78. ^ Jump up to:a b Damodaran, Aswath (2005). “The Promise and Peril of Real Options” (PDF).
NYU Working Paper (SDRP0502). Archived (PDF) from the original on 20010613. Retrieved 20161214.
79. ^ Smith, James E.; Nau, Robert F. (1995). “Valuing Risky Projects: Option Pricing Theory and Decision Analysis” (PDF). Management Science.
41 (5): 795–816. doi:10.1287/mnsc.41.5.795. Archived (PDF) from the original on 20100612. Retrieved 20170817.
80. ^ See for example: Magee, John F. (1964). “Decision Trees for Decision Making”. Harvard Business Review. July 1964: 795–816. Archived
from the original on 20170816. Retrieved 20170816.
81. ^ Kritzman, Mark (2017). “An Interview with Nobel Laureate Harry M. Markowitz”. Financial Analysts Journal. 73 (4): 16–21. doi:10.2469/faj.v73.n4.3. S2CID 158093964.
82. ^ Jump up to:a
b See Kruschwitz and Löffler under Bibliography.
83. ^ “Capital Budgeting Applications and Pitfalls” Archived 20170815 at the Wayback Machine. Ch 13 in Ivo Welch (2017). Corporate Finance: 4th Edition
84. ^ George Chacko and Carolyn Evans (2014).
Valuation: Methods and Models in Applied Corporate Finance. FT Press. ISBN 0132905221
85. ^ See Jensen and Smith under “External links”, as well as Rubinstein under “Bibliography”.
86. ^ Jensen, Michael; Meckling, William (1976). “Theory of the
firm: Managerial behavior, agency costs and ownership structure”. Journal of Financial Economics. 3 (4): 305–360. doi:10.1016/0304405X(76)90026X.
87. ^ Corporate Finance: First Principles, from Aswath Damodaran (2022). Applied Corporate Finance:
A User’s Manual. Wiley. ISBN 9781118808931
88. ^ Kenneth D. Garbade (2001). Pricing Corporate Securities as Contingent Claims. MIT Press. ISBN 9780262072236
89. ^ 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.
90. ^ O’Hara, Maureen, Market
Microstructure Theory, Blackwell, Oxford, 1995, ISBN 1557864438, p.1.
91. ^ 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 95119
92. ^ Randi Næs, Johannes Skjeltorp (2006). “Is the market microstructure of stock markets important?”. Norges Bank Economic Bulletin 3/06 (Vol. 77)
93. ^ See, e.g., Westerhoff, Frank H.
(2008). “The Use of AgentBased Financial Market Models to Test the Effectiveness of Regulatory Policies”, Journal of Economics and Statistics
94. ^ See, e.g., Mizuta, Takanobu (2019). “An agentbased model for designing a financial market that
works well”. 2020 IEEE Symposium Series on Computational Intelligence (SSCI).
95. ^ Jump up to:a b For a survey see: LeBaron, Blake (2006). “Agentbased Computational Finance”. Handbook of Computational Economics. Elsevier
96. ^ Jump up to:a b
Katalin Boer, Arie De Bruin, Uzay Kaymak (2005). “On the Design of Artificial Stock Markets”. Research In Management ERIM Report Series
97. ^ Jump up to:a b c LeBaron, B. (2002). “Building the Santa Fe artificial stock market”. Physica A, 1, 20.
98. ^
Mandelbrot, Benoit (1963). “The Variation of Certain Speculative Prices” (PDF). The Journal of Business. 36 (Oct): 394–419. doi:10.1086/294632.
99. ^ Jump up to:a b Nassim Taleb and Benoit Mandelbrot. “How the Finance Gurus Get Risk All Wrong” (PDF).
Archived from the original (PDF) on 20101207. Retrieved 20100615.
100. ^ Jump up to:a b Black, Fischer (1989). “How to use the holes in BlackScholes”. Journal of Applied Corporate Finance. 1 (Jan): 67–73. doi:10.1111/j.17456622.1989.tb00175.x.
101. ^
See for example III.A.3, in Carol Alexander, ed. (January 2005). The Professional Risk Managers’ Handbook. PRMIA Publications. ISBN 9780976609704
102. ^ Hagan, Patrick; et al. (2002). “Managing smile risk”. Wilmott Magazine (Sep): 84–108.
103. ^
See for example Pg 217 of: Jackson, Mary; Mike Staunton (2001). Advanced modelling in finance using Excel and VBA. New Jersey: Wiley. ISBN 0471499226.
104. ^ 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 multimoment approximate option pricing models: a general comparison (Part 1)”. Working paper, London School of Economics and Political
Science.
105. ^ 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
106. ^ Pablo Fernandez (2019). “Common Sense and Illogical Models: Finance and Financial Economics”. SSRN 2906887.
107. ^ From The New Palgrave Dictionary of Economics, Online Editions, 2011, 2012, with abstract links:
• “regulatory responses to the financial crisis: an interim assessment” Archived 20130529 at the Wayback Machine by Howard Davies
• “Credit Crunch Chronology: April 2007–September 2009” Archived 20130529 at the Wayback Machine by The Statesman’s
Yearbook team
• “Minsky crisis” Archived 20130529 at the Wayback Machine by L. Randall Wray
• “euro zone crisis 2010” Archived 20130529 at the Wayback Machine by Daniel Gros and Cinzia Alcidi.
• Carmen M. Reinhart and Kenneth S. Rogoff,
2009. This Time Is Different: Eight Centuries of Financial Folly, Princeton. Description Archived 20130118 at the Wayback Machine, ch. 1 (“Varieties of Crises and their Dates”. pp. 320) Archived 20120925 at the Wayback Machine, and chapterpreview
links.
108. ^ William F. Sharpe (1991). “The Arithmetic of Active Management” Archived 20131113 at the Wayback Machine. Financial Analysts Journal Vol. 47, No. 1, January/February
109. ^ William F. Sharpe (2002). Indexed Investing: A Prosaic
Way to Beat the Average Investor Archived 20131114 at the Wayback Machine. Presentation: Monterey Institute of International Studies. Retrieved May 20, 2010.
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