financial economics

 

  • 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 certainty-case 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 mean-variance efficient portfolios can be formed simply
    as a combination of holdings of the risk-free 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 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.

  • 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: Post-modern 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 (Pareto-optimal 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 risk-free 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 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.

  • 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 equilibrium-based result – and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitrage-free 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 macro-economic 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 short-rate 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 risk-neutral 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”, long-term 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 state-prices; 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 non-rationality 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 arbitrage-free 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 European-styled 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 Arbitrage-free 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 risk-neutral probability (or arbitrage-pricing 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 § Arbitrage-free 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 down-states, and , are multiplied through by and , and are then discounted
    at the risk-free interest rate; per the second equation above.

  • 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.

  • • 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 risk-neutral 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 continuous-time
processes from 1969. [38]
12. ^ The single-index 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
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Photo credit: https://www.flickr.com/photos/belindah/7187035716/’]