Editing Financial economics (section)

===Derivative pricing===
{{Further|Mathematical finance#Derivatives pricing: the Q world}}
{{see also|Quantitative analyst#History}}
[[File:Arbre Binomial Options Reelles.png|thumb|right| Binomial Lattice with [[Binomial options pricing model#STEP 1: Create the binomial price tree|CRR formulae]] ]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
| {{smalldiv|1=PDE for a zero-coupon bond:
:<math>\frac{1}{2}\sigma(r)^{2}\frac{\partial^2 P}{\partial r^2}+[a(r)+\sigma(r)+\varphi(r,t)]\frac{\partial P}{\partial r}+\frac{\partial P}{\partial t} = rP</math>}}
{{small|[[Bond valuation#Stochastic calculus approach|Interpretation:]] Analogous to Black–Scholes, 
<ref>For a derivation see, for example, [https://www.math.fsu.edu/~dmandel/Primers/Understanding%20Market%20Price%20of%20Risk.pdf "Understanding Market Price of Risk"] (David Mandel, [[Florida State University]], 2015)</ref>
arbitrage arguments describe the instantaneous change in the bond price <math>P</math> for changes in the (risk-free) short rate <math>r</math>; the analyst selects the specific [[short-rate model]] to be employed.
}}
|}
[[Image:volatility smile.svg|thumb|right|Stylized volatility smile: showing the (implied) volatility by strike-price, for which the [[Black–Scholes formula]] returns market prices.]]
In pricing derivatives, the [[binomial options pricing model]] provides a discretized version of Black–Scholes, useful for the valuation of [[American option|American styled options]]. Discretized models of this type are built – at least implicitly – using state-prices ([[#State prices|as above]]); relatedly, a large number of researchers [[contingent claim analysis|have used options]] to extract state-prices for a variety of other applications in financial economics.<ref name="Rubinstein"/><ref name="Chance1"/><ref name="Chance2">Don M. Chance (2008). [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf "Option Prices and State Prices"] {{Webarchive|url=https://web.archive.org/web/20120209215717/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf |date=2012-02-09 }}</ref> For [[Option style#Non-vanilla path-dependent "exotic" options|path dependent derivatives]], [[Monte Carlo methods for option pricing]] are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various [[Option (finance)#Model implementation|other numeric techniques]] have also been developed. The theoretical framework too has been extended such that [[martingale pricing]] is now the standard approach.
{{NoteTag|
The binomial model was first proposed by [[William F. Sharpe|William Sharpe]] in the 1978 edition of ''Investments'' ({{ISBN|013504605X}}), 
and in 1979 formalized by [[John Carrington Cox|Cox]], [[Stephen Ross (economist)|Ross]] and [[Mark Rubinstein|Rubinstein]] 
<ref>{{Cite journal |last1=Cox |first1=J. C. |authorlink1=John Carrington Cox |last2=Ross |first2=S. A. |authorlink2=Stephen Ross (economist)|last3=Rubinstein |first3=M. |authorlink3=Mark Rubinstein|doi=10.1016/0304-405X(79)90015-1 |title=Option pricing: A simplified approach |journal=[[Journal of Financial Economics]]|volume=7 |issue=3 |page=229 |year=1979|citeseerx=10.1.1.379.7582 }}</ref> 
and by Rendleman and Bartter.
<ref>Richard J. Rendleman, Jr. and Brit J. Bartter. 1979. "Two-State Option Pricing". ''[[Journal of Finance]]'' 24: 1093-1110.  {{doi|10.2307/2327237}}</ref>
[[Finite difference methods for option pricing]] were due to [[Eduardo Schwartz]] in 1977.<ref name="Schwartz">{{cite journal |last=Schwartz  |first= E.|date=January 1977|title= The Valuation of Warrants: Implementing a New Approach|journal= [[Journal of Financial Economics]]|volume= 4|pages= 79–94 |url= http://ideas.repec.org/a/eee/jfinec/v4y1977i1p79-93.html|doi=10.1016/0304-405X(77)90037-X }}</ref> 
[[Monte Carlo methods for option pricing]] were originated by [[Phelim Boyle]] in 1977;
<ref>{{cite journal|last1=Boyle |first1=Phelim P. |url=http://ideas.repec.org/a/eee/jfinec/v4y1977i3p323-338.html |access-date=June 28, 2012 |title=Options: A Monte Carlo Approach |journal=Journal of Financial Economics |volume=4 |number=3 |year=1977 |pages=323–338 |doi=10.1016/0304-405x(77)90005-8}}</ref>
In 1996, methods were developed for [[American option|American]]
<ref>{{Cite journal|title=Valuation of the early-exercise price for options using simulations and nonparametric regression |last=Carriere|first=Jacques|date=1996|journal=Insurance: Mathematics and Economics |doi=10.1016/S0167-6687(96)00004-2|volume=19|pages=19–30}}</ref>
and [[Asian option]]s.
<ref>{{cite journal|last1=Broadie |first1=M. |first2=P. |last2=Glasserman |url=http://www.columbia.edu/~mnb2/broadie/Assets/bg_ms_1996.pdf |access-date=June 28, 2012 |title=Estimating Security Price Derivatives Using Simulation |journal=Management Science |volume=42 |issue=2 |year=1996 |pages=269–285 |doi=10.1287/mnsc.42.2.269|citeseerx=10.1.1.196.1128 }}</ref>
}}

Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "[[contingent claim analysis]]"). [[Real options valuation]] allows that option holders can influence the option's underlying; models for [[Employee stock option#Valuation|employee stock option valuation]] explicitly assume non-rationality on the part of option holders; [[Credit derivative]]s allow that payment obligations or delivery requirements might not be honored. [[Exotic derivative]]s are now routinely valued. Multi-asset underlyers are handled via simulation or [[Copula (probability theory)#Quantitative finance|copula based analysis]].

Similarly, the various [[short-rate model]]s allow for an extension of these techniques to [[Fixed income#Derivatives|fixed income-]] and [[interest rate derivative]]s. (The [[Vasicek model|Vasicek]] and [[Cox–Ingersoll–Ross model|CIR]] models are equilibrium-based, while [[Ho–Lee model|Ho–Lee]] and subsequent models are based on arbitrage-free pricing.) The more general [[Heath–Jarrow–Morton framework|HJM Framework]] describes the dynamics of the full [[forward rate|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 security|hybrid securities]], where credit risk is combined with uncertainty re future rates;  see {{section link|Bond valuation|Stochastic calculus approach}} and {{section link|Lattice model (finance)|Hybrid securities}}. 
{{NoteTag|
[[Oldrich Vasicek]] developed his pioneering short-rate model in 1977.
<ref>{{cite journal |last=Vasicek |first=O. |date=1977 |title=An equilibrium characterization of the term structure |journal=[[Journal of Financial Economics]] |volume=5 |issue=2 |pages=177–188 |doi=10.1016/0304-405X(77)90016-2 |citeseerx=10.1.1.164.447 }}</ref> 
The HJM framework originates from the work of [[David Heath (probabilist)|David Heath]], [[Robert A. Jarrow]], and Andrew Morton in 1987.
<ref>David Heath, Robert A. Jarrow, and Andrew Morton (1987). [https://www.jstor.org/stable/2951677 ''Bond pricing and the term structure of interest rates: a new methodology'']  – working paper, Cornell University</ref>
}}

Following the [[Black Monday (1987)|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 volatility|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 [[#Departures from normality|in the next section]].

After the [[2008 financial crisis]], a further development:<ref name="Youmbi">Didier Kouokap Youmbi (2017). "[https://ssrn.com/abstract=2511585  Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach]". [[Bank of England]] – [[Prudential Regulation Authority (United Kingdom)|Prudential Regulation Authority]]</ref> as outlined, ([[Over-the-counter (finance)|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. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. 
Addressing this, therefore, issues such as [[counterparty credit risk]], funding costs and costs of capital are now additionally considered when pricing,<ref>[http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf "Post-Crisis Pricing of Swaps using xVAs"] {{Webarchive|url=https://web.archive.org/web/20160917015231/http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf |date=2016-09-17 }}, Christian Kjølhede & Anders Bech, Master thesis, [[Aarhus University]]</ref> and a [[credit valuation adjustment]], or CVA – and potentially other ''valuation adjustments'', collectively [[xVA]] – is generally added to the risk-neutral derivative value.
The standard economic arguments can be extended to incorporate these various adjustments.<ref name="Hull_White_2">John C. Hull and Alan White (2014). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2212953 Collateral and Credit Issues in Derivatives Pricing]. Rotman School of Management Working Paper No. 2212953</ref>

A related, and perhaps more fundamental change, is that discounting is now on the [[Overnight index swap|Overnight Index Swap]] (OIS) curve, as opposed to [[LIBOR]] as used previously.<ref name="Youmbi"/> This is because post-crisis, the [[overnight rate]]  is considered a better proxy for the "risk-free rate".<ref>{{cite journal |title=LIBOR vs. OIS: The Derivatives Discounting Dilemma |first1=John |last1=Hull |first2=Alan |last2=White |journal=[[Journal of Investment Management]] |volume=11 |issue=3 |year=2013 |pages=14–27}}</ref> (Also, practically, the interest paid on cash [[collateral (finance)|collateral]] is usually the overnight rate; OIS discounting is then, sometimes, referred to as "[[Credit Support Annex|CSA]] discounting".) [[Swap (finance)#Valuation|Swap pricing]] – and, therefore, [[yield curve]] construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "[[multi-curve framework]]" where "forecast curves" are constructed for each floating-leg [[Libor#Maturities|LIBOR tenor]], with discounting on the ''common'' OIS curve.