Editing Interest rate cap and floor (section)

===Black model===
The simplest and most common valuation of interest rate caplets is via the [[Black model]]. Under this model we assume that the underlying rate is [[log-normal distribution|distributed log-normally]] with [[Volatility (finance)|volatility]] <math>\sigma</math>. Under this model, a caplet on a [[LIBOR]] expiring at t and paying at T has present value

:<math> V = P(0,T)\left(F N(d_1) - K N(d_2)\right), </math>
where
:''P''(0,''T'') is today's [[discount factor]] for ''T''
:''F'' is the [[forward price]] of the rate. For LIBOR rates this is equal to <math> {1\over \alpha }\left(\frac{P(0,t)}{P(0,T)} - 1\right)</math>
:''K'' is the strike
:''N'' is the standard normal CDF.
:<math>d_1 = \frac{\ln(F/K) + 0.5 \sigma^2t}{\sigma\sqrt{t}}</math>
and
:<math>d_2 = d_1 - \sigma\sqrt{t}</math>

Notice that there is a one-to-one mapping between the volatility and the present value of the option. Because all the other terms arising in the equation are indisputable, there is no ambiguity in quoting the price of a caplet simply by quoting its volatility. This is what happens in the market. The volatility is known as the "Black vol" or [[implied volatility|implied vol]].

As negative interest rates became a possibility and then reality in many countries at around the time of [[Quantitative Easing]], so the Black model became increasingly inappropriate (as it implies a zero probability of negative interest rates). Many substitute methodologies have been proposed, including shifted log-normal, normal and Markov-Functional, though a new standard is yet to emerge.<ref>{{Cite web |url=http://www.d-fine.com/fileadmin/d-fine/hochgeladen/Fachartikel/WhitePaper_Vols_NegIR_V1_1_en.pdf |title=Archived copy |access-date=2016-01-30 |archive-date=2016-02-03 |archive-url=https://web.archive.org/web/20160203015755/http://www.d-fine.com/fileadmin/d-fine/hochgeladen/Fachartikel/WhitePaper_Vols_NegIR_V1_1_en.pdf |url-status=dead }}</ref>