Editing Interest rate cap and floor (section)

==Valuation of interest rate caps==
The size of cap and floor premiums are impacted by a wide range of factors, as follows; the price calculation itself is performed by one of several approaches discussed below.
* The relationship between the strike rate and the prevailing 3-month LIBOR
** premiums are highest for in the money options and lower for at the money and out of the money options
* Premiums increase with maturity.
** The option seller must be compensated more for committing to a fixed-rate for a longer period of time.
* Prevailing economic conditions, the shape of the [[yield curve]], and the volatility of interest rates.
** upsloping [[yield curve]]—caps will be more expensive than floors.
** the steeper is the slope of the [[yield curve]], [[ceteris paribus]], the greater are the cap premiums.
** floor premiums reveal the opposite relationship.

===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>

===As a bond put===
It can be shown that a cap on a LIBOR from ''t'' to ''T'' is equivalent to a multiple of a ''t''-expiry put on a ''T''-maturity bond. Thus if we have an interest rate model in which we are able to value bond puts, we can value interest rate caps. Similarly a floor is equivalent to a certain bond call. Several popular [[short-rate model]]s, such as the [[Hull–White model]] have this degree of tractability. Thus we can value caps and floors in those models.

===Valuation of CMS Caps===
Caps based on an underlying rate (like a Constant Maturity Swap Rate) cannot be valued using simple techniques described above. The methodology for valuation of CMS Caps and Floors can be referenced in more advanced papers.