formulasearchengine - User contributions [en]

Dihedral group

2015-01-10T15:16:08Z

131.111.185.41: /* Equivalent definitions */ corrected notation, as the old one was ambiguous, for instance, dih_12 could be written as dih_6

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Square-cube law

2015-01-08T14:02:17Z

131.111.185.44: /* Biomechanics */ This placement seems to imply that the paragraph is a justification for Godzilla and King Kong, which it isn't

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Field of view in video games

2014-12-04T13:09:40Z

131.111.185.44: /* Low FOV and illness */

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Rescorla–Wagner model

2013-11-07T10:51:49Z

131.111.185.74:

In [[financial mathematics]], the '''Hull–White model''' is a [[mathematical model|model]] of future [[interest rate]]s. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a [[Lattice model (finance)|tree or lattice]] and so [[interest rate derivative]]s such as [[bermudan swaption]]s can be valued in the model.

The first Hull–White model was described by [[John C. Hull]] and [[Alan White (economist)|Alan White]] in 1990. The model is still popular in the market today.

==The model==

===One-factor model===
The model is a [[short-rate model]]. In general, it has dynamics

:<math>dr(t) = (\theta(t) - \alpha(t) r(t))\,dt + \sigma\, dW(t)\,\!</math>

There is a degree of ambiguity amongst practitioners about exactly which parameters in the model are time-dependent or what name to apply to the model in each case.
The most commonly accepted hierarchy has

:''θ'' and ''α'' constant – '''the [[Vasicek model]]'''
:''θ'' has ''t'' dependence – '''the Hull-White model'''
:''θ'' and ''α'' also time-dependent – '''the extended [[Vasicek model]]'''

===Two-factor model===
The two-factor Hull–White model {{harvcol|Hull|2006|pp=657–658}} contains an additional disturbance term whose mean reverts to zero, and is of the form:

:<math>d\,f(r(t)) = \left [\theta(t) + u - \alpha(t)\,f(r(t))\right ]dt + \sigma_1(t)\, dW_1(t)\!</math>

where <math>\displaystyle u</math> has an initial value of 0 and follows the process:

:<math>du = -bu\,dt + \sigma_2\,dW_2(t)</math>

==Analysis of the one-factor model==
For the rest of this article we assume only <math>\theta </math> has t-dependence.
Neglecting the stochastic term for a moment, notice that the change in r is negative if r is currently "large" (greater than θ(''t'')/α) and positive if the current value is small. That is, the stochastic process is a [[mean reversion|mean-reverting]] [[Ornstein–Uhlenbeck process]].

θ is calculated from the initial [[yield curve]] describing the current term structure of interest rates. Typically α is left as a user input (for example it may be estimated from historical data). σ is determined via [[calibration]] to a set of [[caplet]]s and [[swaption]]s readily tradeable in the market.

When <math>\alpha</math>, <math>\theta</math> and <math>\sigma</math> are constant, [[Itô's lemma]] can be used to prove that

:<math> r(t) = e^{-\alpha t}r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right) + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u)\,\!</math>

which has distribution
:<math>r(t) \sim N\left(e^{-\alpha t} r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right), \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)\right).</math>

where <math>N( \cdot ,\cdot )</math> is the [[normal distribution]].

==Bond pricing using the Hull–White model==

It turns out that the time-''S'' value of the ''T''-maturity [[discount bond]] has distribution (note the ''affine term'' structure here!)

:<math>P(S,T) = A(S,T)\exp(-B(S,T)r(S))\!</math>

where

:<math> B(S,T) = \frac{1-\exp(-\alpha(T-S))}{\alpha} \,</math>
:<math> A(S,T) = \frac{P(0,T)}{P(0,S)}\exp\left( \, -B(S,T) \frac{\partial\log(P(0,S))}{\partial S} - \frac{\sigma^2(\exp(-\alpha T)-\exp(-\alpha S))^2(\exp(2\alpha S)-1)}{4\alpha^3}\right) \,</math>

Note that their terminal distribution for ''P''(''S'',''T'') is [[log-normal distribution|distributed log-normally]].

==Derivative pricing==

By selecting as [[numeraire]] the time-''S'' bond (which corresponds to switching to the S-forward measure), we have from the [[fundamental theorem of arbitrage-free pricing]], the value at time 0 of a derivative which has payoff at time ''S''.

:<math>V(t) = P(t,S)\mathbb{E}_S[V(S)| \mathcal{F}(t)].\,</math>

Here, <math>\mathbb{E}_S</math> is the expectation taken with respect to the [[forward measure]]. Moreover that standard arbitrage arguments show
that the time ''T'' forward price <math>F_V(t,T)</math> for a payoff at time ''T'' given by ''V(T)'' must satisfy <math>F_V(t,T) = V(t)/P(t,S)</math>, thus
:<math>F_V(t,T) = \mathbb{E}_T[V(T)|\mathcal{F}(t)].\,</math>

Thus it is possible to value many derivatives ''V'' dependent solely on a single bond ''P''(''S'',''T'') analytically when working in the Hull–White model. For example in the case of a [[put option|bond put]]
:<math>V(S) = (K-P(S,T))^+.\,</math>

Because ''P''(''S'',''T'') is lognormally distributed, the general calculation
used for Black-Scholes shows that

:<math>{E}_S[(K-P(S,T))^{+}] = KN(-d_2) - F(t,S,T)N(d_1)\,</math>

where

:<math>d_1 = \frac{\log(F/K) + \sigma_P^2S/2}{\sigma_P \sqrt{S}}\,</math>

and

:<math>d_2 = d_1 - \sigma_P \sqrt{S}.\,</math>

Thus today's value (with the ''P''(0,''S'') multiplied back in) is:

:<math>P(0,S)KN(-d_2) - P(0,T)N(-d_1)\,</math>

Here σ''P'' is the standard deviation of the
log-normal distribution for ''P''(''S'',''T''). A fairly substantial amount
of algebra shows that it is related to the original parameters via

:<math>\sqrt{S}\sigma_P
=\frac{\sigma}{\alpha}(1-\exp(-\alpha(T-S)))\sqrt{\frac{1-\exp(-2\alpha S)}{2\alpha}}\,</math>

Note that this expectation was done in the S-bond measure, whereas we did not specify a measure at all for the original Hull-White process. This does not matter — the volatility is all that matters and is measure-independent.

Because [[interest rate caps/floors]] are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model. [[Jamshidian's trick]] applies to Hull-White (as today's value of a swaption in HW is a [[monotonic function]] of today's short rate). Thus knowing how to price caps is also sufficient for pricing swaptions.

The swaptions can also be priced directly as described in Henrard (2003). The direct implementation is usually more efficient.

==Trees and lattices==
However, valuing vanilla instruments such as caps and swaptions is useful primarily for calibration. The real use of the model is to value somewhat more [[exotic derivatives]] such as [[bermudan swaption]]s on a [[Lattice model (finance)|lattice]], or other derivatives in a multi-currency context such as Quanto Constant Maturity Swaps, as explained for example in Brigo and Mercurio (2001).

==See also==
*[[Vasicek model]]
*[[Cox–Ingersoll–Ross model]]
*[[Black-Karasinski model]]

==References==

'''Primary references'''
*John Hull and Alan White, "Using Hull-White interest rate trees," Journal of Derivatives, Vol. 3, No. 3 (Spring 1996), pp 26–36
*John Hull and Alan White, "Numerical procedures for implementing term structure models I," Journal of Derivatives, Fall 1994, pp 7–16
*John Hull and Alan White, "Numerical procedures for implementing term structure models II," Journal of Derivatives, Winter 1994, pp 37–48
*John Hull and Alan White, "The pricing of options on interest rate caps and floors using the Hull–White model" in Advanced Strategies in Financial Risk Management, Chapter 4, pp 59–67.
*John Hull and Alan White, "One factor interest rate models and the valuation of interest rate derivative securities," Journal of Financial and Quantitative Analysis, Vol 28, No 2, (June 1993) pp 235–254
*John Hull and Alan White, "Pricing interest-rate derivative securities", The Review of Financial Studies, Vol 3, No. 4 (1990) pp. 573–592

'''Other references'''
*{{cite book |last= Hull|first= John C.|authorlink= John C. Hull|title= Options, Futures, and Other Derivatives|edition= 6th|year=2006 |publisher= [[Prentice Hall]] |location= [[Upper Saddle River, New Jersey|Upper Saddle River, N.J]] |isbn= 0-13-149908-4 |oclc= 60321487 |pages= 657–658 |chapter= Interest Rate Derivatives: Models of the Short Rate |ref=harv |lccn= 2005047692 }}
*{{cite book | title = Interest Rate Models — Theory and Practice with Smile, Inflation and Credit| author = [[Damiano Brigo]], [[Fabio Mercurio]] | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
*Henrard, Marc (2003). Explicit Bond Option and Swaption Formula in Heath-Jarrow-Morton One Factor Model, ''International Journal of Theoretical and Applied Finance'', 6(1), 57-72. [http://ssrn.com/abstract=434860 Preprint SSRN].
*Henrard, Marc (2009). Efficient swaptions price in Hull-White one factor model, arXiv, 0901.1776v1. [http://arxiv.org/abs/0901.1776 Preprint arXiv].
*Eugen Puschkarski, [http://web.archive.org/web/*/www.angelfire.com/ny/financeinfo/Diplomnew.ppt ''Implementation of Hull-White´s No-Arbitrage Term Structure Model''], Diploma Thesis, Center for Central European Financial Markets
*Letian Wang, [http://letianwang.net/Fixed_Income/09_Hull-White_Model.htm ''Hull-White Model''], Fixed Income Quant Group, DTCC (detailed numeric example and derivation)

'''Online utilities'''
*[http://lombok.demon.co.uk/financialTap/interestrates/hwtrinomialtree Hull-White Trinomial Tree], Dr. S.H. Man, Turaz.
*[http://lombok.demon.co.uk/financialTap/montecarlo/hullwhite Short Rates Simulation using Hull White Model], Dr. S.H. Man, Turaz.

{{Bond market}}
{{Stochastic processes}}

{{DEFAULTSORT:Hull-White model}}
[[Category:Finance theories]]
[[Category:Mathematical finance]]
[[Category:Interest rates]]
[[Category:Fixed income analysis]]
[[Category:Short-rate models]]

Reflective subcategory

2013-05-07T13:53:26Z

131.111.185.88: fixed typo

Originally, both [[Deutsche Bundesbahn]] and [[Deutsche Reichsbahn of the GDR|Deutsche Reichsbahn]] continued the classification system of the [[Deutsche Reichsbahn-Gesellschaft|Deutsche Reichsbahn (DRG)]] - see also a short overview of the [[Numbering scheme of the German railways|numbering system of the German railways]]. When [[International Union of Railways|UIC]] introduced a new classification system that could be processed by the computers of the late 1960s, DB did a major modification of their system, effective 1 January 1968. This system is still in use and now includes the engines of the former GDR railways as well. (See [[List of Deutsche Bahn AG locomotives and railbuses]] for a current list.)

==Basics==
Since January 1, 1968, all vehicles are denoted by a seven-digit vehicle number. It consists of a three-digit class number, a three-digit serial number and a [[check digit]] separated by a dash. The check digit is calculated by multiplying the first six digits by 1 and 2 alternately, the difference of the result to the next multitude of ten is the check digit. The check digit is used to perform a cross-check to ensure the correct number, for example in computer systems.

Series that contain more than 1,000 vehicles are assigned ascending numbers.

The first number denotes the type of the vehicle, replacing the DRG convention of using letters to differentiate between the vehicle types.

{| class="wikitable"
|-
! Number || Letter(s) || Vehicle type
|-
| align="center" | 0 ||   || [[Steam locomotive]]s
|-
| align="center" | 1 || E || [[Electric locomotive]]s
|-
| align="center" | 2 || V || [[Diesel locomotive]]s
|-
| align="center" | 3 || K || Small [[Shunt (rail)|shunting]] locomotives
|-
| align="center" | 4 || ET || [[Electric multiple units]], not including battery powered
|-
| align="center" | 5 || ETA || Battery powered [[railcar]]s
|-
| align="center" | 6 || VT || [[Diesel multiple units]]
|-
| align="center" | 7 ||   || [[Railbus]]es and work vehicles
|-
| align="center" | 8 || ES, EB || Cab cars and accompanying cars to electric railcars
|-
| align="center" | 9 || VS, VB || Cab cars and accompanying cars to diesel railcars and railbuses
|}

The letters of the DRG schema are not officially used any more since the introduction of computer numbers, however they can often be found in informal writing and spoken conversation. So if somebody is talking of an E110, they likely mean an E10 / 110, if a V216 is mentioned, it is likely the author is intending the V160 / 216.

Tenders are not assigned a number of their own, as they are regarded a part of the vehicle they are coupled to.
[[Cab car]]s and intermediate cars of [[multiple unit]]s receive the numbers 8 if they belong to an [[Electrical Multiple Unit|EMU]], 9 if they belong to a [[Diesel multiple unit|DMU]]. If another powered car or engine is present in the multiple unit, its serial number usually is offset by 500, but it is still assigned into the original class.

For example, a two-car [[DB Class 628|Class 628]] DMU would look like:
* 628 210-7 (powered cab car)
* 928 210-4 (non-powered cab, hence 9 instead of 6)

A [[DB Class 420|class 420]] commuter EMU, for example, would look like this:
* 420 210-1 (powered cab car)
* 421 210-2 (powered intermediate car)
* 420 710-7 (powered cab car, +500)

===Example: 110 494-2===
[[Image:Bahn Logo.jpg|thumb|Electric class 110 locomotive - 110 494-2]]

{| class="wikitable"
|-
! colspan="2" | Meaning of 110 494-2
|- valign="top"
| <tt>110</tt> || Class number → first number (<tt>1</tt>) meaning [[electric locomotive]]
|- valign="top"
| <tt>494</tt> || Serial number
|- valign="top"
| <tt>2</tt> || Check digit
|-
! colspan="2" | How to calculate the check digit
|- valign="top"
| <math>1\!+\!0\!+\!9\,=\,10</math> || Add 1st, 3rd and 5th number.
|- valign="top"
| <math>2\!\cdot\!1\,+\,2\!\cdot\!4\,+\,2\!\cdot\!4\,=\,18</math> || Digit sum of the double of 2nd, 4th and 6th number.
|- valign="top"
| <math>10\!+\!18\,=\,28</math> || Add both results.
|- valign="top"
| <math>30\!-\!28\,=\,2</math> || Difference of the result to the next multiple of ten is the check digit.
|- valign="top"
| <math>\,2</math> || check digit
|- valign="top"
|}

==Steam locomotives==
See also: [[DRG locomotive classification]] for the origins of the system.

The number '''0''' was assigned to steam locomotives, as their end of duty was already foreseeable. The schematics are largely derivative of the DRG system of classification, therefore like in the older system the class numbers were grouped by locomotive types:

{| class="wikitable"
|-
! Number || Vehicle types
|-
| align="center" | 001–019 || [[Express train]] [[tender locomotive]]s
|-
| align="center" | 020–039 || Other [[Passenger train]] tender locomotives
|-
| align="center" | 040–059 || [[Freight train]] tender locomotives
|-
| align="center" | 060–079 || Passenger train [[tank locomotives]]
|-
| align="center" | 080–096 || Freight train tank locomotives
|-
| align="center" | 097|| [[Rack railway|Rack]] locomotives (in theory only - none were left by 1968)
|-
| align="center" | 098 || ''Lokalbahn'' (branch line) engines
|-
| align="center" | 099 || [[narrow gauge railway|narrow gauge]] locomotives
|-
|}
[[Image:Class 55 2967 Front.jpg|right|thumb|200px|Class 55 engine 2967 in late 1950s West Germany.]]
Steam locomotives that had been converted to oil fuelling received separate class numbers from their coal-fuelled counterparts, e. g. oil-fuelled units from the old [[DRG Class 44|class 44]] were assigned to class 043 while coal-fuelled ones were assigned to class 044 under the new numbering plan. The [[DRB Class 50]] (with originally more than 3,000 units, of which DB still had more than 1,000 by the end of 1967) gave rise to BR 050, 051, 052 and 053.

A noteworthy change from the old system was that four-digit engine numbers were no longer possible. Locomotives with such a number would either be put into an overflow class (e. g. ''50 3097'' became ''053 097-2'') or have their number abbreviated to three digits. The latter solution could lead to number conflicts in cases where the original (sub)class had comprised more than 1,000 units. These were dealt with by adjusting one of the numbers involved, e. g., ''38 3711'' became ''038 711-8'' while ''38 1711'' became ''038 710-0''. This also means that for some classes it is not possible to reconstruct the old number from the new one in a purely systematic way.

The last steam locomotives in the [[Deutsche Bundesbahn]] were decommissioned on 26 October 1977 at ''Rheine und Emden'' locomotive depot (''[[Bahnbetriebswerk]]'' or ''Bw''). These were the 043 903, 043 315 and 043 196 .

==Electric engines==
Electric engines were considered the most important method of traction, and hence were assigned the number '''1'''.
They usually were relabelled by replacing the letter "E" with the number "1".
For example, the ''[[Einheits-Elektrolokomotive]]n'' were relabelled from E 40 into 140, E 10 into 110 and E 10.12 into 112.

==Diesel engines==
[[Image:V200 048.jpg|right|thumb|140px|V200 No. 048 in West Germany, circa 1961.]]
In the pre-1968 scheme, diesel engines were assigned the letter "V" (as in the German term ''Verbrennungsmotor'' for combustion engine). The old numbers were directly proportional with the engine power, so that two- and three-digit codes existed. Two-digit codes were transposed one by one (Examples: Class V60 became [[DB Class 260|Class 260]], Class V80 became [[DB Class 280|Class 280]]). Three digit codes generally lost their last digit (V 160 became [[DBAG Class 210, 215 - 218, 219 old|DB Class 216]]). Variants of the previously same class were also assigned individual subclasses, the [[DBAG class 210, 215 - 218, 219 old|V 160 family]], consisting of V 160, V 160 long (V 160.3, V 168), V 162, V 164 and V 169 were assigned the numbers 210, 215, 216, 217, 218 and 219, for example.)

==Small locomotives==
Small shunting locomotives were assigned the number "3". The second number indicates the engine power (according to 1955 standards). The third number differentiates between brakes and transmission (chain drive or Cardan shafts). Small locomotives of the former Ka series were given the new class numbers 381 (pre-war models) and 382 (new models). The existing [[narrow gauge railway|narrow gauge]] locomotives of the [[Wangerooge Island Railway]] became Class 329. In 1987 the [[DB Class V 60|DB Class 260/261]] were also assigned to the small locomotives and labelled as Class 360/361.

==See also==
*[[History of rail transport in Germany]]
*[[Deutsche Bundesbahn]]
*[[UIC classification]]

{{German locomotives}}

{{DEFAULTSORT:Db Locomotive Classification}}
[[Category:Locomotives of Germany]]
[[Category:Locomotive classification systems]]