The standard CRR btree is always drawn with all straight lines, equally spaced vertically and equally spaced horizontally. Therefore you always see nothing but a strict pattern of identical diamonds. Let's zoom into this "geometry".

First, let's set the stage for the discussion. In this conceptual "world", a price (say IBM) can only be observed/sampled at periodic discrete moments, either once a second, or once a day, though the interval should be small relative to time to maturity. Price may change mid-interval, but we can't observe that. Further, during each interval, the price either moves up or down. It can remain unchanged only in a trinomial tree -- not popular in industry.

Why diamonds? Because of interlocking/recombinant. See http://bigblog.tanbin.com/2011/06/option-pricing-recombinant-binomial.html

Why equally spaced horizontally? Because the intervals are fixed and constant -- at each clock tick, the variable must either rise or fall, never stay flat like a trinomial.

Why equally spaced vertically? Because the y-axis is log(price). An interesting feature of the CRR btree. If after n-1 intervals you plot the n price values in a bar chart, they don't fit a straight line -- but try plotting log(price).

Why are all the diamonds identical? Because the nodes are equally spaced both vertically and horizontally.

I feel the regularity is a great simplification and helps us focus on the real issue -- the probability of an upswing at each node -- the transition probability function, which is individually determined at each node position.

## Wednesday, April 18, 2012

### binomial tree - why identical diamonds

Labels: fMathOption, original_content

## my favorite topics (labels)

_fuxi
(302)
_misLabel
(13)
_orig?
(3)
_rm
(2)
_vague
(2)
clarified
(58)
cpp
(39)
cpp_const
(22)
cpp_real
(76)
cpp/java/c#
(101)
cppBig4
(54)
cppSmartPtr
(35)
cppSTL
(33)
cppSTL_itr
(27)
cppSTL_real
(26)
cppTemplate
(28)
creditMkt
(14)
db
(65)
db_sybase
(43)
deepUnder
(31)
dotnet
(20)
ECN
(27)
econ/bank`
(36)
fin/sys_misc
(43)
finGreek
(34)
finReal
(45)
finRisk
(30)
finTechDesign
(46)
finTechMisc
(32)
finVol
(66)
FixedIncom
(28)
fMath
(7)
fMathOption
(33)
fMathStoch
(67)
forex
(39)
gr8IV_Q
(46)
GTD_skill
(15)
GUI_event
(30)
inMemDB
(42)
intuit_math
(41)
intuitFinance
(57)
javaMisc
(68)
javaServerSide
(13)
lambda/delegate
(22)
marketData
(28)
math
(10)
mathStat
(55)
memIssue
(8)
memMgmt
(66)
metaProgram`
(6)
OO_Design
(84)
original_content
(749)
polymorphic/vptr
(40)
productive
(21)
ptr/ref
(48)
py
(28)
reflect
(8)
script`/unix
(82)
socket/stream
(39)
subquery/join
(30)
subvert
(13)
swing/wpf
(9)
sysProgram`
(16)
thread
(164)
thread_CAS
(15)
thread_cpp
(28)
Thread*
(22)
timeSaver
(80)
transactional
(23)
tune
(24)
tuneDB
(40)
tuneLatency
(30)
z_ajax
(9)
z_algoDataStruct
(41)
z_arch
(26)
z_arch_job
(27)
z_automateTest
(17)
z_autoTrad`
(19)
z_bestPractice
(39)
z_bold
(83)
z_bondMath
(35)
z_book
(18)
z_boost
(19)
z_byRef^Val
(32)
z_c#GUI
(43)
z_c#misc
(80)
z_cast/convert
(28)
z_container
(67)
z_cStr/arr
(39)
z_Favorite*
(8)
z_FIX
(15)
z_forex
(48)
z_fwd_Deal
(18)
z_gz=job
(33)
z_gzBig20
(13)
z_gzMgr
(13)
z_gzPain
(20)
z_gzThreat
(19)
z_hib
(19)
z_IDE
(52)
z_ikm
(5)
z_IR_misc
(36)
z_IRS
(26)
z_javaWeb
(28)
z_jdbc
(10)
z_jobFinTech
(46)
z_jobHunt
(20)
z_jobRealXp
(10)
z_jobStrength
(15)
z_jobUS^asia
(27)
z_letter
(42)
z_linq
(10)
z_memberHid`
(11)
z_MOM
(54)
z_nestedClass
(5)
z_oq
(24)
z_PCP
(12)
z_pearl
(1)
z_php
(20)
z_prodSupport
(7)
z_py
(31)
z_quant
(14)
z_regex
(8)
z_rv
(38)
z_skillist
(48)
z_slic`Problem
(6)
z_SOA
(14)
z_spring
(25)
z_src_code
(8)
z_swingMisc
(50)
z_swingTable
(26)
z_unpublish
(2)
z_VBA/Excel
(8)
z_windoz
(17)
z_wpfCommand
(9)

## About Me

- familyman
- New York (Time Square), NY, United States
- http://www.linkedin.com/in/tanbin

## No comments:

Post a Comment