Monday, December 12, 2011

recombinant binomial tree, price relative and lognormal

I find the recombinant/interlock concept a nice simplification. I was told in practice, all pricing btrees are interlocked – much easier without loss of generality. wikipedia says

"The CRR method ensures that the tree is *recombinant*, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u) -- here the two paths merge or recombine."

Often (u,d) will 100% cancel each other. At each tiny step, either

newPrice = oldPrice * u (typically 1.0101010101010101) or
newPrice = oldPrice * d (typically 0.99)
u * d == 1

Note the standard definition of u and d [[CFA]] as __PriceRelative__ of day2closing/day1closing. (Useful in h-vol...)

For a simple example, u = 2, so our underlying price either doubles or halves at each step. Same result if it double-then-half, or half-then-double. Consistent with the lognormal model...

Warning: many texts use illustrations of u=1.01000000000, d=0.99, which violates lognormal assumption but still recombinant...

In binomial tree, the next value of u is kind of random. In other words, the multiply factor could change from 2.0 to 1.01 to 1.002... usually just above 1 if step is small.

More precisely, u is computed using the underlying volatility, σ, and the time duration of a step. In a complete model, the pdf of random variable "u" can be derived.

Q: in industry, are most binomial trees recombinant?
%%A: I think so.

No comments:

Total Pageviews

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

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