Monday, May 28, 2012

how to make volatility values ANNUALIZED

(Let’s assume a flat forward curve i.e. 0 drift, 0 dividends, 0 interests.) Suppose an implied vol for a 1-year option is 20%. If we record ln(PR) i.e. log of daily price relatives until expiry, we expect 68% of the 200+ daily readings to fall between -0.2 and 0.2. That’s because ln(PR) is supposed to follow a normal distribution.

Note we aren’t 68% sure about the expiration underlier price i.e. S(t=T) or S(T) for short. This S(T) has a lognormal distribution[2], so no 68% rule. However, we do know something about the S(t=T) because the end-to-end ln(PR) is the sum of ln(daily PR), and due to central limit theorem, the overall ln(PR) has a normal distribution with a variance = sum(variance of ln(daily PR)). We always assume the individual items in the sum() are independent and "identical", variance of ln(daily PR) is therefore 0.04/252days.

Also, Since ln(overal PR) = ln[S(T)/S(0)] has normal distribution, S(T) has a lognormal distribution. That's the reason for [2].

To answer any option pricing question, we invariably need to convert quoted, annualized vol to what I call raw-sigma or stdev.

Rule #1 -- we assume one-period variance will persist to 2 periods, 3 periods, 4 periods... (eg: a year consists of 12 one-month periods.)

Example 1: If one-year variance is 0.04, then a four-year raw-variance would be .04 * 48/12 = .16. The corresponding stdev i.e. raw-sigma would be 40%. This value is what goes into BS equation to price options with this maturity.

Example 2: If one-year variance is 0.04, then a three-month raw-variance would be .04 *3/12 = .01. The stdev i.e. raw-sigma would be sqrt(.01) = 10%. This value is what goes into BS equation to price options with this maturity. By Rule #1, we assume the same 3-month variance would persist in 2nd 3-month, the 3rd 3-month and 4th 3-month periods.

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