Tuesday, June 9, 2015

discounted asset price is MG but "discount" means...@@

The Fundamental Theorem

A financial market with time horizon T and price processes of the risky asset and riskless bond (I would say a money-market-account) given by S1, ..., ST and B0, ..., BT, respectively, is arbitrage-free under the real world probability P if and only if there exists an equivalent probability measure Q (i.e. risk neutral measure) such that
The discounted price process, X0 := S0/B0, ..., XT := ST/BT is a martingale under Q.

#1 Key concept – divide the current stock price by the current MMA value. This is the essence of "discounting", different from the usual "discount future cashflow to present value"
#2  key concept – the alternative interpretation is "using MMA as currency, then any asset price S(t) is a martingale"

I like the discrete time-series notation, from time_0, time_1, time_2... to time_T.

I like the simplified (not simplistic:) 2-asset world.

This theorem is generalized with stochastic interest rate on the riskless bond:)

There's an implicit filtration. The S(T) or B(T) are prices in the future i.e. yet to be revealed [1]. The expectation of future prices is taken against the filtration.

[1] though in the case of T-forward measure, B(T) = 1.0 known in advance.


--[[Hull]] P 636 has a concise one-pager (I would skip the math part) that explains the numeraire can be just "a tradable", not only the MMA. A few key points:

) both S and B must be 2 tradables, not something like "fwd rate" or "volatility"
) the measure is the measure related to the numeraire asset
) what market forces ensure this ratio is a MG? Arbitragers!

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