Equiy Derivaives Teach-In Modeling Risk Dr. Hans Buehler, Head of EDG QR Asia Singapore, Augus s, 009 hans.x.buehler@jpmorgan.com 0
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Conens Inroducion, Overview, Produc Lifecycle Producs Modeling Risk Basics Local Volailiy Sochasic Volailiy Crash Risk Numerical Mehods
Conens Challenges in Equiy Modelling Deerminisic Implied Volailiy Ineres raes Dividend marks and handling (proporional vs. discree) FX Volailiies Equiy Correlaion Equiy/FX Correlaion Repo, borrow, basis, ec Firs Order Local Volailiy CDS/Credi risk Second Order Sochasic Volailiy / Jumps Sochasic Dividends Sochasic Ineres raes Equiy/IR correlaion Third Order Sochasic Correlaion Sochasic Credi Risk Equiy/Credi correlaion Fourh Order Transacion Coss, Liqudiy Forward curve models 3
Modeling Risk Basics 4
Modeling Risk Basics Seing Given is a Sochasic Differenial Equaion Brownian Moion d = ( d s (, ) dw ( Drif Volailiy Forward In he risk-neural world, he drif is implied by E[] = F( as df( ( d = r( ( F( Black & Scholes [3]: s= s BS is a consan number. 5
Modeling Risk Basics Black & Scholes 400 S&P500 versus Geomeric Brownian Moion 300 00 00 000 900 800 One of hese pahs is he acual S&P 500, he oher wo are simulaed using Black & Scholes. 8/06/003 06/0/003 4/0/004 3/04/004 0/08/004 09//004 7/0/005 8/05/005 05/09/005 4//005 6
Modeling Risk Basics Black & Scholes For his paricular choice, he SDE has he analyical soluion S BS ( = F( exp s BSW ( s BS which allows compuing vanilla opion prices and, indeed, many more opion prices analyically. The famous Black&Scholes formula for he value of an European call is: FV( T,Soday) : = DF( T) = DF( T) E S BS ( T) K F(T)N ( d ) KN( d ) d = ln( F( T) / K) s BS T s BS T 7
Modeling Risk Basics Black & Scholes The value of each payoff under Black&Scholes is a funcion of he curren spo level S oday and ime-o-mauriy T: FV( T,Soday) : = DF( T) E F( S ( T), S) : = FV(, S) S The seminal paper of Black&Scholes has shown ha if he marke is behaving like he model prediced, hen perfec replicaion is possible: BS Payoff a mauriy T T) K = FV(0, S ) Fair Value of he opion oday (ime 0) T oday (, ) d 0 Dela-Hedging (*) This simplified formula holds if Ineres raes are zero The sock pays mo dividends The sock earns/pays no borrow 8
Modeling Risk Basics Black & Scholes Wha does his formula mean? T) K = FV(0, S ) (, ) d I ells us how o execue he hedge: FV( oday d, )) = FV(0, Soday) (0, Soday) ) T 0 S oday Buy unis of shares oday and sell hem omorrow. Tomorrow s opion value For wo days: Today s opion value 9
Modeling Risk Basics Black & Scholes Wha does his formula mean? For wo days: FV( = FV(,, = FV(0, S T) K = FV(0, S ) (, ) d )) oday )) (, ) (0, S )) oday ) oday ) ) T 0 S ( ) S (, )) S ( ) ) oday 0
Modeling Risk Basics Black & Scholes The formula is undersood as Change of opion value Change of Dela posiion value FV( d, d) FV(, ) dfv( = = (, ) ( d d Praciioners ofen look a cash dela raher han dela. I reflecs he cash value of he dela posiion: This gives $ ( : = dfv(, ) = ( $ d ( Reurn on he sock invesmen
-00% -80% -60% -40% -0% -00% -80% -60% -40% -0% 0% 0% 40% 60% 80% 00% 0% 40% 60% 80% 00% Fair Value Modeling Risk Basics 4 Hedging a European Call (3d o mauriy, 45% Volailiy) Price T 3 Price(S0) + Dela T Price T+ Break-Even Poin a -00% Break-Even Poin a +00% 0 Sock price move in sandard deviaions (relaive o one-day volailiy)
Daily Reurn / Iniial Fair Value Modeling Risk Basics 40% Naked European Call (maures 99% OTM) 30% 0% 0% 0% -0% -0% Using acual -30% HSI marke 04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 daa 04-May-09 3
Daily Reurn / Iniial Fair Value Modeling Risk Basics 40% Dela-Hedging an European Call 30% 0% Unhedged Hedged 0% 0% -0% -0% -30% 04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 04-May-09 4
Daily Reurn / Iniial Fair Value Modeling Risk Basics 40% Dela-Hedging an European Call 500% 30% 0% LogSqRe Unhedged 450% 400% 350% 0% Hedged 300% 50% 0% 00% -0% -0% High daily volailiy leads o deerioraion of hedging performance. 50% 00% 50% -30% 04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 04-May-09 0% 5
Probabiliy Modeling Risk Basics 70% Effec of Dela Hedging on he Reurn Disribuion for ATM Calls 60% 50% 40% 30% Unhedged w Unhedged m Unhedged m Hedged w Hedged m Hedged m 0% 0% Marked reducion in ail risk Marked reducion in ail risk 0% -50% -43% -37% -30% -3% -7% -0% -3% 3% 0% 7% 3% 30% 37% 43% 50% Period Feb 06 April 07 6
Modeling Risk Basics Black & Scholes Wha abou he error if we only hedge daily (no insananeous)? Black&Scholes PDE: dfv( $ ( d = $ d d Cash Gamma $ := S Thea Key message Thea compensaes for Gamma (hea is negaive he opion value decays in ime) The hedge works perfecly if Thea and he Gamma erm cancel. 7
Modeling Risk Basics Black & Scholes Break Even Vol In a Black&Scholes world, he hedge works perfecly, Hence, Thea mus cancel if he asse squared reurn is jus as anicipaed by he Black&Scholes model his means: dfv( ( d Define he Break Even Volailiy $ s break even = = $ $ d s s realized d : = $ d BS d 8
Modeling Risk Basics Black & Scholes Break Even Vol Break Even Vol Noe ha for pracical compuaion, needs o be compued carefully; he so-called opionaliy hea is he raw hea opionali y = simpleshif {dividends funding clien cashflows} In oher words, s break even opionali y : = $ d 9
Modeling Risk Basics Black & Scholes Summary Break Even Vol The mos basic equaion of Modeling Risk dfv( $ ( d = $ s realized s model d Holds for all one-facor sock price models (*) d = ( d s (, ) dw ( A similar formula will be shown for sochasic volailiy We say we pay Thea for he Gamma of he opion. (*) assuming no daily recalibraion. 0
Modeling Risk Local Volailiy
Modeling Risk Local Volailiy Black & Scholes Implied Volailiy Recall he Black&Scholes formula for a European call: FV( T,Soday) : = DF( T) = DF( T) E S BS ( T) K F(T)N ( d ) KN( d ) d = ln( F( T) / K) s BS T s BS T
8346 9737 8 59 390 530 669 8083 9474 Implied Vol Modeling Risk Local Volailiy Black & Scholes Implied Volailiy Using he BS volailiy s BS as a free parameer, we can inver he BS formula o back ou he so-called BS implied volailiy surface of he Marke marke. 70% 60% 50% 40% 30% Y Y 3Y 3M Srike Example: HSI Mar 5, 009 @ 3,90 3
Modeling Risk Local Volailiy Black & Scholes Implied Volailiy s Τ Κ = s ATM Τ Κ K K skew* ln convexiy *ln F( T) F( T) approximaely s ( T,05%) s ( T,95%) skew = 0% for 95/05 srikes relaive o forward (wha we used before). Formula does no hold for far OTM srikes. 4
Modeling Risk Local Volailiy Recall: Pricing wih Skew 0.7 Skew Impac on ATM Digial Pus (95-05 skew @ -5%) 0.6 0.5 0.4 0.3 0. 0. 0 Digial Pu Spread 0.0% Pu Spread 0.% Pu Spread 0.5% Pu Spread % w m 3m 6m y 5
8346 9737 8 59 390 530 669 8083 9474 Implied Vol Modeling Risk Local Volailiy Black & Scholes Implied Volailiy Observed marke opion prices are no consisen wih he BS assumpion of a single consan volailiy, and no even wih he assumpion of a imedependen volailiy. Marke 70% 60% 50% 40% 30% Y Y 3Y 3M Srike Any pricing algorihm needs o re-price he observed opion prices (oherwise we would have inernal arbirage). 6
Modeling Risk Local Volailiy Dupire s Local Volailiy Classic Soluion: Dupire s Local Volailiy [] he second mos used model in equiy derivaives Idea: find s as a funcion of spo and ime such ha ds S LV LV ( ( = ( d s LV (, S LV ( ) dw ( reprices all marke prices, ie! S ( K MarkeCallPrice(, K) DF( E LV = 7
8 Modeling Risk Local Volailiy Dupire s Local Volailiy Key poins Dupire s local volailiy is unique wihin he class of diffusions In his class i is well-defined and given as where ) ( ) DF( ) ( MarkeCallPrice : ), ( ~ F,kF k C = ), ( ~ ), ( ~ :, ~ k C k k k C k LV = s = ) (, ~ ):, ( F S S LV s LV s Normalized Call Prices
Modeling Risk Local Volailiy Dupire s Local Volailiy Implied Volailiy SPX Dec 9 005 Local Volailiy SPX 80 80 70 70 60 60 50 50 40 40 30 0 0 0 09-Dec-8 09-Jun-6 09-Dec-3 09-Jun- 09-Dec-08 30 0 0 0 09-Jun-8 09-Dec-5 09-Jun-3 09-Dec-0 09-Jun-08 09-Jun-06 09-Dec-05 Srike/Spo Srike/Spo 9
Implied Volailiy Implied Volailiy Modeling Risk Local Volailiy Dupire s Local Volailiy ime dependency Implied Vol flaens ou in he fuure inside he Local Vol Model Local Vol Calibraion.STOXX50E 0/0/005 Local Vol Calibraion.STOXX50E 0/0/005 shifed forward by years 50.0 50.0 45.0 45.0 40.0 40.0 35.0 35.0 30.0 30.0 5.0 0.0 5.0 0.0 5.0 0.0 70% 80% 90% 00% 0% 0% 30% Srike/Spo m 4m y 5.0 0.0 5.0 0.0 5.0 0.0 70% 80% 90% 00% Srike/Spo 0% 0% 30% m 3m 6m y 30
Modeling Risk Local Volailiy Dupire s Local Volailiy Plus Allows o consisenly price all combinaions of European opions. One-facor model fas o simulae Break Even Vol formula applies Cons Dynamical behaviour is no consisen Implied Vol needs o be sensible Widely used reference model for equiy derivaives. 3
Thank you very much for your aenion. hans.x.buehler@jpmorgan.com 3
Numerical Mehods References [] Gaheral: The Volailiy Surface, Wiley, 006 [] Dupire: Pricing wih a Smile, Risk, 7 (), pp. 8-0, 996 [3] Black, Scholes: The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, 8, pp. 637-59, 973 [4] Carr, Madan: Opion Valuaion Using he Fas Fourier Transform, Journal of Compuaional Finance,, 998 [5] Ren, Madan, Qian: Calibraing and pricing wih embedded local volailiy models, Risk, Sepember 007 [6] Buehler, Volailiy Markes, VDM Verlag, 008 [7] Meron: Opion Pricing When Underlying Sock Reurns are Disconinuous. Journal of Financial Economics 3 (976) pp. 5-44, 976 [8] Con, Tankov: Financial modeling wih Jump Processes, Chapman & Hall / CRC Press, 003 [9] Buehler, Volailiy and Dividends, Working paper, 008, hp://www.mah.uberlin.de/~buehler/ [0] Glasserman, Mone Carlo Mehods in Financial Engineering, Springer 004 [] Andersen, Pierbarg: Momen Explosions in Sochasic Volailiy Models WP~004, hp://ssrn.com/absrac=55948 [] Bermudez, Buehler, Ferraris, Jordinson, Lamnouar, Overhaus: Equiy Hybrid Derivaives, Wiley, 006 33