Introduction to Black-Scholes Model
|
|
- Sharlene McKenzie
- 6 years ago
- Views:
Transcription
1 4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY maxmasuda@maxmasuda.com hp:// December 4 his paper presens everyhing you need o know abou Black-choles model which is ruly single mos imporan revoluionary work in he hisory of quaniaive finance. Alhough B model has is flaws such as he normally disribued (i.e. zero skewness and zero excess kurosis) log reurn densiy and he assumpion of consan volailiy across srike prices and he ime o mauriy, i ouperforms more (so-called) advanced models in numerous cases. 4 azuhisa Masuda All righs reserved.
2 4 azuhisa Masuda All righs reserved. [] andard Brownian Moion: Building Block of B Model A sandard Brownian moion filered probabiliy space ( [, ) B ) ( Ω, F, P) [, ) is a real valued sochasic process defined on a saisfying: () Is incremens are independen. In oher words, for < <... < n <: P ( B... ) B B B B B B n n = P ( B ) ( ) ( )... ( ) P B B P B B B B P n. n () Is incremens are saionary (ime homogeneous): i.e. for h, B+ h B has he same disribuion as B h. In oher words, he disribuion of incremens does no depend on. (3) P( B = ) =. he process sars from almos surely (wih probabiliy ). (4) B Normal(, ). Is incremens follow a Gaussian disribuion wih he mean and he variance. I urns ou ha a sandard Brownian moion ( [, ) B ) saisfies he following condiions: () he process is sochasically coninuous: ε >, lim P ( X X ε ) =. () Is sample pah (rajecory) is coninuous in (i.e. coninuous rcll) almos surely. For more deails abou Brmwoan moion, consul Masuda (5) Inroducion o Brownian Moion. h [] Black-choles Disribuional Assumpions on a ock Price In radiional finance lieraure almos every financial asse price (socks, currencies, ineres raes) is assumed o follow some variaions of Brownian moion wih drif process. B (Black-choles) models a sock price incremen process in an infiniesimal ime inerval d as a log-normal random walk process: d µ d σ db + h = +, () where he drif is µ which is a consan expeced reurn on a sock µ proporional o a sock price and he volailiy is σ which is a consan sock price volailiy σ proporional o a sock price. he reason why he process () is called a log-normal random walk process will be explained very soon. Alernaively, we can sae ha B models a percenage change in a sock price process in an infiniesimal ime inerval d as a Brownian moion wih drif process:
3 4 azuhisa Masuda All righs reserved. d = µ d + σdb, () d ( d / µ d) P ( ) = exp[ ]. πσ d σ d Le be a random variable whose dynamics is given by an Io process: d = a(, ) d + b(, ) db, and V be a funcion dependen on a random variable V(, ) is given by an Io formula: and ime. he dynamics of V V V dv = d + d + b d, (3) or in erms of a sandard Brownian moion process B : V V V dv = d + ( ad + bdb) + b d, V V V V dv = + a + b d b db +. (4) Dynamics of a log sock price process ln can be obained by applying (4) o () as: ln ln ln ln dln = + µ + σ d σ + db. ubsiuing ln =, ln =, and ln = yields: ln = µ σ + σ d d db, (5) or: ln ln = µ σ ( ) + σ( B B) ln = ln + µ σ + σb. (6) 3
4 4 azuhisa Masuda All righs reserved. he equaion (6) means ha B models a log sock price l n as a Brownian moion wih drif process whose probabiliy densiy is given by a normal densiy: ln ln + ( µ σ ) (ln ) exp[ P = ]. (7) πσ σ Alernaively, he equaion (6) means ha B models a log reurn ln ( / ) as a Brownian moion wih drif process whose probabiliy densiy is given by a normal densiy: ln ( / ) = µ σ + σb, ln ( / ) µ σ P ( ln ( / )) = exp[ ]. (8) πσ σ An example of B normal log reurn ln ( / ) densiy of (8) is illusraed in Figure. Of course, B log reurn densiy is symmeric (i.e. zero skewness) and have zero excess kurosis because i is a normal densiy. Densiy log reurn lnh ê L Figure : An Example of B normal log reurn ln( / ) Densiy. Parameers and variables fixed are µ =., σ =., and =.5. Le y be a random variable. If he log of y is normally disribued wih mean variance b such ha ln y N( a, b ), hen y is a log-normal random variable whose densiy is a wo parameer family ( ab), : a and a+ b a+ b b (, ( y Lognormal e e e )), 4
5 4 azuhisa Masuda All righs reserved. { ln y a} P ( y) = exp[ ]. y π b b From he equaion (6), we can sae ha B models a sock price disribued random variable whose densiy is given by: as a log-normally P ( ) Is annualized momens are calculaed as: ln ln + ( µ σ ) exp[ = ]. (9) πσ σ Mean[ ] = e µ, ( ) σ ( ) Variance[ ] e e σ µ =, kewness = e + e, σ [ ] 3 σ σ 4σ Excess urosis[ ] 6 3e e e = An example of B log-normal sock price densiy of (9) is illusraed in Figure. Noice ha B log-normal sock price densiy is posiively skewed..5.4 Densiy ock price Figure : An Example of B Log-Normal Densiy of a ock Price. Parameers and variables fixed are = 5, µ =., σ =., and =.5. able Annualized Momens of B Log-Normal Densiy of A ock Price in Figure Mean andard Deviaion kewness Excess urosis
6 4 azuhisa Masuda All righs reserved. From he equaion (6), we can obain an inegral version equivalen of (): exp[ln ] = exp[ln + µ σ + σb ] = exp[ln ]exp µ σ σb + = exp µ σ + σb. () Equaion () means ha B models a sock price dynamics as a geomeric (i.e. exponenial) process wih he growh rae given by a Brownian moion wih drif process: + B. µ σ σ [3] radiional Black-choles Opion Pricing: PDE Approach by Hedging Consider a porfolio P of he one long opion posiion V(, ) on he underlying sock wrien a ime and a shor posiion of he underlying sock in quaniy o derive opion pricing funcion. P = V(, ). () Porfolio value changes in a very shor period of ime d by: dp = dv (, ) d. () ock price dynamics is given by a log-normal random walk process of he equaion (7.): d µ d σ db = +. (3) Opion price dynamics is given by applying Io formula of he equaion (3): dv = d + d + d. (4) V V V σ Now he change in he porfolio value can be expressed as by subsiuing (3) and (4) ino (): 6
7 4 azuhisa Masuda All righs reserved. V V V σ dp = d + d + d d. (5) eing = V / (i.e. dela hedging) makes he porfolio compleely risk-free (i.e. he randomness d has been eliminaed) and he porfolio value dynamics of he equaion (5) simplifies o: V V dp = + σ d. (6) ince his porfolio is perfecly risk-free, assuming he absence of arbirage opporuniies he porfolio is expeced o grow a he risk-free ineres rae r : EdP [ ] = rpd. (7) Afer subsiuion of () and (6) ino (7) by seing = V /, we obain: V V V + σ d r V = Afer rearrangemen, Black-choles PDE is obained: d. V V V + + r rv(, ) =. (8) (, ) (, ) (, ) σ B PDE is caegorized as a linear second-order parabolic PDE. he equaion (8) is a linear PDE because coefficiens of he parial derivaives of V(, ) (i.e. σ / and r ) are no funcions of V(, ) iself. he equaion (8) is a second-order PDE because i involves he second-order parial derivaive V(, )/. Generally speaking, a PDE of he form: is said o be a parabolic ype if: V V V V V a+ b + c + d + e + g = g 4de=. (9) he equaion (8) is a parabolic PDE because i has g = and e = which saisfies he condiion (9)., 7
8 4 azuhisa Masuda All righs reserved. B solves PDE of (8) wih boundary condiions: ( ( max,) for a plain vanilla call, ) max, for a plain vanilla pu, and obains closed-form soluions of call and pu pricing funcions. Exac derivaion of closed-form soluions by solving B PDE is omied here (i.e. he original B approach). Insead we will provide he exac derivaion by a maringale asse pricing approach (his is much simpler) in he nex secion. [4] radiional Black-choles Opion Pricing: Maringale Pricing Approach Le { B ; } be a sandard Brownian moion process on a space ( Ω, F, P). Under acual probabiliy measure P, he dynamics of B sock price process is given by equaion (9) in he inegral form (i.e. which is a geomeric Brownian moion process): exp. () = µ σ + σb B model is an example of a complee model because here is only one equivalen maringale risk-neural measure Q P under which he discouned asse price process r { e ; } becomes a maringale. B finds he equivalen maringale risk-neural measure Q P B by changing he drif of he Brownian moion process while keeping he volailiy parameer σ unchanged: exp QB = r σ + σb. () B Noe ha B Q is a sandard Brownian moion process on ( Ω, F, Q B ) and he r discouned sock price process { e ; } is a maringale under and wih respec o he filraion { F ; }. hen, a plain vanilla call opion price C (, ) which has a erminal payoff funcion max, is calculaed as: ( ) ( ) C e E r ( ) QB (, ) = max, Q B F. () Le Q( ) (drop he subscrip B for simpliciy) be a probabiliy densiy funcion of in a risk-neural world. From he equaion (9), a erminal sock price is a log-normal random variable wih is densiy of he form: 8
9 4 azuhisa Masuda All righs reserved. Q ( ) ln ln + ( r σ ) τ exp[ = ]. (3) στ πσ τ Using (3), he expecaion erm in () can be rewrien as: max (,) = ( ) Q( ) + ( ) ( ) Q F max (,) F = ( ) Q( F) d. Q E F F d d Q E Using his, we can rewrie () by seing τ as: ( ) ( ) C(, τ ) = e Q F d. (4) ince is a log-normal random variable wih is densiy given by he equaion (3): ln F Normal m ln + ( r σ ) τστ,. (5) For he noaional simpliciy, le ln from a log-normal random variable wih: From (6): Z F ln. We use a change of variable echnique + r σ τ σ τ ln ln ( ) o a sandard normal random variable ln m N (,) ( Z ) σ τ Z exp[ ] Z =. π Z hrough: ormal, (6) = exp( Zσ τ + m). (7) We can rewrie (4) as: ( ( ) ) ( ) C ( τ, ) e r = τ exp Zσ τ + m Z Z dz, (ln m) / σ τ 9
10 4 azuhisa Masuda All righs reserved. and we express his wih more compac form as: C(, ) C C τ =, (8) where = exp( σ τ + ) ( ) C e Z m Z dz Consider C : Z and ( ) (ln m) / σ τ ( σ τ ) exp( ) ( ) C e = exp Z m Z Z dz (ln m) / σ τ C = e Z Z dz. (ln m) / σ τ C = exp r exp ln + ( r ) exp Z Z ( τ) σ τ ( σ τ ) ( ) Z dz (ln m) / σ τ ( ) ( ) Z dz Z C = exp ln στ exp( Z ) exp[ ] σ τ dz (ln m)/ σ τ π C = exp ln στ exp Zσ τ Z (ln m) / σ τ Z Zσ τ C = exp ln στ exp[ ] dz (ln m) / σ τ π ( ) Z σ τ σ τ C = exp ln στ exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp ln στexp στ exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp( ln ) exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp[ ] dz. (9) (ln m) / σ τ π Use he following relaionship: Equaion (9) can be rewrien as: b a ( ) Z c b c Z exp[ ] dz = exp[ ] dz a c π π. Z exp[ ] (ln m)/ σ τ σ τ π C = dz. (3)
11 4 azuhisa Masuda All righs reserved. Le N ( ) be he sandard normal cumulaive densiy funcion. Using he symmery of a normal densiy, (3) can be rewrien as: (ln m)/ σ τ + σ τ Z exp[ ] C = dz π ln + m C = N + σ τ σ τ. (3) From (5), subsiue for m. he equaion (3) becomes: ln + ln + ( r σ ) τ C = N + σ τ σ τ ln ( r ) ln ( r ) σ τ σ τ σ τ C = N = N σ τ σ τ (3) Nex, consider C in (8): (ln m)/ σ τ ( ) ( ) C e r Z dz e τ Z dz = Z = (ln m)/ σ τ Z r ln + m C = e τ N. (33) σ τ From (5), subsiue for m. he equaion (33) becomes: ln + ln + ( r σ ) τ ln ( r ) rτ + σ τ C = e N = e N. (34) σ τ σ τ ubsiue (3) and (34) ino (8) and we obain B plain vanilla call opion pricing formula: ( ) ( ) C(, τ ) = N d e N d, (35) where d ln + ( r + σ ) τ = and σ τ d ln ( ) + r σ τ = = d σ τ. σ τ
12 4 azuhisa Masuda All righs reserved. Following he similar mehod, B plain vanilla pu opion pricing formula can be obained as: ( ) ( ) P(, τ ) = e N d N d. (36) We conclude ha boh PDE approach and maringale approach give he same resul. his is because in boh approaches we move from a hisorical probabiliy measure P o a riskneural probabiliy measure Q. his is very obvious for maringale mehod. Bu in PDE approach because he source of randomness can be compleely eliminaed by forming a porfolio of opions and underlying socks, his porfolio grows a a rae equal o he riskfree ineres rae. hus, we swich o a measure Q. For more deails, we recommend Nefci () pages 8-8 and [5] Alernaive Inerpreaion of Black-choles Formula: A ingle Inegraion Problem Under an equivalen maringale measure Q P under which he discouned asse price r process { e ; } becomes a maringale, a plain vanilla call and pu opion price which has a erminal payoff funcion as: ( ) and max (,) max, ( ) ( ) C e E r ( ) Q (, ) = max, P e E r ( ) Q (, ) = max, are calculaed F, (37) F. (38) Noe ha an expecaion operaor E [ ] is under a probabiliy measure Q and wih respec o he filraion F. Le Q( F ) be a condiional probabiliy densiy funcion of a erminal sock price. For he noaional simpliciy we use Q ( F ) Q( ) and τ. he expeced erminal payoffs in he equaions (37) and (38) can be rewrien as: ( ) F = ( ) Q( ) Q E max, ( ) F = ( ) Q( ) Q E max, Using hese, we can rewrie (37) and (38) as: d, d. ( ) Q( ) C(, τ ) = e d, (39)
13 4 azuhisa Masuda All righs reserved. ( ) ( ) P(, τ ) = e Q d. (4) B assumes ha a erminal sock price densiy of he form: is a log-normal random variable wih is ln ln + ( r σ ) τ Q ( ) = exp[ ]. στ πσ τ herefore, B opion pricing formula comes down o a very simple single inegraion problem: ln ln + ( r σ ) τ = ( ) στ πσ τ C( τ, ) e exp[ ] d ln ln + ( r σ ) τ = ( ) στ πσ τ P( τ, ) e exp[ ] d, (4). (4) his implies ha as far as a risk-neural condiional densiy of he erminal sock price Q F ) is known, plain vanilla opion pricing reduces o a simple inegraion problem. ( [6] Black-choles Model as an Exponenial Lévy Model he equaion () ells us ha B models a sock price process as an exponenial Brownian moion wih drif process: which means: = +, exp µ σ σb =, e L where he sock price process { : } is modeled as an exponenial of a Lévy process { L ; }. Black and choles choice of he Lévy process is a Brownian moion wih drif (coninuous diffusion process): 3
14 4 azuhisa Masuda All righs reserved. + L µ σ σb. (43) he fac ha an sock price is modeled as an exponenial of Lévy process L means ha is log-reurn ln( ) is modeled as a Lévy process such ha: ln( ) = L = + µ σ σb. B model can be caegorized as he only coninuous exponenial Lévy model apparenly because a Brownian moion wih drif process is he only coninuous (i.e. no jumps) Lévy process. his indicas ha he Lévy measure of a Brownian moion wih drif process is zero: ( dx ) =, and obviously is arrival rae of jumps is zero: ( dx ) =. References Black, F. and choles, M., 973, he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 3. Hull, J. C.,, Opions, Fuures, and Oher Derivaives (5h Ediion), Prenice Hall. Masuda,., 5, Inroducion o Brownian Moion. Nefci,. N.,, An Inroducion o he Mahemaics of Financial Derivaives, Academic Press. Wilmo, P., 998, Derivaives, John Wiley & ons. 4
The Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations
The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone
More informationBlack-Scholes Model and Risk Neutral Pricing
Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s
More informationMAFS Quantitative Modeling of Derivative Securities
MAFS 5030 - Quaniaive Modeling of Derivaive Securiies Soluion o Homework Three 1 a For > s, consider E[W W s F s = E [ W W s + W s W W s Fs We hen have = E [ W W s F s + Ws E [W W s F s = s, E[W F s =
More informationIJRSS Volume 2, Issue 2 ISSN:
A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural
More informationEquivalent Martingale Measure in Asian Geometric Average Option Pricing
Journal of Mahemaical Finance, 4, 4, 34-38 ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang
More informationTentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.
Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following
More informationMatematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
Maemaisk saisik Tenamen: 8 5 8 kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula.
More informationModels of Default Risk
Models of Defaul Risk Models of Defaul Risk 1/29 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed
More informationPricing FX Target Redemption Forward under. Regime Switching Model
In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok
More informationMay 2007 Exam MFE Solutions 1. Answer = (B)
May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have 0.0 0.05 0.03 4.50 =.45 + 5.00 D e D e 50 e = 54.45 D (
More informationBrownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that
Advanced Financial Models Example shee 4 - Michaelmas 8 Michael Tehranchi Problem. (Hull Whie exension of Black Scholes) Consider a marke wih consan ineres rae r and wih a sock price modelled as d = (µ
More informationPricing formula for power quanto options with each type of payoffs at maturity
Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih
More informationOption pricing and hedging in jump diffusion models
U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy Maser
More informationAn Analytical Implementation of the Hull and White Model
Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque,
More informationThe Binomial Model and Risk Neutrality: Some Important Details
The Binomial Model and Risk Neuraliy: Some Imporan Deails Sanjay K. Nawalkha* Donald R. Chambers** Absrac This paper reexamines he relaionship beween invesors preferences and he binomial opion pricing
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your
More informationAMS Q03 Financial Derivatives I
AMS Q03 Financial Derivaives I Class 08 Chaper 3 Rober J. Frey Research Professor Sony Brook Universiy, Applied Mahemaics and Saisics frey@ams.sunysb.edu Lecure noes for Class 8 wih maerial drawn mainly
More informationVALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION
Aca Universiais Mahiae Belii ser. Mahemaics, 16 21, 17 23. Received: 15 June 29, Acceped: 2 February 21. VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION TOMÁŠ BOKES
More informationOption Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka
Opion Valuaion of Oil & Gas E&P Projecs by Fuures Term Srucure Approach March 9, 2007 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion
More informationComputations in the Hull-White Model
Compuaions in he Hull-Whie Model Niels Rom-Poulsen Ocober 8, 5 Danske Bank Quaniaive Research and Copenhagen Business School, E-mail: nrp@danskebank.dk Specificaions In he Hull-Whie model, he Q dynamics
More informationSome Remarks on Derivatives Markets (third edition, 2013)
Some Remarks on Derivaives Markes (hird ediion, 03) Elias S. W. Shiu. The parameer δ in he Black-Scholes formula The Black-Scholes opion-pricing formula is given in Chaper of McDonald wihou proof. A raher
More informationINSTITUTE OF ACTUARIES OF INDIA
INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on
More informationErratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index
Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends.
More informationAMS Computational Finance
AMS 54 - Compuaional Finance European Opions Rober J. Frey Research Professor Sony Brook Universiy, Applied Mahemaics and Saisics frey@ams.sunysb.edu Feb 2006. Pu-Call Pariy for European Opions A ime T
More informationPricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.
Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 pklein@sfu.ca (604) 268-7922 Pricing ulnerable American Opions Absrac We exend
More informationLeveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model. Dale L. Domian, Marie D. Racine, and Craig A.
Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Dale L. Domian, Marie D. Racine, and Craig A. Wilson Deparmen of Finance and Managemen Science College of Commerce Universiy of
More informationAvailable online at ScienceDirect
Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches',
More informationSystemic Risk Illustrated
Sysemic Risk Illusraed Jean-Pierre Fouque Li-Hsien Sun March 2, 22 Absrac We sudy he behavior of diffusions coupled hrough heir drifs in a way ha each componen mean-revers o he mean of he ensemble. In
More informationA pricing model for the Guaranteed Lifelong Withdrawal Benefit Option
A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica Index Main References Survey of he Variable
More informationUCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory
UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All
More informationMA Advanced Macro, 2016 (Karl Whelan) 1
MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese
More informationA Note on Forward Price and Forward Measure
C Review of Quaniaive Finance and Accouning, 9: 26 272, 2002 2002 Kluwer Academic Publishers. Manufacured in The Neherlands. A Noe on Forward Price and Forward Measure REN-RAW CHEN FOM/SOB-NB, Rugers Universiy,
More informationOptimal Early Exercise of Vulnerable American Options
Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee. Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk
More informationResearch Article A General Gaussian Interest Rate Model Consistent with the Current Term Structure
Inernaional Scholarly Research Nework ISRN Probabiliy and Saisics Volume 212, Aricle ID 67367, 16 pages doi:1.542/212/67367 Research Aricle A General Gaussian Ineres Rae Model Consisen wih he Curren Term
More informationVaR and Low Interest Rates
VaR and Low Ineres Raes Presened a he Sevenh Monreal Indusrial Problem Solving Workshop By Louis Doray (U de M) Frédéric Edoukou (U de M) Rim Labdi (HEC Monréal) Zichun Ye (UBC) 20 May 2016 P r e s e n
More informationProceedings of the 48th European Study Group Mathematics with Industry 1
Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 1 ADR Opion Trading Jasper Anderluh and Hans van der Weide TU Delf, EWI (DIAM), Mekelweg 4, 2628 CD Delf jhmanderluh@ewiudelfnl, JAMvanderWeide@ewiudelfnl
More informationTentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl
Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna
More informationOn the multiplicity of option prices under CEV with positive elasticity of variance
Rev Deriv Res (207) 20: 3 DOI 0.007/s47-06-922-2 On he mulipliciy of opion prices under CEV wih posiive elasiciy of variance Dirk Veesraeen Published online: 4 April 206 The Auhor(s) 206. This aricle is
More informationMORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion
More informationFinancial Markets And Empirical Regularities An Introduction to Financial Econometrics
Financial Markes And Empirical Regulariies An Inroducion o Financial Economerics SAMSI Workshop 11/18/05 Mike Aguilar UNC a Chapel Hill www.unc.edu/~maguilar 1 Ouline I. Hisorical Perspecive on Asse Prices
More informationChange of measure and Girsanov theorem
and Girsanov heorem 80-646-08 Sochasic calculus I Geneviève Gauhier HEC Monréal Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P :
More informationOnce we know he probabiliy densiy funcion (pdf) φ(s ) of S, a European call wih srike is priced a C() = E [e r d(s ) + ] = e r d { (S )φ(s ) ds } = e
Opion Basics Conens ime-dependen Black-Scholes Formula Black-76 Model Local Volailiy Model Sochasic Volailiy Model Heson Model Example ime-dependen Black-Scholes Formula Le s begin wih re-discovering he
More informationMarket Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009
s Praciioner Course: Ineres Rae Models March 29, 2009 In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae,
More informationPricing options on defaultable stocks
U.U.D.M. Projec Repor 2012:9 Pricing opions on defaulable socks Khayyam Tayibov Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Juni 2012 Deparmen of Mahemaics Uppsala Universiy Pricing
More informationJarrow-Lando-Turnbull model
Jarrow-Lando-urnbull model Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul
More informationRisk-Neutral Probabilities Explained
Risk-Neural Probabiliies Explained Nicolas Gisiger MAS Finance UZH ETHZ, CEMS MIM, M.A. HSG E-Mail: nicolas.s.gisiger @ alumni.ehz.ch Absrac All oo ofen, he concep of risk-neural probabiliies in mahemaical
More informationPARAMETER ESTIMATION IN A BLACK SCHOLES
PARAMETER ESTIMATIO I A BLACK SCHOLES Musafa BAYRAM *, Gulsen ORUCOVA BUYUKOZ, Tugcem PARTAL * Gelisim Universiy Deparmen of Compuer Engineering, 3435 Isanbul, Turkey Yildiz Technical Universiy Deparmen
More informationOn the Impact of Inflation and Exchange Rate on Conditional Stock Market Volatility: A Re-Assessment
MPRA Munich Personal RePEc Archive On he Impac of Inflaion and Exchange Rae on Condiional Sock Marke Volailiy: A Re-Assessmen OlaOluwa S Yaya and Olanrewaju I Shiu Deparmen of Saisics, Universiy of Ibadan,
More informationValuation and Hedging of Correlation Swaps. Mats Draijer
Valuaion and Hedging of Correlaion Swaps Mas Draijer 4298829 Sepember 27, 2017 Absrac The aim of his hesis is o provide a formula for he value of a correlaion swap. To ge o his formula, a model from an
More informationEconomic Growth Continued: From Solow to Ramsey
Economic Growh Coninued: From Solow o Ramsey J. Bradford DeLong May 2008 Choosing a Naional Savings Rae Wha can we say abou economic policy and long-run growh? To keep maers simple, le us assume ha he
More informationMoney in a Real Business Cycle Model
Money in a Real Business Cycle Model Graduae Macro II, Spring 200 The Universiy of Nore Dame Professor Sims This documen describes how o include money ino an oherwise sandard real business cycle model.
More informationBond Prices and Interest Rates
Winer erm 1999 Bond rice Handou age 1 of 4 Bond rices and Ineres Raes A bond is an IOU. ha is, a bond is a promise o pay, in he fuure, fixed amouns ha are saed on he bond. he ineres rae ha a bond acually
More informationA UNIFIED PDE MODELLING FOR CVA AND FVA
AWALEE A UNIFIED PDE MODELLING FOR CVA AND FVA By Dongli W JUNE 2016 EDITION AWALEE PRESENTATION Chaper 0 INTRODUCTION The recen finance crisis has released he counerpary risk in he valorizaion of he derivaives
More informationMacroeconomics II A dynamic approach to short run economic fluctuations. The DAD/DAS model.
Macroeconomics II A dynamic approach o shor run economic flucuaions. The DAD/DAS model. Par 2. The demand side of he model he dynamic aggregae demand (DAD) Inflaion and dynamics in he shor run So far,
More informationAN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES
Inernaional Journal of Pure and Applied Mahemaics Volume 76 No. 4 212, 549-557 ISSN: 1311-88 (prined version url: hp://www.ijpam.eu PA ijpam.eu AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM
More informationMEAN-VARIANCE ASSET ALLOCATION FOR LONG HORIZONS. Isabelle Bajeux-Besnainou* James V. Jordan** January 2001
MEAN-VARIANCE ASSE ALLOCAION FOR LONG HORIZONS Isabelle Bajeux-Besnainou* James V. Jordan** January 1 *Deparmen of Finance he George Washingon Universiy 3 G S., NW Washingon DC 5-994-559 (fax 514) bajeux@gwu.edu
More informationAdvanced Tools for Risk Management and Asset Pricing
MSc. Finance/CLEFIN 214/215 Ediion Advanced Tools for Risk Managemen and Asse Pricing May 215 Exam for Non-Aending Sudens Soluions Time Allowed: 13 minues Family Name (Surname) Firs Name Suden Number (Mar.)
More informationProblem Set 1 Answers. a. The computer is a final good produced and sold in Hence, 2006 GDP increases by $2,000.
Social Analysis 10 Spring 2006 Problem Se 1 Answers Quesion 1 a. The compuer is a final good produced and sold in 2006. Hence, 2006 GDP increases by $2,000. b. The bread is a final good sold in 2006. 2006
More informationFINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004
FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 This exam has 50 quesions on 14 pages. Before you begin, please check o make sure ha your copy has all 50 quesions and all 14 pages.
More informationA Method for Estimating the Change in Terminal Value Required to Increase IRR
A Mehod for Esimaing he Change in Terminal Value Required o Increase IRR Ausin M. Long, III, MPA, CPA, JD * Alignmen Capial Group 11940 Jollyville Road Suie 330-N Ausin, TX 78759 512-506-8299 (Phone) 512-996-0970
More informationarxiv:math/ v2 [math.pr] 26 Jan 2007
arxiv:mah/61234v2 [mah.pr] 26 Jan 27 EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION VICTOR GOODMAN AND KYOUNGHEE KIM Absrac. We find a simple expression for he probabiliy densiy of R exp(bs
More informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationExtended MAD for Real Option Valuation
Exended MAD for Real Opion Valuaion A Case Sudy of Abandonmen Opion Carol Alexander Xi Chen Charles Ward Absrac This paper exends he markeed asse disclaimer approach for real opion valuaion. In sharp conras
More informationRoger Mercken 1, Lisette Motmans 2, Ghislain Houben Call options in a nutshell
No more replicaing porfolios : a simple convex combinaion o undersand he ris-neural valuaion mehod for he muli-sep binomial valuaion of a call opion Roger Mercen, Lisee Momans, Ghislain Houben 3 Hassel
More informationBlack-Scholes and the Volatility Surface
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Black-Scholes and he Volailiy Surface When we sudied discree-ime models we used maringale pricing o derive he Black-Scholes
More informationAn Extended Model of Asset Price Dynamics
Iranian In. J. Sci. 6(), 005, p.15-5 An Exended Model of Asse Price Dynamics M. H. Nojumi Deparmen of Maemaical Sciences, Sarif Universiy of Tecnology P.O. Box 11365-9415, Teran, Iran. e-mail: nojumi@sina.sarif.edu
More informationHull-White one factor model Version
Hull-Whie one facor model Version 1.0.17 1 Inroducion This plug-in implemens Hull and Whie one facor models. reference on his model see [?]. For a general 2 How o use he plug-in In he Fairma user inerface
More informationCoupling Smiles. November 18, 2006
Coupling Smiles Valdo Durrleman Deparmen of Mahemaics Sanford Universiy Sanford, CA 94305, USA Nicole El Karoui Cenre de Mahémaiques Appliquées Ecole Polyechnique 91128 Palaiseau, France November 18, 2006
More informationEXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION
EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION VICTOR GOODMAN AND KYOUNGHEE KIM Absrac. We find a simple expression for he probabiliy densiy of R exp(b s s/2ds in erms of is disribuion funcion
More informationSingle Premium of Equity-Linked with CRR and CIR Binomial Tree
The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences,
More informationLIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg
LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in
More informationImplied volatility phenomena as market's aversion to risk
Implied volailiy phenomena as marke's aversion o risk S.Siyanko, Universiy College London, Gower Sree, London WC1E 6B, UK, M. Ponier, Insiu Mahémaiques de oulouse, France June 27, 2017 Absrac In his aricle,
More informationOptimal Consumption and Investment with Habit Formation and Hyperbolic discounting. Mihail Zervos Department of Mathematics London School of Economics
Oimal Consumion and Invesmen wih Habi Formaion and Hyerbolic discouning Mihail Zervos Dearmen of Mahemaics London School of Economics Join work wih Alonso Pérez-Kakabadse and Dimiris Melas 1 The Sandard
More informationA Two-Asset Jump Diffusion Model with Correlation
A Two-Asse Jump Diffusion Model wih Correlaion Mahew Sephen Marin Exeer College Universiy of Oxford A hesis submied for he degree of MSc Mahemaical Modelling and Scienific Compuing Michaelmas 007 Acknowledgemens
More informationVolatility and Hedging Errors
Volailiy and Hedging Errors Jim Gaheral Sepember, 5 1999 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of
More informationFunding beyond discounting: collateral agreements and derivatives pricing
cuing edge. DERIVAIVES PRICING Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo
More informationMidterm Exam. Use the end of month price data for the S&P 500 index in the table below to answer the following questions.
Universiy of Washingon Winer 00 Deparmen of Economics Eric Zivo Economics 483 Miderm Exam This is a closed book and closed noe exam. However, you are allowed one page of handwrien noes. Answer all quesions
More informationFAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004
FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES
More informationCredit Spread Option Valuation under GARCH. Working Paper July 2000 ISSN :
Credi Spread Opion Valuaion under GARCH by Nabil ahani Working Paper -7 July ISSN : 6-334 Financial suppor by he Risk Managemen Chair is acknowledged. he auhor would like o hank his professors Peer Chrisoffersen
More informationLecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing
Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic
More informationPricing floating strike lookback put option under heston stochastic volatility
Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 0974-3200 Volume 9, Number 3 (2017), pp. 427 439 Inernaional Research Publicaion House hp://www.irphouse.com Pricing floaing srike lookback
More informationPrinciples of Finance CONTENTS
Principles of Finance CONENS Value of Bonds and Equiy... 3 Feaures of bonds... 3 Characerisics... 3 Socks and he sock marke... 4 Definiions:... 4 Valuing equiies... 4 Ne reurn... 4 idend discoun model...
More informationR e. Y R, X R, u e, and. Use the attached excel spreadsheets to
HW # Saisical Financial Modeling ( P Theodossiou) 1 The following are annual reurns for US finance socks (F) and he S&P500 socks index (M) Year Reurn Finance Socks Reurn S&P500 Year Reurn Finance Socks
More informationwhere r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wi
Hull-Whie Model Conens Hull-Whie Model Hull-Whie Tree Example: Hull-Whie Tree Calibraion Appendix: Ineres Rae Derivaive PDE Hull-Whie Model This secion is adaped from Brigo and Mercurio (006). As an exension
More informationSTATIONERY REQUIREMENTS SPECIAL REQUIREMENTS 20 Page booklet List of statistical formulae New Cambridge Elementary Statistical Tables
ECONOMICS RIPOS Par I Friday 7 June 005 9 Paper Quaniaive Mehods in Economics his exam comprises four secions. Secions A and B are on Mahemaics; Secions C and D are on Saisics. You should do he appropriae
More informationApplications of Interest Rate Models
WDS'07 Proceedings of Conribued Papers, Par I, 198 204, 2007. ISBN 978-80-7378-023-4 MATFYZPRESS Applicaions of Ineres Rae Models P. Myška Charles Universiy, Faculy of Mahemaics and Physics, Prague, Czech
More informationBruno Dupire. Banque Paribas Swaps and Options Research Team 33 Wigmore Street London W1H 0BN United Kingdom
ARBIRAGE PRICING WIH SOCHASIC VOLAILIY Bruno Dupire Banque Paribas Swaps and Opions Research eam 33 Wigmore Sree London W1H 0BN Unied Kingdom Firs version: March 199 his version: May 1993 Absrac: We address
More informationOn Monte Carlo Simulation for the HJM Model Based on Jump
On Mone Carlo Simulaion for he HJM Model Based on Jump Kisoeb Park 1, Moonseong Kim 2, and Seki Kim 1, 1 Deparmen of Mahemaics, Sungkyunkwan Universiy 44-746, Suwon, Korea Tel.: +82-31-29-73, 734 {kisoeb,
More informationVariance Dependent Pricing Kernels in GARCH Models
U.U.D.M. Projec Repor 01:0 Variance Dependen Pricing Kernels in GARCH Models Amir Hossein Khalilzadeh Examensarbee i maemaik, 30 hp Handledare och examinaor: Maciej Klimek epember 01 Deparmen of Mahemaics
More informationCENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T. J. KEHOE MACROECONOMICS I WINTER 2011 PROBLEM SET #6
CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T J KEHOE MACROECONOMICS I WINTER PROBLEM SET #6 This quesion requires you o apply he Hodrick-Presco filer o he ime series for macroeconomic variables for he
More informationStandard derivatives pricing theory (see, for example, Hull,
Cuing edge Derivaives pricing Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo
More informationSan Francisco State University ECON 560 Summer 2018 Problem set 3 Due Monday, July 23
San Francisco Sae Universiy Michael Bar ECON 56 Summer 28 Problem se 3 Due Monday, July 23 Name Assignmen Rules. Homework assignmens mus be yped. For insrucions on how o ype equaions and mah objecs please
More informationFundamental Basic. Fundamentals. Fundamental PV Principle. Time Value of Money. Fundamental. Chapter 2. How to Calculate Present Values
McGraw-Hill/Irwin Chaper 2 How o Calculae Presen Values Principles of Corporae Finance Tenh Ediion Slides by Mahew Will And Bo Sjö 22 Copyrigh 2 by he McGraw-Hill Companies, Inc. All righs reserved. Fundamenal
More informationEXPLICIT OPTION PRICING FORMULA FOR A MEAN-REVERTING ASSET IN ENERGY MARKET
Journal of Numerical and Applied Mahemaics Vol.1 (96), 8, pp.16-33 ANATOLY SWISHCHUK EXPLICIT OPTION PRICING FORMULA FOR A MEAN-REVERTING ASSET IN ENERGY MARKET Some commodiy prices, like oil and gas,
More informationMany different investment objectives and
The Risk and Rewards of Minimizing Shorfall Probabiliy The risk may be worhwhile. Sid Browne 76 SID BROWNE is vice presiden of firmwide risk a Goldman, Sachs and Co. in New York (NY 10005), and a professor
More informationModeling of Tradeable Securities with Dividends
Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 June 15, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are
More informationFinancial Econometrics (FinMetrics02) Returns, Yields, Compounding, and Horizon
Financial Economerics FinMerics02) Reurns, Yields, Compounding, and Horizon Nelson Mark Universiy of Nore Dame Fall 2017 Augus 30, 2017 1 Conceps o cover Yields o mauriy) Holding period) reurns Compounding
More informationOnline Appendix. Using the reduced-form model notation proposed by Doshi, el al. (2013), 1. and Et
Online Appendix Appendix A: The concep in a muliperiod framework Using he reduced-form model noaion proposed by Doshi, el al. (2013), 1 he yearly CDS spread S c,h for a h-year sovereign c CDS conrac can
More informationForeign Exchange, ADR s and Quanto-Securities
IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Foreign Exchange, ADR s and Quano-Securiies These noes consider foreign exchange markes and he pricing of derivaive
More informationYou should turn in (at least) FOUR bluebooks, one (or more, if needed) bluebook(s) for each question.
UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has
More information